Open Access

Complicated asymptotic behavior of solutions for a porous medium equation with nonlinear sources

Boundary Value Problems20132013:35

DOI: 10.1186/1687-2770-2013-35

Received: 12 July 2012

Accepted: 4 February 2013

Published: 21 February 2013

Abstract

In this paper, we investigate the complicated asymptotic behavior of the solutions to the Cauchy problem of a porous medium equation with nonlinear sources when the initial value belongs to a weighted L https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq1_HTML.gif space.

AMS Subject Classification:35K55, 35B40.

Keywords

complexity asymptotic behavior porous medium equation

1 Introduction

In this paper, we consider the asymptotic behavior of solutions for the Cauchy problem of the porous medium equation with nonlinear sources
u t Δ u m = u p , in  R N × ( 0 , ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ1_HTML.gif
(1.1)
u ( x , 0 ) = u 0 ( x ) , in  R N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ2_HTML.gif
(1.2)

where m , p > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq2_HTML.gif and u 0 L ( ρ σ ) { φ ; φ ρ σ L ( R N ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq3_HTML.gif with ρ σ ( x ) = ( 1 + | x | 2 ) σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq4_HTML.gif.

It is well known that any positive solutions of problem (1.1)-(1.2) blow up in finite time if 1 < p p c m + 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq5_HTML.gif [13], while positive global solutions do exist if p > p c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq6_HTML.gif [47]. In 2000, Mukai, Mochizuki and Huang in [6] found that if p > m + 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq7_HTML.gif and 2 p m < σ < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq8_HTML.gif and 0 φ C b ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq9_HTML.gif satisfies lim sup | x | | x | σ φ ( x ) < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq10_HTML.gif, then there exists a constant η ( φ ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq11_HTML.gif such that for 0 < η < η ( φ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq12_HTML.gif, the solutions u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq13_HTML.gif of problem (1.1)-(1.2) with the initial value u 0 = η φ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq14_HTML.gif are global and the following estimate holds:
u ( t ) L ( R N ) C t σ σ ( m 1 ) + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ3_HTML.gif
(1.3)
Moreover, if lim | x | | x | σ φ ( x ) = M 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq15_HTML.gif, then
t σ σ ( m 1 ) + 2 u ( t 1 σ ( m 1 ) + 2 x , t ) t S ( 1 ) w 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equa_HTML.gif
uniformly on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif, where w 0 ( x ) = M | x | σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq17_HTML.gif. Here S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq18_HTML.gif is a semigroup generated by the Cauchy problem of the porous medium equation
w t Δ w m = 0 , in  R N × ( 0 , ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ4_HTML.gif
(1.4)
w ( x , 0 ) = w 0 ( x ) , in  R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ5_HTML.gif
(1.5)

and w 0 ( x ) = η M 0 | x | σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq19_HTML.gif.

On the other hand, regarding problem (1.4)-(1.5), in 2002, Vázquez and Zuazua [8] found that for any bounded sequence { φ n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq20_HTML.gif in L ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq21_HTML.gif, there exists an initial value u 0 L ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq22_HTML.gif and a sequence t n k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq23_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq24_HTML.gif such that lim k S ( t n k ) u 0 ( t n k 1 2 x ) = S ( 1 ) φ n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq25_HTML.gif uniformly on any compact subsets of R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif. In our previous papers [9], for any bounded sequence { φ n } n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq26_HTML.gif in C 0 + ( R N ) { φ C 0 ( R N ) ; ϕ ( x ) 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq27_HTML.gif, we have shown that there exists an initial value u 0 C 0 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq28_HTML.gif and a sequence t n k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq29_HTML.gif as k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq24_HTML.gif such that
lim k t n k μ 2 S ( t n k ) u 0 ( t n k β x ) = S ( 1 ) φ n ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equb_HTML.gif

uniformly on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif, where 0 < μ < N N ( m 1 ) + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq30_HTML.gif and β = 2 μ ( m 1 ) 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq31_HTML.gif. For more details on the study of complicated asymptotic behavior of solutions for the heat equation and other evolution equations, we refer the readers to [1014].

In this paper, we are quite interested in the above mentioned same topic for the equation with strongly nonlinear sources, namely equation (1.1) with p > m + 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq32_HTML.gif. We will show that for any M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq33_HTML.gif, there is a constant η ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq34_HTML.gif and an initial value u 0 C η ( M ) σ , + { φ C ( R N ) ; φ B η ( M ) σ , + } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq35_HTML.gif with 2 p m < σ < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq36_HTML.gif such that for any φ C η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq37_HTML.gif, there exists a sequence t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq38_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq39_HTML.gif satisfying
lim n t n σ σ ( m 1 ) + 2 S ( t n ) u 0 ( t n 1 σ ( m 1 ) + 2 x ) = S ( 1 ) φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equc_HTML.gif
uniformly on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif. Here B η ( M ) σ , + { ϕ = η φ ; 0 φ L ( ρ σ ) , ( 1 + | | 2 ) σ 2 φ ( ) L ( R N ) M  and  0 η η ( M ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq40_HTML.gif. For this purpose, we first show that if the initial value u 0 B η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq41_HTML.gif, then the solutions u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq13_HTML.gif are global and satisfy
u ( x , t ) C ( 1 + t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ6_HTML.gif
(1.6)
One can easily see that (1.6) captures (1.3). From this, we can follow the framework by Kamin and Peletier [15] to prove that
lim t t 1 σ ( m 1 ) + 2 u ( t σ σ ( m 1 ) + 2 , t ) S ( t ) u 0 ( t 1 σ ( m 1 ) + 2 ) L ( R N ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ7_HTML.gif
(1.7)

So, we can get our results by following the framework in [9] and using (1.6)-(1.7).

The rest of this paper is organized as follows. The next section is devoted to giving a sufficient condition for the global existence of solutions for problem (1.1)-(1.2) and the upper bounded estimates on these solutions. In the last section, we investigate the complicated asymptotic behavior of solutions.

2 Preliminaries and estimates

In this section we state the definition of a weak solution of problem (1.1)-(1.2) and give the upper bounded estimates on the global solutions. We begin with the definition of the weak solution of problem (1.1)-(1.2).

Definition 2.1 [16, 17]

By a weak solution of problem (1.1)-(1.2) in R N × [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq42_HTML.gif, we mean a function u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq13_HTML.gif in R N × [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq42_HTML.gif such that
  1. 1.

    u ( x , t ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq43_HTML.gif in R N × [ 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq42_HTML.gif and u ( x , t ) C ( R N × ( 0 , τ ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq44_HTML.gif for each 0 < τ < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq45_HTML.gif.

     
  2. 2.
    For 0 < τ < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq46_HTML.gif and any nonnegative φ ( x , t ) C 2 , 1 ( R N × [ 0 , T ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq47_HTML.gif which vanishes for large | x | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq48_HTML.gif, the following equation holds:
    R N u ( x , τ ) φ ( x , τ ) d x R N u 0 ( x ) φ ( x , 0 ) d x = 0 τ R N u m ( x , t ) Δ φ ( x , t ) d x d t + 0 τ R N u ( x , t ) φ t ( x , t ) d x d t + 0 τ R N u p ( x , t ) φ ( x , t ) d x d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ8_HTML.gif
    (2.1)
     

A supersolution [or subsolution] is similarly defined with equality of (2.1) replaced by ≥ [or ≤]. The weak solutions for problem (1.4)-(1.5) can be defined in a similar way as above. It is well known that problem (1.1)-(1.2) has a unique, nonnegative and bounded weak solution, at least locally in time [16, 17]. Now we state the comparison principle for problem (1.1)-(1.2).

Lemma 2.1 [16, 17]

Suppose that for 0 < τ < T https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq49_HTML.gif, u ¯ ( x , t ) , u ̲ ( x , t ) C ( R N × [ 0 , T ) ) L ( R N × [ 0 , τ ] ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq50_HTML.gif are supersolution and subsolution of the problem (1.1)-(1.2), respectively. If
u ¯ ( x , 0 ) u ̲ ( x , 0 ) for  x R N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equd_HTML.gif
then, for all ( x , t ) R N × ( 0 , T ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq51_HTML.gif,
u ¯ ( x , t ) u ̲ ( x , t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Eque_HTML.gif
To study the asymptotic behavior of solutions for problem (1.1)-(1.2), we adopt the space X 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq52_HTML.gif and L ( ρ σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq53_HTML.gif as that in [1618]. For any σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq54_HTML.gif and ρ σ ( x ) = ( 1 + | x | 2 ) σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq55_HTML.gif, the L ( ρ σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq53_HTML.gif is defined as
L ( ρ σ ) { φ ; φ ρ σ L ( R N ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equf_HTML.gif
with the obvious norm φ L ( ρ σ ) = φ ρ σ L ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq56_HTML.gif and the X 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq52_HTML.gif is given by
X 0 { φ L loc 1 ( R N ) ; φ 1 <  and  ( φ ) = 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equg_HTML.gif
with the norm 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq57_HTML.gif. Here
φ r = sup R r R N ( m 1 ) + 2 m 1 { | x | R } | φ ( x ) | d x and ( φ ) = lim r φ r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equh_HTML.gif

Hence they are both Banach spaces. The existence and uniqueness of a weak solution of problem (1.4)-(1.5) with the initial-value u 0 X 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq58_HTML.gif is shown in [16, 17], and this solution satisfies the following proposition.

Proposition 2.1 [17]

Problem (1.4)-(1.5) generates a continuous bounded semigroup in X 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq52_HTML.gif given by
S ( t ) : w 0 w ( x , t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equi_HTML.gif

In other words, S ( t ) w 0 C ( [ 0 , ) ; X 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq59_HTML.gif. Moreover, if u 0 L 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq60_HTML.gif, then the semigroup S ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq18_HTML.gif is a contraction.

We now introduce the definitions of scalings and the commutative relations between the semigroup operators and the dilation operators. For any μ , β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq61_HTML.gif and v ( x ) X 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq62_HTML.gif, the dilation D λ μ , β https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq63_HTML.gif is defined as follows:
D λ μ , β w ( x ) λ μ w ( λ β x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equj_HTML.gif
From the definitions of the dilation operator and the semigroup operator, we can get that for μ , β > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq61_HTML.gif and w 0 X 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq64_HTML.gif,
D λ μ , β [ S ( λ 2 t ) w 0 ] ( x ) = S ( λ 2 μ ( m 1 ) 2 β t ) [ D λ μ , β w 0 ] ( x ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ9_HTML.gif
(2.2)

see details in [19, 20].

In the rest of this section, we give a sufficient condition for the existence of global solutions of problem (1.1)-(1.2) and establish the upper bounded estimates of these solutions.

Theorem 2.1 Let 2 p m < σ < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq36_HTML.gif and M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq33_HTML.gif. There exists a constant η ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq34_HTML.gif such that for any 0 η η ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq65_HTML.gif, ϕ ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq66_HTML.gif and ϕ L ( ρ σ ) M https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq67_HTML.gif, the solutions u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq13_HTML.gif of problem (1.1)-(1.2) with the initial value u 0 ( x ) = η ϕ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq68_HTML.gif are global. Moreover, the following estimate holds:
0 u ( x , t ) C ( M , η ) ( 1 + t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ10_HTML.gif
(2.3)

where C ( M , η ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq69_HTML.gif is a constant dependent only on M and η.

Remark 2.1 Notice that if 0 φ C b ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq9_HTML.gif and lim sup | x | φ ( x ) | x | σ < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq70_HTML.gif, then φ L ( ρ σ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq71_HTML.gif. So, our results capture Theorem 3 in [6]. Here we use some ideas of them.

Proof To prove this theorem, we need the fact that if v 0 = M | x | σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq72_HTML.gif, then
S ( t ) v 0 ( x ) C ( M ) ( t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ11_HTML.gif
(2.4)
which has been given in Lemma 2.6 of [20]. We give the proof here for completeness. In fact,
v 0 r = sup R r R N ( m 1 ) + 2 m 1 B R A | x | σ d x C r σ 2 m 1 0 as  r . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equk_HTML.gif
This means that v 0 X 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq73_HTML.gif. Therefore, from Proposition 2.1, we obtain that S ( t ) v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq74_HTML.gif is well defined. Taking μ = 2 σ σ ( m 1 ) + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq75_HTML.gif and β = 2 σ ( m 1 ) + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq76_HTML.gif in (2.2), we have
λ 2 σ σ ( m 1 ) + 2 [ S ( λ 2 s ) v 0 ] ( λ 2 σ ( m 1 ) + 2 x ) = S ( s ) [ λ 2 σ σ ( m 1 ) + 2 v 0 ( λ 2 σ ( m 1 ) + 2 ) ] ( x ) = S ( s ) v 0 ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ12_HTML.gif
(2.5)
Now taking s = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq77_HTML.gif, λ = t 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq78_HTML.gif and g ( x ) = S ( 1 ) v 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq79_HTML.gif in (2.5), we obtain that
S ( t ) v 0 ( x ) = t σ σ ( m 1 ) + 2 g ( t 1 σ ( m 1 ) + 2 x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ13_HTML.gif
(2.6)
The fact that ϕ C ( R N { 0 } ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq80_HTML.gif clearly means that
S ( t ) v 0 C ( [ 0 , ) × R N { ( 0 , 0 ) } ) C α 2 , α ( ( 0 , ) × R N ) for some  α > 0 ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ14_HTML.gif
(2.7)
see [21]. This implies that for | x | = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq81_HTML.gif, the following limit holds:
t σ σ ( m 1 ) + 2 g ( t 1 σ ( m 1 ) + 2 x ) = S ( t ) v 0 ( x ) ϕ ( x ) = M | x | σ = M as  t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equl_HTML.gif
Let
y = t 1 σ ( m 1 ) + 2 x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equm_HTML.gif
So,
| y | as  t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equn_HTML.gif
Therefore,
| y | σ g ( y ) M 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equo_HTML.gif
as | y | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq82_HTML.gif. This means that there exists an M 1 > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq83_HTML.gif such that if | y | M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq84_HTML.gif, then
g ( y ) 2 M | y | σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ15_HTML.gif
(2.8)
From (2.7), for | y | M 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq85_HTML.gif, there exists a constant C such that
g ( y ) C . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ16_HTML.gif
(2.9)
Combining (2.8) and (2.9), we have
g ( x ) C ( M ) ( 1 + | x | 2 ) σ 2 for  x R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equp_HTML.gif
By (2.6), we thus obtain that
S ( t ) v 0 ( x ) C ( M ) ( t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equq_HTML.gif
So, we complete the proof of (2.4). Now taking
φ ( x ) = M ( 1 + | x | 2 ) σ 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equr_HTML.gif
we get that
0 < φ ( x ) v 0 ( x ) = M | x | σ for  x 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equs_HTML.gif
Therefore, by the comparison principle and (2.4), for all t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq86_HTML.gif, we have
S ( t ) φ ( x ) S ( t ) v 0 ( x ) C ( M ) ( t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ17_HTML.gif
(2.10)
Since S ( t ) φ ( x ) C ( [ 0 , ) × R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq87_HTML.gif (see [17, 21]), there exists a t 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq88_HTML.gif such that for all | x | 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq89_HTML.gif and 0 t t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq90_HTML.gif,
S ( t ) φ ( x ) C φ ( x ) C ( M ) ( 1 + | x | 2 ) σ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equt_HTML.gif
Combining this with (2.6) and using the comparison principle, we can get
S ( t ) ϕ ( x ) S ( t ) φ ( x ) C ( M ) ( 1 + t 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equu_HTML.gif
In other words,
S ( t ) ϕ ( x ) C ( M ) ( ( 1 + t ) 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ18_HTML.gif
(2.11)
If η = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq91_HTML.gif, (2.3) clearly holds. In the rest of proof, we can assume that η > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq92_HTML.gif. The hypothesis 2 p m < σ < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq36_HTML.gif indicates
σ ( p m ) 2 > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equv_HTML.gif
Let
η ( M ) m p = 2 C p 1 ( M ) ( p m ) 0 ( 1 + t ) σ ( p 1 ) σ ( m 1 ) + 2 d t = 2 C ( M ) p 1 ( σ ( m 1 ) + 2 ) ( p m ) σ ( p m ) 2 > 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equw_HTML.gif
where C ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq93_HTML.gif is the constant given by (2.11). For 0 < η η ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq94_HTML.gif, taking
α ( t ) = [ ( η m p C ( M ) p 1 ( p m ) 0 t ( 1 + s ) σ ( p 1 ) σ ( m 1 ) + 2 d x ) ] 1 p m , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equx_HTML.gif
and
w ( x , t ) = S ( t ) ϕ ( x ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equy_HTML.gif
we obtain from (2.11) that α ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq95_HTML.gif is an increasing function satisfying
{ a ( 0 ) = η , a ( t ) 2 1 p m η ( M ) for all  t 0 , a ( t ) = C ( M ) p 1 a ( t ) p m + 1 ( 1 + t ) σ ( p 1 ) σ ( m 1 ) + 2 a ( t ) p m + 1 w ( t ) L ( R N ) p 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ19_HTML.gif
(2.12)
Now letting b ( t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq96_HTML.gif to satisfy
{ b ( t ) = a ( b ( t ) ) m 1 , b ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ20_HTML.gif
(2.13)
and then taking
w ¯ ( x , t ) = a ( b ( t ) ) w ( x , t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equz_HTML.gif
one can see that w ¯ ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq97_HTML.gif is a supersolution of the following problem:
u t Δ u m = u p , ( x , t ) R N × ( 0 , ) ; u ( x , 0 ) = u 0 = η ϕ ( x ) , x R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equaa_HTML.gif
Therefore,
u ( x , t ) a ( b ( t ) ) w ( x , t ) 2 1 p m η ( M ) w ( x , b ( t ) ) C ( η , M ) ( 1 + b ( t ) 2 σ ( m 1 ) + 2 + | x | 2 ) σ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ21_HTML.gif
(2.14)
(2.12) and (2.13) clearly mean that
η m 1 t b ( t ) 2 m 1 p m η ( M ) m 1 t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equab_HTML.gif

From this and (2.14), we can get (2.3). So, we complete the proof of this theorem. □

3 Complicated asymptotic behavior

For any M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq33_HTML.gif, let η ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq34_HTML.gif be as given by Theorem 2.1. We introduce
B η ( M ) σ , + { φ ( x ) = η ϕ ( x ) : ϕ ( x ) 0 , ϕ L ( ρ σ ) M  and  0 η η ( M ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equac_HTML.gif
and
C η ( M ) σ , + { φ C ( R N ) ; φ B η ( M ) σ , + } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equad_HTML.gif

In the rest of this section, we show that the complexity may occur in the asymptotic behavior of solutions of problem (1.1)-(1.2) with u 0 C η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq98_HTML.gif. Our main result is the following theorem.

Theorem 3.1 Let p > m + 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq32_HTML.gif and 2 p m < σ < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq36_HTML.gif. Then there is a function u 0 C η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq98_HTML.gif such that for any φ C η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq37_HTML.gif, there exists a sequence t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq99_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq39_HTML.gif such that
lim n t n σ σ ( m 1 ) + 2 u ( t n 1 σ ( m 1 ) + 2 x , t n ) = S ( 1 ) φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equae_HTML.gif

uniformly on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif. Here u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq13_HTML.gif is the solution of problem (1.1)-(1.2).

To get this theorem, we need to prove the following lemma first.

Lemma 3.1 Suppose p > m + 2 N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq32_HTML.gif and M > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq33_HTML.gif. Let u be a solution of problem (1.1)-(1.2). If 0 u 0 B η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq100_HTML.gif with 2 p m < σ < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq36_HTML.gif, then
lim t t σ 2 + σ ( m 1 ) u ( t 1 2 + σ ( m 1 ) , t ) [ S ( t ) u 0 ] ( t 1 2 + σ ( m 1 ) ) L ( R N ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equaf_HTML.gif

Proof

We first define the functions
u λ ( x , t ) = D λ μ , β u ( x , λ t ) = λ μ u ( λ β x , λ 2 t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equag_HTML.gif
and
w λ ( x , t ) = D λ μ , β w ( x , λ t ) = λ μ w ( λ β x , λ 2 t ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equah_HTML.gif
where μ = 2 σ σ ( m 1 ) + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq75_HTML.gif and β = 2 σ ( m 1 ) + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq101_HTML.gif. Using the comparison principle, we know that for ( x , t ) R N × ( 0 , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq102_HTML.gif,
w ( x , t ) u ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equai_HTML.gif
and for all λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq103_HTML.gif,
w λ ( x , t ) u λ ( x , t ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equaj_HTML.gif
The results of Theorem 2.1 imply that
u λ ( x , t ) C λ μ [ ( 1 + λ 2 t ) β + λ 2 β | x | 2 ] σ 2 C ( ( λ 2 + t ) 2 β + | x | 2 ) σ 2 C ( λ 2 + t ) μ ( 1 + ( λ 2 + t ) 2 β | x | 2 ) σ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ22_HTML.gif
(3.1)
Here we have used the fact μ = β σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq104_HTML.gif. So,
0 τ B 1 u λ ( x , t ) d x d t C λ 2 τ + λ 2 s N β μ d s 0 s β r N σ 1 d r C τ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equak_HTML.gif
Now we estimate the integral
0 τ B 1 u λ ( x , t ) q d x d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equal_HTML.gif
with q > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq105_HTML.gif in several steps. For any τ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq106_HTML.gif, we take λ large enough to satisfy λ 2 τ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq107_HTML.gif and assume, without loss of generality, that ( τ + λ 2 ) β > 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq108_HTML.gif in the rest of this proof. Then using the same method as above, we have
0 τ B 1 u λ ( x , t ) q d x d t { C τ 2 γ β + C τ if  γ > 0  and  N σ q , C τ + C τ ln 1 τ if  γ > 0  and  N = σ q , C τ + C ln ( 1 + λ 2 τ ) if  γ = 0 , C τ + C λ 2 β γ if  γ < 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ23_HTML.gif
(3.2)
where γ = N + σ ( m 1 ) σ q + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq109_HTML.gif. Similarly, we can get the integral estimates for w λ ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq110_HTML.gif, which have been given in [22]. By using the same methods as in [15], we can get that for T > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq111_HTML.gif
u λ ( T ) w λ ( T ) 0 as  λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ24_HTML.gif
(3.3)
uniformly on any compact subset of R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif. For any T , λ , ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq112_HTML.gif, we can obtain from (3.1) that there exists a constant R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq113_HTML.gif satisfying
u λ ( T ) L ( R N B R ) φ λ ( T ) L ( R N B R ) < ϵ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ25_HTML.gif
(3.4)
and
w λ ( T ) L ( R N B R ) u λ ( , T ) L ( R N B R ) < ϵ 3 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ26_HTML.gif
(3.5)
where φ λ ( x , t ) = C λ μ [ ( 1 + λ 2 t ) 2 σ ( m 1 ) + 1 + | λ β x | 2 ] σ 2 = C [ ( λ 2 + t ) 2 σ ( m 1 ) + 2 + | x | 2 ] σ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq114_HTML.gif and B R { x R N ; | x | R } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq115_HTML.gif. Taking R as given by (3.4), from (3.3), there exists λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq116_HTML.gif such that for all λ λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq117_HTML.gif,
u λ ( T ) w λ ( T ) L ( B R ) < ϵ 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ27_HTML.gif
(3.6)
Therefore, from (3.4)-(3.6), we have
lim λ u λ ( T ) w λ ( T ) L ( R N ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ28_HTML.gif
(3.7)
Now letting T = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq118_HTML.gif and λ = t 1 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq78_HTML.gif in (3.7), we get that
lim t t σ 2 + σ ( m 1 ) [ u ( t 1 2 + σ ( m 1 ) , t ) w ( t 1 2 + σ ( m 1 ) , t ) ] L ( R N ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equam_HTML.gif

So, we complete the proof of this lemma. □

Now we can prove our main result.

Proof of Theorem 3.1

Let
μ = 2 σ σ ( m 1 ) + 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equan_HTML.gif
and
β = 2 σ ( m 1 ) + 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equao_HTML.gif
From the definition of C η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq119_HTML.gif, we obtain that there exists a countable set F such that
F C η ( M ) σ , + L 1 ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equap_HTML.gif
and for any ϵ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq120_HTML.gif and φ C η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq37_HTML.gif, there exists a function ϕ ϵ F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq121_HTML.gif satisfying
φ ϵ φ L ( R N ) < ϵ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ29_HTML.gif
(3.8)
Therefore, there exists a sequence { φ j } j 1 F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq122_HTML.gif such that
  1. I.
    For any ϕ F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq123_HTML.gif, there exists a subsequence { φ j k } k 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq124_HTML.gif of the sequence { φ j } j 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq125_HTML.gif satisfying
    φ j k ( x ) = ϕ for all  k 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equaq_HTML.gif
     
  2. II.
    There exists a constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq126_HTML.gif satisfying
    max ( φ j L ( R N ) , φ j L 1 ( R N ) ) C j for  j 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equar_HTML.gif
     
Now we can follow the methods given in [9] to construct an initial value as follows. Let
u 0 ( x ) = j = 1 λ j μ χ j ( x / λ j β ) φ j ( x / λ j β ) = j = 1 D λ j 1 μ , β [ χ j ( x ) φ j ( x ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ30_HTML.gif
(3.9)
Here
λ j = { 2 for  j = 1 , max ( j 4 N ( m 1 ) + 8 2 N μ [ N ( m 1 ) + 2 ] λ j 1 4 β N 2 μ 2 N μ [ N ( m 1 ) + 2 ] , ( 2 j λ j 1 ) 1 μ , λ ¯ j ) for  j > 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ31_HTML.gif
(3.10)
χ j ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq127_HTML.gif is the cut-off function defined on { x R N ; 2 j < | x | < 2 j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq128_HTML.gif relatively to { x R N ; 2 j + 1 < | x | < 2 j 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq129_HTML.gif, and λ ¯ j https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq130_HTML.gif is selected large enough to satisfy
D λ j μ , β [ S ( λ j 2 t ) u 0 ( x ) ] = D λ j μ , β [ S ( λ j 2 t ) n = 1 j 1 λ n μ χ n ( x / λ n β ) φ n ( x / λ n β ) ] + D λ j μ , β [ S ( λ j 2 t ) λ j μ χ j ( x / λ j β ) φ j ( x / λ j β ) ] + D λ j μ , β [ S ( λ j 2 t ) n = j + 1 λ n μ χ n ( x / λ n β ) φ n ( x / λ n β ) ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equas_HTML.gif
Notice first that if φ C η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq37_HTML.gif, then
φ L ( R N ) η ( M ) , φ L ( ρ σ ) η ( M ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equat_HTML.gif
and
φ C 0 ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equau_HTML.gif
By (3.9) and (3.10), we have
u 0 L ( R N ) u 0 L ( ρ σ ) sup j 1 λ j μ χ j ( x / λ j β ) φ j ( x / λ j β ) L ( ρ σ ) η ( M ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equav_HTML.gif
So, we have
u 0 C η ( M ) σ , + C 0 ( R N ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equaw_HTML.gif
Using the same method as that in [9], we can get that for any φ F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq131_HTML.gif, there exists a sequence t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq132_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq39_HTML.gif such that
t n σ σ ( m 1 ) + 2 [ S ( t n ) u 0 ] ( t n 1 σ ( m 1 ) + 2 x ) n S ( 1 ) φ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ32_HTML.gif
(3.11)
uniformly on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif. For any ϕ C η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq133_HTML.gif, from (1.2), we know that there exists a sequence { φ k } F https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq134_HTML.gif such that
φ k ϕ as  k . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equax_HTML.gif
Therefore,
S ( 1 ) φ k S ( 1 ) ϕ as  k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ33_HTML.gif
(3.12)
uniformly on any compact subset of R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif. This uses the fact that the map S ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq135_HTML.gif is regularizing since the images of bounded sets are relatively compact subsets of C α https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq136_HTML.gif for some α > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq137_HTML.gif in compact sets of R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif [21]. And notice that φ k , ϕ C η ( M ) σ , + B η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq138_HTML.gif. We thus obtain from Theorem 2.1 that for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq139_HTML.gif, there exists R > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq113_HTML.gif such that if | x | > R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq140_HTML.gif, then
S ( 1 ) ϕ ( x ) < ε 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ34_HTML.gif
(3.13)
and
S ( 1 ) φ k ( x ) < ε 3 for all  k 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ35_HTML.gif
(3.14)
Combining (3.12), (3.13) with (3.14), we thus have that
S ( 1 ) φ k S ( 1 ) ϕ as  k https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equ36_HTML.gif
(3.15)
uniformly on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif. By Lemma 3.1, (3.11) and (3.15), we can get that for any ϕ C η ( M ) σ , + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq133_HTML.gif, there exists a sequence t n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq99_HTML.gif as t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq141_HTML.gif such that
lim n t n σ σ ( m 1 ) + 2 u ( t n 1 σ ( m 1 ) + 2 x , t n ) = S ( 1 ) ϕ ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_Equay_HTML.gif

uniformly on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-35/MediaObjects/13661_2012_Article_300_IEq16_HTML.gif. So, we complete the proof of Theorem 3.1. □

Declarations

Acknowledgements

This work is supported by NSFC, the Research Fund for the Doctoral Program of Higher Education of China, the Natural Science Foundation Project of ‘CQ CSTC’ (cstc2012jjA00013), the Scientific and Technological Projects of Chongqing Municipal Commission of Education (KJ121105).

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Chongqing Three Gorges University
(2)
School of Mathematical Sciences, South China Normal University

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© Wang and Yin; licensee Springer. 2013

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