Open Access

Singular degenerate problems occurring in biosorption process

Boundary Value Problems20132013:30

DOI: 10.1186/1687-2770-2013-30

Received: 22 November 2012

Accepted: 5 February 2013

Published: 15 February 2013

Abstract

The boundary value problems for singular degenerate arbitrary order differential-operator equations with variable coefficients are considered. The uniform coercivity properties of ordinary and partial differential equations with small parameters are derived in abstract L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq1_HTML.gif spaces. It is shown that corresponding differential operators are positive and also are generators of analytic semigroups. In application, well-posedeness of the Cauchy problem for an abstract parabolic equation and systems of parabolic equations are studied in mixed L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq2_HTML.gif spaces. These problems occur in fluid mechanics and environmental engineering.

MSC:34G10, 35J25, 35J70.

Keywords

differential-operator equations degenerate equations semigroups of operators Banach-valued function spaces coercive problems operator-valued Fourier multipliers interpolation of Banach spaces

0 Introduction

Boundary value problems (BVPs) for differential-operator equations (DOEs) in H-valued (Hilbert space valued) function spaces have been studied extensively by many researchers (see, e.g., [114] and the references therein). A comprehensive introduction to DOEs and historical references may be found in [6] and [14]. The maximal regularity properties for DOEs have been studied, e.g., in [3, 1019].

In this work, singular degenerate BVPs for arbitrary order DOEs with parameters are considered. This problem has numerous applications. The parameter-dependent BVPs occur in different situations of fluid mechanics and environmental engineering etc.

In Section 2, the BVP for the following singular degenerate ordinary DOE with a small parameter:
L u = ( 1 ) m t a ( x ) u [ 2 m ] + A ( x ) u + k = 0 2 m 1 t k 2 m A k ( x ) u [ k ] = f ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equa_HTML.gif
is considered, where
D x [ i ] u = u [ i ] ( x ) = [ x γ d d x ] i u ( x ) , x ( 0 , 1 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equb_HTML.gif
t is a small parameter, a ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq3_HTML.gif is a complex-valued function, γ > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq4_HTML.gif, A = A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq5_HTML.gif is a principal, A 1 = A 1 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq6_HTML.gif and A 2 = A 2 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq7_HTML.gif are subordinate linear operators in a Banach space E. Several conditions for the uniform coercivity and the resolvent estimates for this problem are given in abstract L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq1_HTML.gif-spaces. We prove that the problem has a unique solution u W p , γ [ 2 ] ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq8_HTML.gif for f L p ( 0 , 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq9_HTML.gif, | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq10_HTML.gif, 0 φ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq11_HTML.gif with sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif and the following uniform coercive estimate holds:
i = 0 2 | λ | 1 i 2 t σ i u [ i ] L p ( 0 , 1 ; E ) + A u L p ( 0 , 1 ; E ) C f L p ( 0 , 1 ; E ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equc_HTML.gif
where
σ i = 1 2 p ( γ 1 ) + i 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equd_HTML.gif
In Section 3, the partial DOE with small parameters
k = 1 n ( 1 ) m k t k a k ( x ) D x k [ 2 m k ] u + A ( x ) u + | α : 2 m | < 1 k = 1 n t k α k 2 m k A α ( x ) D [ α ] u = f ( x ) , x = ( x 1 , x 2 , , x n ) G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Eque_HTML.gif
is considered in a mixed L p ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq13_HTML.gif space, where a k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq14_HTML.gif are complex-valued functions, A and A α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq15_HTML.gif are linear operators in E, λ is a complex and t k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq16_HTML.gif are positive parameters, G is an n-dimensional rectangular domain, p = ( p 1 , p 2 , , p n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq17_HTML.gif. Here we prove that for f L p ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq18_HTML.gif, | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq10_HTML.gif with sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif, this problem has a unique solution u that belongs to the Sobolev space W p [ 2 ] ( G ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq19_HTML.gif with a mixed p norm and the following coercive uniform estimate holds:
k = 1 n i = 0 2 m k | λ | 1 i 2 m k t k σ k i D x k [ i ] u L p ( G ; E ) + A u L p ( G ; E ) C f L p ( G ; E ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equf_HTML.gif
where
D x k [ i ] = [ x k α k x k ] i , α k > 1 , σ k i = 1 2 m k p ( α k 1 ) + i 2 m k , k = 1 , 2 , , n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equg_HTML.gif
In Section 4, the uniform well-posedeness of the mixed problem for the following singular degenerate abstract parabolic equation:
u y + k = 1 n ( 1 ) m k t k a k ( x ) D x k [ 2 m k ] u + A ( x ) u = f ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equh_HTML.gif
is obtained. Particularly, the above problem occurs in atmospheric dispersion of pollutants and evolution models for phytoremediation of metals from soils. In application, particularly, by taking E = R 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq20_HTML.gif, A ( x ) = [ a i j ( x ) ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq21_HTML.gif, u = ( u 1 , u 2 , u 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq22_HTML.gif, i , j = 1 , 2 , 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq23_HTML.gif, we consider the mixed problem for the system of the following parabolic equations with parameters:
u i y + k = 1 n ( 1 ) m k t k a k ( x ) D x k [ 2 m k ] u i + j = 1 3 a i j ( x ) u j = f i ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equi_HTML.gif

which arises in phytoremediation process, where a i j ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq24_HTML.gif are real-valued functions and f i ( x , y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq25_HTML.gif are data. The maximal regularity properties of this problem are studied. Note that the maximal regularity properties for undegenerate DOEs were investigated, e.g., in [110, 1416, 19, 20]. Regular degenerate DOEs in Banach spaces were treated in [1113, 15, 1719, 21]. It should be noted that contrary to these results, here high-order singular degenerated BVPs with small parameters are considered. Moreover, principal coefficients depend on space variables. The proofs are based on abstract harmonic analysis, operator theory, interpolation of Banach spaces, theory of semigroups of linear operators, microlocal analysis, embedding and trace theorems in vector-valued Sobolev-Lions spaces.

1 Notations and background

Let γ = γ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq26_HTML.gif, x = ( x 1 , x 2 , , x n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq27_HTML.gif be a positive measurable function on a domain Ω R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq28_HTML.gif. L p , γ ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq29_HTML.gif denotes the space of strongly measurable E-valued functions that are defined on Ω with the norm
f L p , γ = f L p , γ ( Ω ; E ) = ( f ( x ) E p γ ( x ) d x ) 1 p , 1 p < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equj_HTML.gif

For γ ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq30_HTML.gif, the space L p , γ ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq31_HTML.gif will be denoted by L p = L p ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq32_HTML.gif.

The Banach space E is called a UMD-space if the Hilbert operator
( H f ) ( x ) = lim ε 0 | x y | > ε f ( y ) x y d y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equk_HTML.gif

is bounded in L p ( R , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq33_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq34_HTML.gif (see, e.g., [22]). UMD spaces include, e.g., L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq1_HTML.gif, l p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq35_HTML.gif spaces and Lorentz spaces L p q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq36_HTML.gif, p , q ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq37_HTML.gif.

Let C denote the set of complex numbers and
S φ = { λ ; λ C , | arg λ | φ } { 0 } , 0 φ < π . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equl_HTML.gif
A linear operator A is said to be φ-positive in a Banach space E with bound M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq38_HTML.gif if D ( A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq39_HTML.gif is dense on E and ( A + λ I ) 1 B ( E ) M ( 1 + | λ | ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq40_HTML.gif for any λ S φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq41_HTML.gif, 0 φ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq11_HTML.gif, where I is the identity operator in E, B ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq42_HTML.gif is the space of bounded linear operators in E. Sometimes A + λ I https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq43_HTML.gif will be written as A + λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq44_HTML.gif and denoted by A λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq45_HTML.gif. It is known [23], Section 1.15.1] that a positive operator A has well-defined fractional powers A θ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq46_HTML.gif. Let E ( A θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq47_HTML.gif denote the space D ( A θ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq48_HTML.gif with the norm
u E ( A θ ) = ( u p + A θ u p ) 1 p , 1 p < , 0 < θ < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equm_HTML.gif

Let E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq49_HTML.gif and E 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq50_HTML.gif be two Banach spaces continuously embedded in a locally convex space. By ( E 1 , E 2 ) θ , p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq51_HTML.gif, 0 < θ < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq52_HTML.gif, 1 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq53_HTML.gif, we denote the interpolation spaces obtained from { E 1 , E 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq54_HTML.gif by the K-method [23], Section 1.3.2].

Let C ( Ω ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq55_HTML.gif denote the space of E-valued uniformly bounded continuous functions on the domain Ω R n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq28_HTML.gif.

Let N denote the set of natural numbers and { r j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq56_HTML.gif be a sequence of independent symmetric { 1 , 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq57_HTML.gif-valued random variables on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq58_HTML.gif (see [22]). A set W h B ( E 1 , E 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq59_HTML.gif is called uniform R-bounded with respect to h (see, e.g., [16]) if there is a constant C independent of h Q R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq60_HTML.gif such that for all T 1 ( h ) , T 2 ( h ) , , T m ( h ) W h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq61_HTML.gif and u 1 , u 2 , , u m E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq62_HTML.gif, m N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq63_HTML.gif,
0 1 j = 1 m r j ( y ) T j ( h ) u j E 2 d y C 0 1 j = 1 m r j ( y ) u j E 1 d y . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equn_HTML.gif

The smallest C for which the above estimate holds is called an R-bound of the collection W h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq64_HTML.gif and is denoted by R ( W h ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq65_HTML.gif.

A φ-positive operator A is said to be R-positive in E if the set L A = { ξ ( A + ξ I ) 1 : ξ S φ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq66_HTML.gif, 0 φ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq11_HTML.gif, is R-bounded.

Note that for Hilbert spaces H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq67_HTML.gif, H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq68_HTML.gif, all norm-bounded sets are R-bounded (see, e.g., [16]). Therefore, in Hilbert spaces all positive operators are R-positive. If A is a generator of a contraction semigroup on L q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq69_HTML.gif, 1 q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq70_HTML.gif, or A has the bounded imaginary powers with A i t B ( E ) C e ν | t | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq71_HTML.gif, ν < π 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq72_HTML.gif in E UMD https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq73_HTML.gif, then those operators are R-positive (e.g., see [16], Section 4.3]).

The operator A ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq74_HTML.gif, t σ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq75_HTML.gif is said to be φ-positive in E uniformly with respect to t σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq76_HTML.gif if D ( A ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq77_HTML.gif is independent of t, D ( A ( t ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq78_HTML.gif is dense in E and ( A ( t ) + λ ) 1 M 1 + | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq79_HTML.gif for all λ S ( φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq80_HTML.gif, 0 φ < π https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq81_HTML.gif, where M does not depend on t.

Let E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq82_HTML.gif and E be two Banach spaces and E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq82_HTML.gif be continuously and densely embedded into E. Let m be a positive integer. W p , γ m ( a , b ; E 0 , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq83_HTML.gif denotes an E 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq82_HTML.gif-valued function space defined by
W p , γ m ( a , b ; E 0 , E ) = { u : u L p , γ ( a , b ; E 0 ) , u ( m ) L p , γ ( a , b ; E ) } , u W p , γ m = u W p , γ m ( a , b ; E 0 , E ) = u L p , γ ( a , b ; E 0 ) + u ( m ) L p , γ ( a , b ; E ) < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equo_HTML.gif
Let t be a positive parameter. We define a parameterized norm in W p , γ m ( a , b ; E 0 , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq84_HTML.gif as follows:
u W p , γ , t m = u W p , γ , t m ( a , b ; E 0 , E ) = u L p , γ ( a , b ; E 0 ) + t u ( m ) L p , γ ( a , b ; E ) < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equp_HTML.gif
Let p = ( p 1 , p 2 , , p n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq85_HTML.gif, G = k = 1 n ( 0 , b k ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq86_HTML.gif, L p ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq87_HTML.gif denote the space of all p-summable E-valued functions with a mixed norm (see, e.g., [24], Section 8] for scalar case), i.e., the space of all measurable E-valued functions f defined on G, for which
f L p ( G ; E ) = ( 0 b 1 ( 0 b 2 ( 0 b n f ( x ) p n d x n ) p n 1 p n d x n 1 ) d x 1 ) 1 p 1 < . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equq_HTML.gif
Let l k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq88_HTML.gif be positive integers, l = ( l 1 , l 2 , , l n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq89_HTML.gif, t k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq16_HTML.gif be positive parameters and t = ( t 1 , t 2 , , t n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq90_HTML.gif.
D x k [ i ] u = [ x k α k x k ] i u ( x ) , α k > 0 , α = ( α 1 , α 2 , , α n ) , k = 1 , 2 , , n . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equr_HTML.gif
Consider the following weighted spaces of functions:
W p , α [ l ] ( G ; E ( A ) , E ) = { u : u L p ( G ; E ( A ) ) , D k [ l k ] u L p ( G ; E ) } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equs_HTML.gif
with the mixed norm
u W p , α [ l ] ( G ; E ( A ) , E ) = u L p ( G ; E 0 ) + k = 1 n D k [ l k ] u L p ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equt_HTML.gif
and with the parameterized norm
u W p , α , t [ l ] ( G ; E ( A ) , E ) = u L p ( G ; E 0 ) + k = 1 n t k D k [ l k ] u L p ( G ; E ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equu_HTML.gif

respectively.

Consider the BVP for DOE
( L + λ ) u = ( 1 ) m t u [ 2 m ] ( x ) + ( A + λ ) u ( x ) = f ( x ) , x ( 0 , 1 ) , L k u = i = 0 m k [ α k i u [ i ] ( 1 ) + i = 1 N δ k i u [ i ] ( x k i ) ] = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ1_HTML.gif
(1)

where u [ i ] = [ x γ d d x ] i u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq91_HTML.gif, α k i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq92_HTML.gif, δ k i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq93_HTML.gif are complex numbers and x k i ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq94_HTML.gif, m k { 0 , 1 , , 2 m 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq95_HTML.gif; A is a possible unbounded operator in E.

In a similar way as in [17], Theorem 5.1], we obtain the following.

Theorem A1 Let the following conditions be satisfied:
  1. (1)

    α k i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq92_HTML.gif, δ k j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq96_HTML.gif are complex numbers, α k m k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq97_HTML.gif, t is a small positive parameter and σ i = 1 2 m p ( γ 1 ) + i 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq98_HTML.gif;

     
  2. (2)

    E is a UMD space, γ > 1 + 1 + 4 p + 1 2 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq99_HTML.gif, 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq100_HTML.gif;

     
  3. (3)

    A is an R-positive operator in E.

     
Then problem (1) for f L p ( 0 , 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq9_HTML.gif and | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq10_HTML.gif with sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif has a unique solution u W p , γ [ 2 m ] ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq101_HTML.gif. Moreover, the following uniform coercive estimate holds:
i = 0 2 m | λ | 1 i 2 m t σ i u [ i ] L p ( 0 , 1 ; E ) + A u L p ( 0 , 1 ; E ) C f L p ( 0 , 1 ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equv_HTML.gif

By reasoning as in [17], Theorem 2.3], we obtain the following.

Theorem A2 Let the following conditions be satisfied:
  1. (1)

    γ ( x ) = x γ 1 γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq102_HTML.gif, γ > 1 + 1 + 4 p + 1 2 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq103_HTML.gif, p ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq104_HTML.gif, 0 < t < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq105_HTML.gif;

     
  2. (2)

    E is a UMD space and A is an R-positive operator in E;

     
  3. (3)

    there exists a bounded linear extension operator from W p , γ m ( 1 , ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq106_HTML.gif to W p , γ m ( R ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq107_HTML.gif.

     
Then the embedding D j W p , γ m ( 1 , ; E ( A ) , E ) L p , γ ( 1 , ; E ( A 1 j m μ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq108_HTML.gif is continuous and for 0 μ 1 j m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq109_HTML.gif, the uniform estimate
t j m u ( j ) L p , γ ( 1 , ; E ( A 1 j m μ ) ) h μ u W p , γ , t m ( 1 , ; E ( A ) , E ) + h ( 1 μ ) u L p , γ ( 1 , ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equw_HTML.gif

holds for all u W p , γ m ( 1 , ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq110_HTML.gif and 0 < h h 0 < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq111_HTML.gif.

Let
D [ α ] = D 1 [ α 1 ] D 2 [ α 2 ] D n [ α n ] , t = ( t 1 , t 2 , , t n ) , ψ ( t ) = k = 1 n t k α k 2 m k , 0 < t k < 1 , G = k = 1 n ( 0 , b k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equx_HTML.gif
Theorem A3 Let the following conditions be satisfied:
  1. (1)

    p = ( p 1 , p 2 , , p n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq85_HTML.gif, p k ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq112_HTML.gif, γ k ( x ) = x k γ k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq113_HTML.gif;

     
  2. (2)

    E is a UMD space and A is an R-positive operator in E;

     
  3. (3)
    α = ( α 1 , α 2 , , α n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq114_HTML.gif and l = ( l 1 , l 2 , , l n ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq115_HTML.gif are n-tuples of a nonnegative integer such that
    ϰ = | α : l | = k = 1 n α k l k 1 , 1 < p k < , γ k > 1 + 1 + 4 p + 1 2 p ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equy_HTML.gif
     
  4. (4)

    there exists a bounded linear extension operator from W p , γ [ l ] ( G ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq116_HTML.gif to W p , γ [ l ] ( R n ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq117_HTML.gif.

     
Then the embedding D α W p , γ [ l ] ( G ; E ( A ) , E ) L p ( G ; E ( A 1 ϰ μ ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq118_HTML.gif is continuous. Moreover, there is a constant h > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq119_HTML.gif such that for u W p , γ [ l ] ( Ω ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq120_HTML.gif, the following uniform estimate holds:
ψ ( t ) D [ α ] u L p ( G ; E ( A 1 ϰ μ ) ) h μ u W p , γ , t [ l ] ( G ; E ( A ) , E ) + h ( 1 μ ) u L p ( G ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equz_HTML.gif

2 Singular degenerate DOEs with parameter

Consider the BVP for the following differential-operator equation with parameter:
L u = ( 1 ) m t a ( x ) u [ 2 m ] + A λ ( x ) u + k = 0 2 m 1 t k 2 m A k ( x ) u [ k ] = f ( x ) , L k u = i = 0 m k [ α k i u [ i ] ( 1 ) + i = 1 N δ k i u [ i ] ( x k i ) ] = 0 , k = 1 , 2 , , m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ2_HTML.gif
(2)
on the domain ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq121_HTML.gif, where t is a positive parameter and λ is a complex parameter; α k i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq92_HTML.gif, δ k i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq93_HTML.gif are complex numbers and x k i ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq94_HTML.gif, m k { 0 , 1 , , 2 m k 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq122_HTML.gif, a ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq3_HTML.gif is a complex-valued function on [ 0 , 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq58_HTML.gif; A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq123_HTML.gif and A k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq124_HTML.gif are linear operators in a Banach space E and A λ ( x ) = A ( x ) + λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq125_HTML.gif. Note that
0 1 z γ d z = . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equaa_HTML.gif

A function u W p , γ [ 2 m ] ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq126_HTML.gif satisfying equation (2) a.e. on ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq121_HTML.gif is said to be the solution of equation (2) on ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq121_HTML.gif.

Remark 1

Let
y = x 1 z γ d z . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ3_HTML.gif
(3)
Under the substitution (3), spaces L p ( 0 , 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq127_HTML.gif and W p , γ [ 2 m ] ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq128_HTML.gif are mapped isomorphically onto weighted spaces
L p , γ ˜ ( , 0 ; E ) , W p , γ ˜ 2 ( , 0 ; E ( A ) , E ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equab_HTML.gif
respectively, where
γ ˜ = γ ˜ ( x ( y ) ) = [ 1 ( γ 1 ) y ] γ 1 γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equac_HTML.gif
Moreover, under the substitution (3), problem (2) is transformed into the following non-degenerate problem:
L u = ( 1 ) m t a ( y ) u ( 2 m ) ( y ) + A λ ( y ) u ( y ) + k = 0 2 m 1 t k 2 m A k ( y ) u ( k ) ( y ) = f ( y ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ4_HTML.gif
(4)
L k u = i = 0 m k [ α k i u ( i ) ( 0 ) + i = 1 N δ k i u [ i ] ( y k i ) ] = 0 , k = 1 , 2 , , m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ5_HTML.gif
(5)

in the weighted space L p , γ ˜ ( , 0 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq129_HTML.gif, where a ˜ ( y ) = a ( γ ˜ ( x ( y ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq130_HTML.gif, A ˜ ( y ) = A ( γ ˜ ( x ( y ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq131_HTML.gif, A ˜ k ( y ) = A k ( γ ˜ ( x ( y ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq132_HTML.gif, γ ˜ ( y ) = γ ( γ ˜ ( x ( y ) ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq133_HTML.gif are again denoted by a ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq134_HTML.gif, A ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq135_HTML.gif, A k ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq136_HTML.gif, γ after the substitution (3), respectively.

Let us consider boundary value problem (4)-(5).

Theorem 1 Let the following conditions be satisfied:
  1. (1)

    α k m k 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq137_HTML.gif, a ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq134_HTML.gif is a positive uniformly bounded continuous function on ( , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq138_HTML.gif;

     
  2. (2)

    E is a UMD space, γ > 1 + 1 + 4 p + 1 2 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq139_HTML.gif, 1 < p < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq100_HTML.gif, and σ i = 1 2 m p ( γ 1 ) + i 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq140_HTML.gif;

     
  3. (3)

    A ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq135_HTML.gif is R-positive in E uniformly with respect to y [ , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq141_HTML.gif and A ( y ) A 1 ( y 0 ) C ( ( , 0 ) ; B ( E ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq142_HTML.gif, y 0 [ , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq143_HTML.gif;

     
  4. (4)
    for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq144_HTML.gif, there is a positive C ( ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq145_HTML.gif such that
    A k ( y ) u ε u ( E ( A ) , E ) k 2 m , + C ( ε ) u for u ( E ( A ) , E ) k 2 m , , k = 0 , 1 , , 2 m 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equad_HTML.gif
     
Then problem (4)-(5) has a unique solution u W p , γ 2 ( , 0 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq146_HTML.gif for f L p , γ ˜ ( , 0 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq147_HTML.gif and | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq10_HTML.gif with sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif. Moreover, the following uniform coercive estimate holds:
i = 0 2 m | λ | 1 i 2 m t σ i u ( i ) p , γ + A u p , γ C f p , γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ6_HTML.gif
(6)
Proof Let G 1 , G 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq148_HTML.gif be bounded intervals in ( , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq149_HTML.gif and φ 1 , φ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq150_HTML.gif correspond to a partition of unit that functions φ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq151_HTML.gif are smooth on ( , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq149_HTML.gif, σ j = supp φ j G j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq152_HTML.gif and j = 1 φ j ( y ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq153_HTML.gif. Then, for all u W p , γ 2 m ( , 0 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq154_HTML.gif, we have u ( y ) = j = 1 u j ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq155_HTML.gif, where u j ( y ) = u ( y ) φ j ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq156_HTML.gif. For u W p , γ 2 ( , 0 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq157_HTML.gif, from (4) we obtain
( L + λ ) u j = ( 1 ) m t a ( y ) u j ( 2 m ) ( y ) + A λ ( y ) u j ( y ) = f j ( y ) , L k u j = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ7_HTML.gif
(7)
where
f j = f φ j ( 1 ) m a i = 0 2 m 1 C 2 m i u ( i ) φ j ( 2 m i ) + k = 1 2 m 1 i = 0 k C k i t k 2 m u ( i ) φ j ( k i ) A k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ8_HTML.gif
(8)
Since a is uniformly bounded on ( , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq138_HTML.gif for all small ρ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq158_HTML.gif, there is a large r 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq159_HTML.gif such that | a ( y ) a ( ) | δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq160_HTML.gif for all | y | r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq161_HTML.gif. Let
G 0 = R n σ r 0 ( 0 ) , σ r 0 ( 0 ) = { y ( , 0 ) , | y | r 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equae_HTML.gif
Cover σ r 0 ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq162_HTML.gif by finitely many intervals G j = σ r j ( y 0 j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq163_HTML.gif such that
| a ( y ) a ( y 0 j ) | δ  for  | y y 0 j | r j , j = 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equaf_HTML.gif
Define coefficients of local operators, i.e.,
a 0 ( y ) = { a ( y ) , y σ r 0 ( 0 ) a ( r 0 2 y | y | 2 ) , y σ r 0 ( 0 ) } , a j ( y ) = { a ( y ) , y σ r j ( y 0 j ) a ( y 0 j + r 0 2 y y 0 j | y y 0 j | 2 ) , y σ r j ( y 0 j ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equag_HTML.gif
and
A 0 ( y ) A 1 ( y 0 j ) = { A ( y ) A 1 ( y 0 j ) , y σ r 0 ( 0 ) A ( r 0 2 y | y | 2 ) A 1 ( y 0 j ) , y σ r 0 ( 0 ) } , A j ( y ) A 1 ( y 0 j ) = { A ( y ) A 1 ( y 0 j ) , y σ r j ( y 0 j ) A ( y 0 j + r 0 2 y y 0 j | y y 0 j | 2 ) A 1 ( y 0 j ) , y σ r j ( y 0 j ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equah_HTML.gif
for each j = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq164_HTML.gif . Then, for all y ( , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq165_HTML.gif and j = 0 , 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq166_HTML.gif , we get
| a j ( x ) a j ( y 0 j ) | δ and A j ( y ) A 1 ( y 0 j ) A j ( y 0 j ) A 1 ( y 0 j ) B ( E ) < δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equai_HTML.gif
Freezing coefficients in (7) obtain that
( 1 ) m a ( y 0 j ) t u j ( 2 m ) ( y ) + A λ ( y 0 j ) u j ( y ) = F j ( y ) , L k u j = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ9_HTML.gif
(9)
where
F j = f j + [ A ( y 0 j ) A ( y ) ] u j + ( 1 ) m t [ a ( y ) a ( y 0 j ) ] u j ( 2 m ) , j = 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ10_HTML.gif
(10)
Since functions u j ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq167_HTML.gif have compact supports in (9), if we extend u j ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq167_HTML.gif on the outsides of σ j = supp φ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq168_HTML.gif, we obtain BVPs with constant coefficients
t a ( y 0 j ) u j ( 2 m ) + A λ ( y 0 j ) u j = F j , L k u j = 0 , j = 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ11_HTML.gif
(11)
Let G j , p , γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq169_HTML.gif denote E-valued weighted L p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq1_HTML.gif-norms with respect to domains G j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq170_HTML.gif. Let φ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq151_HTML.gif be such that 0 σ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq171_HTML.gif. Then, by virtue of Theorem A1, we obtain that problem (11) has a unique solution u j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq172_HTML.gif and for | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq10_HTML.gif and sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif, the following estimate holds:
i = 0 2 m | λ | 1 i 2 m t σ i u j ( i ) G j , p , γ 1 + A u j G j , p , γ 1 C F j G j , p , γ 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ12_HTML.gif
(12)
Theorem A2 implies that for all ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq144_HTML.gif, there is a continuous function C ( ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq145_HTML.gif such that
F j G j , p , γ ε u j W p , γ , t 2 m ( G j ; E ( A ) , E ) + C ( ε ) f j G j , p , γ , j = 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ13_HTML.gif
(13)
Consequently, by using Theorem A2, from (12)-(13) we get
i = 0 2 m | λ | 1 i 2 m t σ i u j ( i ) G j , p , γ + A u j G j , p , γ C f G j , p , γ + ε u j W p , γ 2 m + C ( ε ) u j G j , p , γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ14_HTML.gif
(14)
Choosing ε < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq173_HTML.gif, from (14) we have
i = 0 2 m | λ | 1 i 2 m t σ i u j ( i ) G j , p , γ + A u j G j , p , γ C [ f G j , p , γ + u j G j , p , γ ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ15_HTML.gif
(15)
Then, by using the equality u ( y ) = j = 1 u j ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq174_HTML.gif and by virtue of (15) for u W p , γ 2 m ( , 0 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq175_HTML.gif, we have
i = 0 2 m | λ | 1 i 2 m t σ i u ( i ) p , γ + A u j p , γ C [ ( L + λ ) u p , γ + u p , γ ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ16_HTML.gif
(16)
Let u W p , γ 2 m ( , 0 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq176_HTML.gif be a solution of problem (4)-(5). For | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq10_HTML.gif, we have
u p , γ = ( L + λ ) u L u p , γ 1 λ [ ( L + λ ) u p , γ + u W p , γ 2 m ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ17_HTML.gif
(17)
By Theorem A2, by virtue of (16) and (17) for sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif, we have
i = 0 2 m | λ | 1 i 2 m t σ i u ( i ) p , γ + A u p , γ C ( L + λ ) u p , γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ18_HTML.gif
(18)
Consider the operator O t λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq177_HTML.gif in L p , γ ( , 0 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq178_HTML.gif generated by problem (4)-(5), i.e.,
D ( O λ ) = W p , γ 2 m ( , 0 ; E ( A ) , E , L k ) , O t λ u = ( 1 ) m t a u ( 2 m ) + A λ u + k = 0 2 m 1 t k 2 m A k ( y ) u ( k ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equaj_HTML.gif
Estimate (18) implies that problem (4)-(5) has only a unique solution and the operator O λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq179_HTML.gif has an invertible operator in its rank space. We need to show that this rank space coincides with the space X = L p , γ ( , 0 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq180_HTML.gif. We consider the smooth functions g j = g j ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq181_HTML.gif with respect to the partition of the unit φ j = φ j ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq182_HTML.gif on ( , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq138_HTML.gif that equals one on supp φ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq183_HTML.gif, where supp g j G j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq184_HTML.gif and | g j ( y ) | < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq185_HTML.gif. Let us construct for all j the function u j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq172_HTML.gif that is defined on the regions Ω j = ( , 0 ) G j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq186_HTML.gif and satisfies problem (4)-(5). Problem (4)-(5) can be expressed in the form
( 1 ) m t a ( y 0 j ) u j ( 2 ) + A λ ( y 0 j ) u j = g j { F j + [ A ( y 0 j ) A ( y ) ] u j + ( 1 ) m t [ a ( y ) a ( y 0 j ) ] u j ( 2 ) } , L k u j = 0 , j = 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ19_HTML.gif
(19)
Consider operators O j t λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq187_HTML.gif in L p , γ ( G j ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq188_HTML.gif generated by BVPs (19). By virtue of Theorem A1 for all f L p , γ ( G j ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq189_HTML.gif, for | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq10_HTML.gif and sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif, we have
i = 0 2 m | λ | 1 i 2 m t σ i d i d y i O j t λ 1 f p , γ + A O j t λ 1 f p , γ C f p , γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ20_HTML.gif
(20)
Extending u j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq172_HTML.gif zero on the outside of supp φ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq183_HTML.gif in equalities (20) and passing substitutions u j = O j λ 1 υ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq190_HTML.gif, we obtain operator equations with respect to υ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq191_HTML.gif
υ j = K j t λ υ j + g j f , j = 1 , 2 , . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ21_HTML.gif
(21)
By virtue of Theorem A2, by estimate (20), in view of the smoothness of the coefficients of K j λ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq192_HTML.gif, for | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq193_HTML.gif and sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif, we have K j λ t < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq194_HTML.gif, where ε is sufficiently small. Consequently, equations (21) have unique solutions
υ j = [ I K j λ t ] 1 g j f . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equak_HTML.gif
Moreover,
υ j p , γ = [ I K j t λ ] 1 g j f p , γ f p , γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equal_HTML.gif
Whence, [ I K j λ t ] 1 g j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq195_HTML.gif are bounded linear operators from X to L p , γ ( G j ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq196_HTML.gif. Thus, we obtain that the functions
u j = U j t λ f = O j t λ 1 [ I K j t λ ] 1 g j f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equam_HTML.gif
are the solutions of equations (21). Consider the linear operator ( U t + λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq197_HTML.gif in X such that
( U t + λ ) f = j = 1 φ j ( y ) U j λ t f . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equan_HTML.gif
It is clear from the constructions U j t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq198_HTML.gif and estimate (20) that operators U j λ t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq199_HTML.gif are bounded linear from X to W p , γ 2 m ( , 0 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq200_HTML.gif and
i = 0 2 m | λ | 1 i 2 m t σ i d i d i y U j t λ 1 f p , γ + A U j t λ 1 f p , γ C f p , γ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ22_HTML.gif
(22)
Therefore, ( U t + λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq197_HTML.gif is a bounded linear operator from X to X. Let L t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq201_HTML.gif denote the operator in L p , γ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq202_HTML.gif generated by BVP (4)-(5). Then the act of ( L t + λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq203_HTML.gif to u = j = 1 φ j U j t λ f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq204_HTML.gif gives ( L t + λ ) u = f + j = 1 Φ j t λ f https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq205_HTML.gif, where Φ j λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq206_HTML.gif is a linear combination of U j λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq207_HTML.gif and d d y U j λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq208_HTML.gif . By virtue of Theorem A2, estimate (22) and in view of the expression Φ j λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq209_HTML.gif, we obtain that operators Φ j λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq210_HTML.gif are bounded linear from X to L p , γ ( G j ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq188_HTML.gif and Φ j λ t < ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq211_HTML.gif. Therefore, there exists a bounded linear invertible operator ( I + j = 1 Φ j t λ ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq212_HTML.gif. Whence, we obtain that for all f X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq213_HTML.gif, BVP (4)-(5) has a unique solution
u ( y ) = ( L t + λ ) 1 f = ( U t + λ ) ( I + j = 1 Φ j t λ ) 1 f = j = 1 φ j ( y ) O j t λ 1 [ I K j t λ ] 1 ( I + j = 1 Φ j t λ ) 1 f . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ23_HTML.gif
(23)

Then, by using the above representation and in view of Theorem A1, we obtain the assertion of Theorem 1. □

Result 1 Theorem 1 implies that the operator L t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq201_HTML.gif has a resolvent ( L t + λ ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq214_HTML.gif for | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq193_HTML.gif and the following estimate holds:
i = 0 2 m | λ | 1 i 2 m t σ i D i ( L t + λ ) 1 B ( X ) + A ( L t + λ ) 1 B ( X ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equao_HTML.gif

Let G t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq215_HTML.gif denote the operator in L p ( 0 , 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq216_HTML.gif generated by BVP (2). By virtue of Theorem 1 and Remark 1, we obtain the following result.

Result 2 Let all conditions of Theorem 1 be satisfied. Then
  1. (a)
    problem (2) has a unique solution u W p , γ [ 2 m ] ( 0 , 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq217_HTML.gif for f L p ( 0 , 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq9_HTML.gif and sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif. Moreover, the following uniform coercive estimate holds:
    i = 0 2 m | λ | 1 i 2 m t σ i u [ i ] p + A u p C f p ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equap_HTML.gif
     
  2. (b)
    G t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq215_HTML.gif has a resolvent operator ( G t + λ ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq218_HTML.gif for | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq219_HTML.gif and
    i = 0 2 m | λ | 1 i 2 m t σ i D [ i ] ( G t + λ ) 1 B ( L p ( 0 , 1 ; E ) ) + A ( G t + λ ) 1 B ( L p ( 0 , 1 ; E ) ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equaq_HTML.gif
     

Theorem 2 Let all conditions of Theorem  1 hold. Then the operator L t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq201_HTML.gif is uniformly R-positive in L p ( 0 , 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq127_HTML.gif, also L t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq201_HTML.gif is a generator of an analytic semigroup.

Proof By virtue of Theorem 1, we obtain that for f L p ( 0 , 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq220_HTML.gif, BVP (4)-(5) has a unique solution expressed in the form
u ( y ) = ( L t + λ ) 1 f = ( U t + λ ) ( I + j = 1 Φ j λ t ) 1 f = j = 1 φ j ( y ) O j t λ 1 [ I K j t λ ] 1 ( I + j = 1 Φ j t λ ) 1 f , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equar_HTML.gif

where O j λ t = O j t + λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq221_HTML.gif are local operators generated by problems (7)-(8) and K j t λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq222_HTML.gif, Φ j t λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq223_HTML.gif are uniformly bounded operators in L p ( 0 , 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq224_HTML.gif. In a similar way as in [1, 11, 17], we obtain that operators O j t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq225_HTML.gif are R-positive. Then, by using the above representation and by virtue of Kahane’s contraction principle, the product and additional properties of the collection of R-bounded operators (see, e.g., [16], Lemma 3.5, Proposition 3.4), we obtain the assertions. □

3 Singular degenerate anisotropic equation with parameters

Consider the following degenerate BVP with parameters:
k = 1 n ( 1 ) m k t k a k ( x ) D x k [ 2 m k ] u + | β : 2 m | < 1 k = 1 n t k α k 2 m k A β ( x ) D [ β ] u + A ( x ) + λ u = f ( x ) , L k j u = i = 0 m k j α k j i u x k [ i ] ( G k b ) = 0 , j = 1 , 2 , , m k , k = 1 , 2 , , n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ24_HTML.gif
(24)
where A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq123_HTML.gif and A α ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq226_HTML.gif are linear operators in a Banach space E,
G = { x = ( x 1 , x 2 , , x n ) , 0 < x k < b k , k = 1 , 2 , , n } , β = ( β 1 , β 2 , , β n ) , m = ( m 1 , m 2 , , m n ) , | β : 2 m | = k = 1 n β k 2 m k , γ k > 1 , G k b = ( x 1 , x 2 , , x k 1 , b k , x k + 1 , , x n ) , G k b = ( x 1 , x 2 , , x k 1 , b k , x k + 1 , , x n ) , 0 m k j 2 m k 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equas_HTML.gif

a k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq227_HTML.gif are complex-valued functions on G, α k j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq228_HTML.gif are complex numbers, t k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq16_HTML.gif are positive and λ is a complex parameter.

Note that BVP (24) is degenerated with different speeds on different directions in general.

The main result of this section is the following.

Theorem 3 Assume the following conditions hold:
  1. (1)

    E is a UMD space, A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq123_HTML.gif is R-positive in E uniformly with respect to x G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq229_HTML.gif and A ( x ) A 1 ( x 0 ) C ( G ¯ ; B ( E ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq230_HTML.gif, x 0 [ , 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq231_HTML.gif;

     
  2. (2)
    for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq144_HTML.gif, there is a positive C ( ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq145_HTML.gif such that
    A β ( x ) u ε u ( E ( A ) , E ) 1 | β : 2 m | , + C ( ε ) u for u ( E ( A ) , E ) 1 | α : 2 m | , ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equat_HTML.gif
     
  3. (3)

    γ k > 1 + 1 + 4 p + 1 2 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq232_HTML.gif, σ k i = 1 2 m k p ( α k 1 ) + i 2 m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq233_HTML.gif, i = 0 , 1 , , 2 m k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq234_HTML.gif, p k ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq235_HTML.gif;

     
  4. (4)

    a k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq227_HTML.gif are continuous positive functions on G ¯ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq236_HTML.gif.

     
Then, for f L p ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq237_HTML.gif, | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq193_HTML.gif and sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif, problem (24) has a unique solution u that belongs to W p , α [ 2 m ] ( G ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq238_HTML.gif and the following coercive uniform estimate holds:
k = 1 n i = 0 2 m | λ | 1 i 2 m t k σ k i D x k [ i ] u L p ( G ; E ) + A u L p ( G ; E ) C f L p ( G ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ25_HTML.gif
(25)

Proof

Consider the BVP
( 1 ) m 1 a 1 ( x 1 ) t 1 D x 1 [ 2 m 1 ] u ( x 1 ) + A ( x ) u ( x 1 ) + λ u ( x 1 ) = f ( x 1 ) , x 1 ( 0 , b 1 ) , L 1 k u = 0 , k = 1 , 2 , , m 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ26_HTML.gif
(26)
where L 1 k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq239_HTML.gif are boundary conditions of type (24) on ( 0 , b 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq240_HTML.gif. By virtue of Result 2, problem (26) has a unique solution u W p 1 , α 1 [ 2 ] ( 0 , b 1 ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq241_HTML.gif for all f L p 1 ( 0 , b 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq242_HTML.gif, | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq10_HTML.gif and sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif. Moreover, the following coercive uniform estimate holds:
i = 0 2 m 1 | λ | 1 i 2 m 1 t 1 σ 1 i u [ i ] L p 1 ( 0 , b 1 ; E ) + A u L p 1 ( 0 , b 1 ; E ) C f L p 1 ( 0 , b 1 ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equau_HTML.gif
Let us now consider in L p 1 , p 2 ( G 2 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq243_HTML.gif the BVP on the domain G 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq244_HTML.gif
k = 1 2 ( 1 ) m k t k a k ( x ) D x k [ 2 m k ] u + λ u = f ( x 1 , x 2 ) , L k j u = 0 , j = 1 , 2 , , m k , k = 1 , 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ27_HTML.gif
(27)
where G 2 = ( 0 , b 1 ) × ( 0 , b 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq245_HTML.gif. Since L p 2 ( 0 , b 2 ; L p 1 ( 0 , b 1 ; E ) ) = L p 1 , p 2 ( G 2 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq246_HTML.gif, then problem (27) can be expressed in the following way:
( 1 ) m 2 t 2 a 2 D x 2 [ 2 m 2 ] u ( x 2 ) + ( B + λ ) u ( x 2 ) = f ( x 2 ) , L 2 k u = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equav_HTML.gif
where B is the differential operator in L p 1 ( 0 , b 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq247_HTML.gif generated by problem (26), i.e.,
D ( B ) = W p 1 α 1 [ 2 m 1 ] ( 0 , b 1 ; L 1 k ) , B u = a 1 ( x 1 ) t 2 D 2 u ( x 1 ) + d u ( x 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equaw_HTML.gif
By virtue of [22], L p 1 ( 0 , b 1 ; E ) UMD https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq248_HTML.gif for p 1 ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq249_HTML.gif provided E UMD https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq73_HTML.gif. Moreover, by virtue of Theorem 2, the operator B is R-positive in L p 1 ( 0 , b 1 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq250_HTML.gif. Hence, by Result 2, we get that problem (27) has a unique solution
u W p 1 , p 2 , α ˜ [ 2 ] ( G 2 ; E ( A ) , E ) , α ˜ = ( α 1 , α 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equax_HTML.gif
for f L p 1 , p 2 ( G 2 ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq251_HTML.gif, | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq193_HTML.gif and sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif, and coercive uniform estimate (25) holds. By continuing this process for k = n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq252_HTML.gif, we obtain that the following problem:
k = 1 n ( 1 ) m k t k a k ( x ) D x k [ 2 m k ] u + A ( x ) u + λ u = f ( x ) , L k j u = 0 , j = 1 , 2 , , m k , k = 1 , 2 , , n , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ28_HTML.gif
(28)
for f L p ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq237_HTML.gif, | arg λ | φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq10_HTML.gif and sufficiently large | λ | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq12_HTML.gif, has a unique solution u W p , α [ 2 m ] ( G ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq253_HTML.gif and the following coercive uniform estimate holds:
k = 1 n i = 0 2 m | λ | 1 i 2 m t k σ k i D x k [ i ] u L p ( G ; E ) + A u L p ( G ; E ) C f L p ( G ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ29_HTML.gif
(29)
Moreover, by virtue of embedding Theorem A3, we have the Ehrling-Nirenberg-Gagliardo type estimate
L 1 u L p ( G ; E ) h μ u W p , α [ 2 ] ( G ; E ( A ) , E ) + h ( 1 μ ) u L p ( G ; E ) , h ( 0 , 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equay_HTML.gif
Let Q t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq254_HTML.gif denote the operator generated by problem (28) and
L 1 u = | β : 2 m | < 1 k = 1 n t k β k 2 m k A β ( x ) D [ β ] u . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equaz_HTML.gif
By using estimate (29), we obtain that there is a δ ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq255_HTML.gif such that
L 1 ( Q t + λ ) 1 L ( L p ( G ; E ) ) < δ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equba_HTML.gif

Then, by using perturbation elements, we obtain the assertion. □

From Theorem 2 and Theorem 3, we obtain the following result.

Result 3 Let all conditions of Theorem 3 hold for φ > π 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq256_HTML.gif and A α = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq257_HTML.gif. Then the operator Q t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq254_HTML.gif is uniformly R-positive in L p ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq258_HTML.gif, it also is a generator of an analytic semigroup.

4 Singular degenerate parabolic DOE

Consider the following mixed problem for a parabolic DOE with parameter:
u y + k = 1 n ( 1 ) m k t k a k ( x ) D x k [ 2 m k ] u + ( A ( x ) + d ) u = f ( y , x ) , y R + , x G L k j u = i = 0 m k j α k j i u x k [ i ] ( G k b ) = 0 , j = 1 , 2 , , m k , u ( 0 , x ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ30_HTML.gif
(30)

where t k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq16_HTML.gif, a k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq259_HTML.gif, G, G k b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq260_HTML.gif, m k j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq261_HTML.gif are defined as in Section 3, d > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq262_HTML.gif.

For p ˜ = ( p 1 , p 2 , , p n , p 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq263_HTML.gif, Δ + = R + × G https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq264_HTML.gif, L p ˜ ( Δ + ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq265_HTML.gif will denote the space of all E-valued p ˜ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq266_HTML.gif-summable functions with a mixed norm. Analogously,
W p ˜ , γ [ 1 , 2 m ] ( Δ + , E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equbb_HTML.gif

denotes the Sobolev space with a corresponding mixed norm (see [24] for a scalar case).

Let Q t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq254_HTML.gif denote a differential operator generated by (28) for λ = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq267_HTML.gif.

Theorem 4 Let all conditions of Theorem  3 hold for A β = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq268_HTML.gif and φ > π 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq269_HTML.gif. Then, for f L p ˜ ( Δ + ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq270_HTML.gif and sufficiently large d > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq262_HTML.gif, problem (30) has a unique solution belonging to W p ˜ , α 1 , [ 2 m ] ( Δ + ; E ( A ) , E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq271_HTML.gif and the following coercive estimate holds:
u y L p ˜ ( G + ; E ) + k = 1 n t k D x k [ 2 m k ] u L p ˜ ( G + ; E ) + A u L p ˜ ( G + ; E ) C f L p ˜ ( G + ; E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equbc_HTML.gif
Proof Problem (30) can be expressed as the following Cauchy problem:
d u d y + ( Q t + d ) u ( y ) = f ( y ) , u ( 0 ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ31_HTML.gif
(31)
Result 3 implies that the operator Q t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq254_HTML.gif is R-positive in F = L p ( G ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq272_HTML.gif. By [23], Section 1.14], Q t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq254_HTML.gif is a generator of an analytic semigroup in F. Then, by virtue of [20], Theorem 4.2], we obtain that for f L p 0 ( R + ; F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq273_HTML.gif problem (31) has a unique solution belonging to W p 0 1 ( R + ; D ( Q t ) , F ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq274_HTML.gif and the following estimate holds:
d u d y L p 0 ( R + ; F ) + Q t u L p 0 ( R + ; F ) C f L p 0 ( R + ; F ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equbd_HTML.gif
Since L p 0 ( R + ; F ) = L p ˜ ( Δ + ; E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq275_HTML.gif, by Theorem 3 we have
( Q t + d ) u L p 0 ( R + ; F ) = D ( Q t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Eqube_HTML.gif

The above estimate proves the hypothesis to be true. □

5 Cauchy problem for infinite systems of degenerate parabolic equations with small parameters

Consider the infinity systems of BVP for the degenerate anisotropic parabolic equation:
u i y + k = 1 n ( 1 ) m k t k a k ( x ) D x k [ 2 m k ] u i + j = 1 N a i j ( x ) u j + d u = f i ( x , y ) , i = 1 , 2 , , N , L k j u = i = 0 m k j β k j ν D x k ( ν ) u i ( G k b , y ) = 0 , j = 1 , 2 , , m k , u ( 0 , x ) = 0 , x G , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equ32_HTML.gif
(32)
where N is finite or infinite natural number, t k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq16_HTML.gif, a k ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq276_HTML.gif, G, G k b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq260_HTML.gif, G + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq277_HTML.gif, m k j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq261_HTML.gif, d are defined as in Sections 3 and 4, a i j ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq278_HTML.gif are real functions and
A ( x ) = { a i j ( x ) } , u = { u j } , A u = { j = 1 N a i j ( x ) u j } , i , j = 1 , 2 , , N , l q ( A ) = { u l q , u l q ( A ) = A u l q = sup i ( j = 1 N | a i j ( x ) u j | q ) 1 q < } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equbf_HTML.gif

From Theorem 4 we obtain the following.

Theorem 5 Let γ k > 1 + 1 + 4 p + 1 2 p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq232_HTML.gif, p k , p 0 ( 1 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq279_HTML.gif, a k , a i j C ( G ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq280_HTML.gif, a k ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq281_HTML.gif, a i j 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq282_HTML.gif and a i j = a j i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq283_HTML.gif. Then for f ( x , y ) = { f i ( x , y ) } 1 N L p ˜ ( G + ; l q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq284_HTML.gif and sufficiently large d > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq262_HTML.gif, problem (32) has a unique solution u = { u i ( x , y ) } 1 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq285_HTML.gif that belongs to the space W p ˜ , α 1 , [ 2 m ] ( G + , l q ( A ) , l q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq286_HTML.gif and the following coercive uniform estimate holds:
u y L p ˜ ( G + ; l q ) + k = 1 n t k D x k [ 2 m k ] u L p ˜ ( G + ; l q ) + A u L p ˜ ( G + ; l q ) C f L p ˜ ( G + ; l q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equbg_HTML.gif
Proof Let E = l q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq287_HTML.gif and A be infinite matrices such that
A = [ a i j ] , i , j = 1 , 2 , , N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_Equbh_HTML.gif

It is clear that the operator A is R-positive in l q https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-30/MediaObjects/13661_2012_Article_295_IEq288_HTML.gif. Problem (32) can be rewritten as problem (30). Then, from Theorem 4, we obtain the assertion. □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)
Department of Environmental Sciences, Okan University, Akfirat
(2)
Department of Mechanical Engineering, Okan University, Akfirat

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© Sahmurova and Shakhmurov; licensee Springer. 2013

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