Skip to content

Advertisement

  • Research
  • Open Access

Asymptotics of solutions of second order parabolic equations near conical points and edges

Boundary Value Problems20142014:252

https://doi.org/10.1186/s13661-014-0252-x

Received: 27 August 2014

Accepted: 21 November 2014

Published: 11 December 2014

Abstract

The authors consider the first boundary value problem for a second order parabolic equation with variable coefficients in a domain with conical points or edges. In the first part of the paper, they study the Green function for this problem in the domain K × R n m , where K is an infinite cone in R m , 2 m n . They obtain the asymptotics of the Green function near the vertex ( n = m ) and edge ( n > m ), respectively. This result is applied in the second part of the paper, which deals with the initial-boundary value problem in this domain. Here, the right-hand side f of the differential equation belongs to a weighted L p space. At the end of the paper, the initial-boundary value problem in a bounded domain with conical points or edges is studied.

1 Introduction

The present paper is concerned with an initial-boundary value problem for a second order parabolic equation in a n-dimensional domain with a ( n m ) -dimensional edge M, n > m 2 . In particular, we are interested in the asymptotics of the solution near the edge. The largest part of the paper deals with the problem
t u ( x , t ) L ( x , t , x ) u ( x , t ) = f ( x , t ) for  x D , t > 0 , u | D × R + = 0 , u | t = 0 = 0
(1)
in the domain D = K × R n m . Here
K = { x R m : ω = x / | x | Ω } ,
is an infinite cone (angle if m = 2 ), Ω is a subdomain of the unit sphere S m 1 with C 1 , 1 boundary Ω, and
L ( x , t , x ) = i , j = 1 n a i j ( x , t ) x i x j + j = 1 n a j ( x , t ) x j + a 0 ( x , t ) ,
(2)

is a linear second order differential operator with variable coefficients.

Initial-boundary value problems for parabolic equations in domains with angular or conical points and edges were studied in a number of papers. Most of these papers deal with the heat equation. Concerning the heat equation in domains with angular or conical points, we mention the papers by Grisvard [1], Kozlov and Maz’ya [2], de Coster and Nicaise [3], where the asymptotics of the solutions near the singular boundary points was studied. For domains with edges, Solonnikov [4], [5] and Nazarov [6] estimated the Green function and proved the existence of solutions of the Dirichlet and Neumann problems for the heat equation in weighted Sobolev spaces. Kozlov and Rossmann [7], [8] and Kweon [9] investigated the asymptotics of solutions of the Dirichlet problem for the heat equation near an edge.

A theory for general parabolic problems with time-independent coefficients in domains with conical points was developed in papers by Kozlov [10]–[12]. This theory includes the asymptotics of solutions in weighed L 2 Sobolev spaces and a description of the singularities of the Green function near the conical points. The goal of the present paper is to extend these results to the case of time-dependent coefficients and to domains with edges. Moreover, we consider solutions in weighted L p Sobolev spaces with arbitrary p ( 1 , ) . However, we restrict ourselves to second order parabolic equations, and we consider only the first terms in the asymptotics. In our previous paper [13], we obtained point estimates for the Green function. These estimates together with results of the theory of elliptic boundary value problems are used in the present paper in order to describe the behavior of solutions near the edge.

We outline the main results of the present paper. For an arbitrary point x = ( x 1 , , x n ) R n , we put x = ( x 1 , , x m ) and x = ( x m + 1 , , x n ) . The same notation is used for multi-indices α = ( α 1 , , α n ) . We assume that a i j = a j i are real-valued functions and that
| a i j ( x , t ) a i j ( 0 , 0 ) | ϵ , | a i ( x , t ) | ϵ | x | 1 , | a 0 ( x , t ) | ϵ | x | 2 ,
(3)
where ϵ is a small positive number. Besides this assumption, we impose some smoothness conditions on the coefficients a i j and a j (see (21) and (22)). The condition (3) ensures in particular that the difference of the operators L ( x , t , x ) and
L 0 ( 0 , 0 , x ) = i , j = 1 n a i j ( 0 , 0 ) x i x j
is small in the operator norm W p ; β 2 , 1 ( D T ) L p ; β ( D T ) . Here W p ; β 2 l , l ( D T ) is the Sobolev space on D T = D × ( 0 , T ) with the norm
u W p ; β 2 l , l ( D T ) = ( 0 T D 2 k + | α | 2 l | x | p ( β 2 l + 2 k + | α | ) | t k x α u ( x , t ) | p d x d t ) 1 / p .
(4)

For l = 0 , this space is denoted by L p ; β ( D T ) .

In Sections 2 and 3, we deal with the asymptotics of the Green function near the edge M of D . In the case of constant coefficients a i j , the asymptotics can easily be obtained by means of the asymptotics of the Green function for the heat equation which is given in [7], [8]. If the coefficients are variable, then the terms in the asymptotics contain the eigenvalues and eigenfunctions of the following operator pencil A ( x , t ; λ ) :
A ( x , t ; λ ) Φ ( ω ) = | x | 2 λ L 0 ( 0 , x , t , x , 0 ) | x | λ Φ ( ω ) , Φ W 2 1 ( Ω ) .
(5)
Let λ 1 + ( x , t ) be the smallest positive eigenvalue and let Φ 1 + ( x , t ; ω ) be the corresponding eigenfunction. As was proved in [13], the Green function G ( x , y , t , τ ) of the problem (1) satisfies the estimate
| G ( x , y , t , τ ) | c ( t τ ) n / 2 ( | x | | x | + t τ ) λ ( | y | | y | + t τ ) λ exp ( κ | x y | 2 t τ )
for 0 < t τ < T , | α | 2 , | γ | 2 , where λ < λ 1 + ( 0 , 0 ) C ϵ . Analogous estimates are valid for the derivatives of G (cf. Theorem 3.1). Using this result, we show in Section 3 (see Theorem 3.2) that G ( x , y , t , τ ) admits the decomposition
G ( x , y , t , τ ) = ψ 1 ( x , y , t , τ ) | x | λ 1 + ( x , t ) Φ 1 + ( x , t ; ω ) + R ( x , y , t , τ ) ,
(6)
where
| R ( x , y , t , τ ) | c ( t τ ) n / 2 ( | x | t τ ) μ ( | y | t τ ) λ exp ( κ ( | y | 2 + | x y | 2 ) t τ )
for 0 < t τ < T and | x | < t τ . Here, μ is a certain number greater than sup λ 1 + ( x , t ) . The coefficient ψ 1 in (6) satisfies the estimate
| ψ 1 ( x , y , t , τ ) | c ( t τ ) ( n + λ 1 + ( x , t ) ) / 2 ( | y | t τ ) λ exp ( κ ( | y | 2 + | x y | 2 ) t τ )
for 0 < t τ < T . Moreover, ψ 1 admits the decomposition
ψ 1 ( x , y , t , τ ) = Ψ 1 , 0 ( x , y , t , τ ) + r 1 ( x , y , t , τ ) ,
where Ψ 1 , 0 is the function (37) and
| r 1 ( x , y , t , τ ) | c ( t τ ) ( n 1 + λ 1 + ( x , t ) ) / 2 ( | y | t τ ) λ exp ( κ ( | y | 2 + | x y | 2 ) t τ )

for 0 < t τ < T .

In Section 4, we apply the results of the foregoing section in order to describe the behavior of the solutions of the problem (1) near the edge M. By Theorem 4.2, the following result holds. Suppose that f L p ; β ( D T ) , where
sup λ 1 + ( x , t ) < 2 β m / p < λ 1 + ( 0 , 0 ) + 1 C ϵ , 2 β m / p < inf λ 2 + ( x , t ) ϵ .
Then the solution of the problem (1) admits the decomposition
u ( x , t ) = ( E h 1 ) ( x , t ) | x | λ 1 + ( x , t ) Φ 1 + ( x , t ; ω ) + v ( x , t ) ,
where
h 1 ( x , t ) = 0 t D ψ 1 ( x , y , t , τ ) f ( y , τ ) d y d τ ,

is the extension operator introduced in Section 4.2, and v W p ; β 2 , 1 ( D T ) . Note that the function h 1 belongs to the anisotropic Sobolev-Slobodetskiĭ space W p s , s / 2 ( R n m × ( 0 , T ) ) , where s is the function s ( x , t ) = 2 β λ 1 + ( x , t ) m / p .

Section 5 in closing deals with the initial-boundary value problem in a bounded domain with an edge. Under some smoothness conditions on the coefficients of the differential operator, we obtain the same decomposition of the weak solution near an edge point as in the case of the previously considered domain D (see Theorem 5.1 at the end of the paper).

2 Green function of parabolic equations with constant coefficients

In this section, we assume that
L 0 ( x ) = i , j = 1 n a i , j x i x j = x T A x ,

where a i j are real numbers, a i j = a j i for all i, j. Here A denotes the square matrix with the elements a i , j , x is the nabla operator and x T its transposed, i.e., x T is the row vector with the components x j .

There exists a coordinate transformation ξ = S x with a constant square matrix S such that the problem
t u ( x , t ) L 0 ( x ) u ( x , t ) = f ( x , t ) for  x D , t R , u | D × R + = 0 ,
(7)
takes the form
t u Δ ξ u = f for  ξ D = K R n m , t R , u | D × R = 0
(8)
in the new coordinates ξ, where K is a certain cone in R m with vertex at the origin. This coordinate transformation can be constructed as follows. Let A be the matrix with the elements a i j , i , j = 1 , , m , A the matrix with the elements a i j , i , j = m + 1 , , n , and B the matrix with the elements a i j , i = 1 , , m , j = m + 1 , , n . Furthermore, let x and x be the nabla operators in the coordinates x and x , respectively. Then the operator L 0 can be written as
L 0 ( x ) = x T ( A x + B x ) + x T ( B T x + A x ) .
There exists an invertible matrix U such that U A U T = I m (the m × m identity matrix). This is true for the matrix U = Λ 1 / 2 V , where Λ is the diagonal matrix of the (positive) eigenvalues of the matrix A and the rows of V are the orthonormalized eigenvectors of A . For the coordinates y = U x , y = x B T A 1 x , we have x = U T y A 1 B y , x = y and, consequently,
L 0 ( x ) = Δ y + y T ( A B T A 1 B ) y .
Obviously, the transformation ( x , x ) ( U x , x B T A 1 x ) = ( y , y ) maps K × R n m onto the set D = K × R n m , where K = U K is a cone in R m . Since A B T A 1 B is a symmetric matrix with only positive eigenvalues, there exists an invertible matrix W such that W ( A B T A 1 B ) W T = I n m . For ξ = y and ξ = W y , we get
L 0 ( x ) = Δ ξ + Δ ξ .
Hence, the equation (7) has the form (8) after the coordinate transformation
ξ = U x , ξ = W ( x B T A 1 x ) .
(9)
We denote the Green function of the problem (8) by G ˜ 0 ( ξ , η , t ) . This means that
( t Δ ξ ) G ˜ 0 ( ξ , η , t ) = δ ( ξ η ) δ ( t ) for  ξ , η D , t R , G ˜ 0 ( ξ , η , t ) = 0 for  ξ D , η D , G ˜ 0 ( ξ , η , t ) | t < 0 = 0 .
Then the function
G 0 ( x , y , t ) = | det S | G ˜ 0 ( S x , S y , t )
(10)

is the Green function of the problem (7).

In order to describe the asymptotics of G 0 near the edge, i.e., for small | x | , we introduce the following notation. We denote by { Λ j } the nondecreasing sequence of eigenvalues of the Beltrami operator −δ on the subdomain Ω = K S m 1 of the unit sphere S m 1 with Dirichlet boundary condition, and by { ϕ j } an orthonormalized (in L 2 ( Ω ) ) system of eigenfunction to the eigenvalues Λ j . Furthermore, let
λ j ± = 2 m 2 ± 1 2 ( 2 m ) 2 + 4 Λ j
be the solutions of the quadratic equation λ ( m 2 + λ ) = Λ j . Then the functions
u ˜ j ( ξ ) = | ξ | λ j + ϕ j ( ω ξ ) and v ˜ j ( ξ ) = 1 2 λ j + + m 2 | ξ | λ j ϕ j ( ω ξ )
are special solutions of the equation Δ ξ u = 0 in K . We also introduce the functions
w ˜ j ( η , t ) = 2 Γ ( λ j + + m / 2 ) ( 4 t ) λ j + m / 2 u ˜ j ( η ) exp ( | η | 2 4 t ) ,

which are special solutions of the heat equation ( t Δ η ) w ˜ ( η , t ) = 0 for η K and t > 0 .

Suppose that μ is a real number satisfying the inequalities λ 1 + < μ < λ 1 + + 1 and μ λ j + for all j. By [8], Theorem 2.1], the Green function G ˜ 0 admits the decomposition
G ˜ 0 ( ξ , η , t ) = Φ ( ξ η , t ) g ˜ ( ξ , η , t ) = Φ ( ξ η , t ) ( λ j + < μ w ˜ j ( η , t ) u ˜ j ( ξ ) + r ˜ ( ξ , η , t ) ) ,
(11)
where Φ ( ξ , t ) = ( 4 π t ) ( m n ) / 2 exp ( | ξ | 2 4 t ) is the fundamental solution of the heat equation in R n m and g ˜ ( ξ , η , t ) is the Green function of the Dirichlet problem for the heat equation in K . The remainder r ˜ ( ξ , η , t ) in (11) satisfies the estimate
| t k ξ α η γ r ˜ ( ξ , η , t ) | c t k ( m + | α | + | γ | ) / 2 ( | ξ | t ) μ | α | ( | η | | η | + t ) λ 1 + | γ | ε × ( d ( ξ ) | ξ | ) ε α ( d ( η ) | η | ) ε γ exp ( κ | η | 2 t )
(12)

for | ξ | < c t , | α | 2 , | γ | 2 . Here, ε α = 0 for | α | 1 , while ε α is an arbitrarily small positive real number if | α | = 2 .

Using the decomposition (11), we obtain an analogous decomposition for the Green function G 0 ( x , y , t ) . For this, we introduce the functions
u j ( x ) = u ˜ j ( U x ) , v j ( x ) = | det U | v ˜ j ( U x )
and
ψ j , 0 ( x , y , t ) = | det S | ( 4 π t ) ( m n ) / 2 w ˜ j ( U y , t ) exp ( | W ( x y + B T A 1 y ) | 2 4 t ) = 2 π ( m n ) / 2 ( 4 t ) λ 1 + n / 2 | det A | 1 / 2 Γ ( λ 1 + + m / 2 ) u j ( y ) exp ( q ( y , x y ) 4 t ) ,
(13)
where q ( y , y ) denotes the quadratic form
q ( y , y ) = | U y | 2 + | W ( y + B T A 1 y ) | 2 .
Note that the form q ( y , y ) is independent of the coordinate transformation since U T U = A 1 and W T W = ( A B T A 1 B ) 1 . Since U and W are invertible matrices, there exists a positive constant κ such that
q ( y , y ) κ ( | y | 2 + | y | 2 ) for all  y , y .
(14)
We furthermore note that both u j ( x ) and v j ( x ) are solutions of the equation
L 0 ( x , 0 ) u = i , j = 1 m a i j x i x j u = 0 in  K
which have the form
u j ( x ) = | x | λ j + Φ j + ( ω x ) , v j ( x ) = | x | λ j Φ j ( ω x ) .
This means in particular that λ j ± are eigenvalues and Φ j ± are eigenfunctions of the pencil A ( λ ) which is defined as
A ( λ ) Φ ( ω ) = | x | 2 λ L 0 ( x , 0 ) | x | λ Φ ( ω ) , Φ W 2 1 ( Ω ) .

Theorem 2.1

Suppose that λ 1 + < μ < λ 1 + + 1 and μ λ j + for all j. Then the Green function G 0 ( x , y , t ) admits the decomposition
G 0 ( x , y , t ) = λ j + < μ ψ j , 0 ( x , y , t ) u j ( x ) + R 0 ( x , y , t ) ,
where
| t k x α y γ R 0 ( x , y , t ) | c t k ( n + | α | + | γ | ) / 2 ( | x | t ) μ | α | ( | y | | y | + t ) λ 1 + | γ | ε × ( d ( x ) | x | ) ε α ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t )
(15)

for | x | < t , α = ( α , α ) , γ = ( γ , γ ) , | α | 2 , and | γ | 2 . Here, ε α = 0 for | α | 1 , while ε α is an arbitrarily small positive real number if | α | = 2 .

Proof

By (10) and (11), we have
G 0 ( x , y , t ) = | det S | G ˜ 0 ( S x , S y , t ) = | det S | Φ ( ξ η , t ) g ˜ ( ξ , η , t ) = | det S | Φ ( ξ η + W B T A 1 x , t ) λ j + < μ w ˜ j ( η , t ) u ˜ j ( ξ ) + R 0 ( x , y , t ) ,
(16)
where ξ = U x , η = U y , ξ = W ( x B T A 1 x ) , and η = W ( y B T A 1 y ) , and
R 0 ( x , y , t ) = | det S | ( Φ ( ξ η , t ) Φ ( ξ η + W B T A 1 x , t ) ) g ˜ ( U x , U y , t ) + | det S | Φ ( ξ η + W B T A 1 x , t ) r ˜ ( ξ , η , t ) .
The right-hand side of (16) is equal to
λ j + < μ ψ j , 0 ( x , y , t ) u j ( x ) + R 0 ( x , y , t ) .

Using (12), one can easily check that R 0 satisfies (15). □

We derive another formula for the coefficient ψ j , 0 ( x , y , t ) in Theorem 2.1. If t > 0 , then
Δ ξ g ˜ ( ξ , η , t ) = t g ˜ ( ξ , η , t )
for ξ , η K . Let V p ; β l ( K ) denote the weighted Sobolev space (closure of C 0 ( K ¯ { 0 } ) ) with the norm
u V p ; β l ( K ) = ( K | α | l | x | p ( β l + | α | ) | x α u ( x ) | p d x ) 1 / p .
It follows from (11) and (12) that t g ˜ ( , η , t ) V p ; β 0 ( K ) for arbitrary p and β such that p ( β + λ 1 + ) > m . Hence, the coefficient w ˜ j ( η , t ) in (11) is given by the formula
w ˜ j ( η , t ) = K v ˜ j ( ξ ) Δ ξ g ˜ ( ξ , η , t ) d ξ
(17)
(cf.[14], Theorem 5.1]). Let
U ( ξ , η , t ) = g ˜ ( ξ , η , t ) ( Φ ( ξ η , t ) Φ ( ξ η W B T U T ξ , t ) ) .
In the integral
w ˜ j ( η , t ) = K v ˜ j ( ξ ) Δ ξ U ( ξ , η , t ) d ξ
(18)
one can integrate by parts for t > 0 . Since Δ ξ v j ( ξ ) = 0 and g ˜ ( ξ , η , t ) = v j ( ξ ) = 0 for ξ K , we conclude that the integral (18) vanishes. Hence, it follows from (17) that
w ˜ j ( η , t ) Φ ( ξ η , t ) = K v ˜ j ( ξ ) Δ ξ g ˜ ( ξ , η , t ) Φ ( ξ η W B T U T ξ , t ) d ξ .
We set ξ = W x , η = U y , η = W ( y B T A 1 y ) , and we substitute ξ = U x in the integral on the right-hand side. Then we obtain
w ˜ j ( U y , t ) Φ ( W ( x y + B T A 1 y ) , t ) = K v j ( x ) L 0 ( x , 0 ) G ˜ 0 ( S x , S y , t ) d x .
Multiplying the last equality by | det S | , we arrive at the formula
ψ j , 0 ( x , y , t ) = K v j ( x ) L 0 ( x , 0 ) G ( x , y , t ) d x .
(19)

3 Green function of parabolic equations with variable coefficients

Now let L ( x , t , x ) be the operator (2) with x- and t-depending coefficients satisfying the condition (3). We consider the Green function G ( x , y , t , τ ) for the operator
L = t L ( x , t , x )
in D = K × R n m , i.e., the solution of the problem
( t L ( x , t , x ) ) G ( x , y , t , τ ) = δ ( x y ) δ ( t τ ) for  x , y D , t , τ R , G ( x , y , t , τ ) = 0 for  x D , y D , t R , G ( x , y , t , τ ) | t < τ = 0 .
(20)
In this section, we will employ an estimate for the Green function which was proved in [13]. For this end, we assume in the following that the coefficients of L satisfy some additional smoothness conditions. To be more precise, we suppose that
x γ a i j C σ , σ / 2 ( D × R ) for  | γ | 2 and a 0 , a j , x i a j C σ , σ / 2 ( D × R )
(21)
with some σ ( 0 , 1 ) for i , j = 1 , , n and that
i , j = 1 n | γ | 2 | x γ x α t k a i j | + j = 1 n | γ | 1 | x γ x α t k a j | + | x α t k a 0 | C for  | α | 4 , k 2 .
(22)

3.1 Estimates of Green function

Let
L 0 ( 0 , x , t , x ) = i , j = 1 n a i j ( 0 , x , t ) x i x j and L 0 ( 0 , x , t , x , 0 ) = i , j = 1 m a i j ( 0 , x , t ) x i x j .
Furthermore, let the pencil A ( x , t ; λ ) be defined by (5), and let λ j ± ( x , t ) be its eigenvalues, where
λ 2 < λ 1 < 2 m 0 < λ 1 + < λ 2 + .

The following estimate for the Green function G ( x , y , t , τ ) is proved in [13], Theorem 4.4].

Theorem 3.1

Suppose that the coefficients of L satisfy the conditions (3), (21), and (22). If T is a positive number and λ < λ 1 + ( 0 , 0 ) C ϵ , then G ( x , y , t , τ ) satisfies the estimate
| t k τ l x α y γ G ( x , y , t , τ ) | c ( t τ ) ( n + 2 k + 2 l + | α | + | γ | ) / 2 ( | x | | x | + t τ ) λ | α | ( | y | | y | + t τ ) λ | γ | × ( d ( x ) | x | ) ε α ( d ( y ) | y | ) ε γ exp ( κ | x y | 2 t τ )
(23)

for 0 < t τ < T , | α | 2 , | γ | 2 , | α | 4 , | γ | 4 , k , l 2 . Here, ε α denotes the same nonnegative number as in Theorem  2.1.

3.2 Asymptotics of Green function

Analogously to the matrix U in Section 2, let U ( x , t ) be a matrix such that
U ( x , t ) A ( x , t ) U T ( x , t ) = I m ,
where A ( x , t ) is the matrix with the elements a i j ( 0 , x , t ) , i , j = 1 , , m . Under our assumptions on the coefficients, we may assume that the elements of U are C 2 -functions. Then the numbers Λ j ( x , t ) = λ j + ( x , t ) λ j ( x , t ) are eigenvalues of the Beltrami operator −δ (with Dirichlet boundary conditions) on the subdomain Ω ( x , t ) = K ( x , t ) S m 1 of the unit sphere, where K ( x , t ) = U ( x , t ) K . As in Section 2, we denote by { ϕ j ( x , t ; ω ) } an orthonormalized system of eigenfunctions corresponding to the eigenvalues Λ j ( x , t ) . Moreover, we set
u ˜ j ( x , t ; ξ ) = | ξ | λ j + ( x , t ) ϕ j ( x , t ; ω ξ ) and v ˜ j ( x , t ; ξ ) = 1 2 λ j + + m 2 | ξ | λ j ( x , t ) ϕ j ( x , t ; ω ξ ) .
Then the functions
u j ( x , t ; x ) = u ˜ j ( x , t ; U x ) , v j ( x ) = | det U | v ˜ j ( x , t ; U x )
(24)
are special solutions of the equation L 0 ( 0 , x , t , x , 0 ) u ( x ) = 0 for x K . By (20), we have
L 0 ( 0 , x , t , x , 0 ) G ( x , y , t , τ ) = t G + ( L 0 ( 0 , x , t , x , 0 ) L ( x , t , x ) ) G ( x , y , t , τ )
(25)
for x D , t > τ . Suppose that p and β are such that
λ 1 + ( 0 , 0 ) < 2 β m / p < λ 1 + ( 0 , 0 ) + 1 C ϵ , 2 β m / p λ j + ( x , t ) for all  j , x , t ,
where C is the same constant as in Theorem 3.1. Then by Theorem 3.1, the right-hand side of (25) belongs to the space V p ; β 0 ( K ) for arbitrary fixed x R n m , y D , t > τ . Applying [15], Theorem 4.2], we obtain
G ( x , x , y , t , τ ) = λ j + < 2 β m / p ψ j ( x , y , t , τ ) u j ( x , t ; x ) + R ( x , y , t , τ ) ,
where R ( , x , y , t , τ ) V p ; β 2 ( K ) . The coefficients ψ j ( x , y , t , τ ) satisfy the equality (cf. (19))
ψ j ( x , y , t , τ ) = K v j ( x , t ; x ) L 0 ( 0 , x , t , x , 0 ) G ( x , y , t , τ ) d x .
(26)

In the following lemma, we give an estimate of these functions.

Lemma 3.1

Suppose that sup λ j + ( x , t ) < λ 1 + ( 0 ) + 1 C ϵ , where C is the same constant as in Theorem  3.1. Then the function (26) satisfies the estimate
| t k τ l x α y γ ψ j ( x , y , t , τ ) | c ( t τ ) k l ( n + | α | + | γ | + λ j + ( x , t ) ) / 2 ( | y | t τ ) λ | γ | ( d ( y ) | y | ) ε γ × exp ( κ ( | y | 2 + | x y | 2 ) t τ )
(27)

for 0 < t τ < T , k 1 , l 1 , and | α | , | γ | , | γ | 2 . Here, κ is a certain positive number, λ is an arbitrary number less than λ 1 + ( 0 ) C ϵ , and ε γ is the same nonnegative number as in Theorem  2.1.

Proof

We define K t = { x K : | x | < t } for t > 0 . Then
t k τ l x α y γ ψ j ( x , y , t , τ ) = K t τ t k τ l x α y γ v j ( x , t ; x ) L 0 ( 0 , x , t , x , 0 ) G ( x , y , t , τ ) d x + K K t τ t k τ l x α y γ v j ( x , t ; x ) L 0 ( 0 , x , t , x , 0 ) G ( x , y , t , τ ) d x .
We estimate the first integral on the right-hand side of the last equality using the decomposition (25). Theorem 3.1 yields
| t ν τ l x σ y γ ( L 0 ( 0 , x , t , x , 0 ) L ( x , t , x ) ) G ( x , y , t , τ ) | c ( t τ ) ν l ( n + | γ | + | σ | + λ + 1 ) / 2 | x | λ 1 ( d ( x ) | x | ) ε ( | y | | y | + t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) 2 ( t τ ) )
for | x | 2 < t τ < T , σ α , and ν k , where ε is an arbitrarily small positive number. Furthermore,
| t k ν x α σ v j ( x , t ; x ) | c | x | λ j ( x , t ) ( 1 + | log | x | | ) k + | α | ν | σ | .
The number λ can be chosen such that sup λ j + ( x , t ) < λ + 1 . Consequently,
| K t τ t k τ l x α v j ( x , t ; x ) y γ ( L 0 ( 0 , x , t , x , 0 ) L ( x , t , x ) ) G ( x , y , t , τ ) d x | c ( t τ ) k l ( n + | α | + | γ | + λ + 1 ) / 2 ( | y | | y | + t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) 2 ( t τ ) ) × K t τ | x | λ λ j + ( x , t ) + 1 m ( 1 + | log | x | | ) l + | α | ( d ( x ) | x | ) ε d x c ( t τ ) k l ( n + | α | + | γ | + λ j + ( x , t ) ) / 2 ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) 4 ( t τ ) ) .
The same estimate holds for
| K t τ t k τ l x α v j ( x , t ; x ) t y γ G ( x , y , t , τ ) d x | .
Thus, we obtain
| K t τ t k x α v j ( x , t ; x ) L 0 ( 0 , x , t , x , 0 ) τ l y γ G ( x , y , t , τ ) d x | c ( t τ ) k l ( n + | α | + | γ | + λ j + ( x , t ) ) / 2 ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) 4 ( t τ ) ) .
Since L 0 ( 0 , x , t , x , 0 ) v j ( x , t ; x ) = 0 for x K and t k τ l x α y γ G ( x , y , t , τ ) is exponentially decaying for large | x | , we get
K K t τ t k x α v j ( x , t ; x ) L 0 ( 0 , x , t , x , 0 ) τ l y γ G ( x , y , t , τ ) d x = S t τ t k x α i , j = 1 n a i , j ( 0 , x , t ) × ( v j ( x , t ; x ) x j τ l y γ G ( x , y , t , τ ) τ l y γ G ( x , y , t , τ ) x j v j ( x , t ; x ) ) cos ( n , x j ) d σ ,
where S t τ is the intersection of K with the sphere | x | = t τ and n is the normal vector to this sphere. By Theorem 3.1, the integrand on the right-hand side of the last equality has the upper bound
c ( t τ ) k l ( n + | α | + | γ | + 1 λ j ( t ) ) / 2 ( | y | | y | + t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ ) .
Therefore,
| K K t τ t k x α v j ( x , t ; x ) L 0 ( 0 , x , t , x ) τ l y γ G ( x , y , t , τ ) d x | c ( t τ ) k l ( n + | α | + | γ | + λ j + ( x , t ) ) / 2 ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) 2 ( t τ ) ) .

This proves the lemma. □

For the estimation of the remainder R ( x , y , t , τ ) , we need the following lemma.

Lemma 3.2

Suppose that u V p ; β 2 ( K ) is a solution of the problem L 0 ( 0 , x , t , x , 0 ) u = f in K, u = 0 on ∂K, where λ j + ( x , t ) + ε < 2 β m / p < λ j + 1 + ( x , t ) ε for a certain integer j 1 . Then
u V p ; β 2 ( K ) c ( x , t ) ε f V p ; β 0 ( K )
(28)

with a constant c ( x , t ) independent of ε.

Proof

First note that the eigenvalues λ j + ( x , t ) and λ j + 1 + ( x , t ) of the pencil A ( x , t ; λ ) have no generalized eigenfunctions (see, e.g., [16], Section 2.2]). Let g ( x , y ) be the Green function of the Dirichlet problem for the operator L 0 ( 0 , x , t , x , 0 ) in the cone K, ζ g ( , y ) V p ; β 2 ( K ) for smooth ζ vanishing in a neighborhood of y . Then
u ( x ) = K g ( x , y ) f ( y ) d y .
By [17], Theorem 2.2], the function g satisfies the estimates
| g ( x , y ) | c | x | λ j + 1 + ( x , t ) | y | 2 n λ j + 1 + ( t ) for  2 | x | < | y | , | g ( x , y ) | c | x | λ j + ( x , t ) | y | 2 n λ j + ( t ) for  | x | > 2 | y | .
Moreover, in the case | y | < 2 | x | < 4 | y | , the estimates | g ( x , y ) | c | x y | 2 m for m > 2 and | g ( x , y ) | c | log | x y | | for m = 2 are valid. For arbitrary integer ν, let χ ν ( x ) = 1 for 2 ν 1 | x | 2 ν , χ ν ( x ) = 0 else. Furthermore, let
u ν ( x ) = K g ( x , y ) χ ν ( y ) f ( y ) d y .
Then it follows from the above estimates for g ( x , y ) and from [18], Lemmas 3.5.1 and 3.5.4] that
χ μ u ν V p ; β 2 0 ( K ) c 2 ( μ ν ) ( λ j + ( x , t ) 2 + β + m / p ) χ ν f V p ; β 0 ( K ) if  μ ν , χ μ u ν V p ; β 2 0 ( K ) c 2 ( μ ν ) ( λ j + 1 + ( x , t ) 2 + β + m / p ) χ ν f V p ; β 0 ( K ) if  μ < ν ,
where c is independent of μ, ν, and f (cf.[18], Lemma 3.5.6]). Consequently,
u V p ; β 2 0 ( K ) p = μ χ μ u V p ; β 2 0 ( K ) p μ ( ν χ μ u ν V p ; β 2 0 ( K ) ) p c μ ( ν 2 ε | μ ν | χ ν f V p ; β 0 ( K ) ) p .
By Hölder’s inequality,
( ν 2 ε | μ ν | χ ν f V p ; β 0 ( K ) ) p ( ν 2 ε | μ ν | χ ν f V p ; β 0 ( K ) p ) ( ν 2 ε | μ ν | ) p 1 = ( 2 ε + 1 2 ε 1 ) p 1 ν 2 ε | μ ν | χ ν f V p ; β 0 ( K ) p .
Thus, we obtain
u V p ; β 2 0 ( K ) p c ( 2 ε + 1 2 ε 1 ) p 1 μ ν 2 ε | μ ν | χ ν f V p ; β 0 ( K ) p = c ( 2 ε + 1 2 ε 1 ) p ν χ ν f V p ; β 0 ( K ) p = c ( 2 ε + 1 2 ε 1 ) p f V p ; β 0 ( K ) p .
The last inequality together with the estimate
u V p ; β 2 ( K ) c ( f V p ; β 0 ( K ) + u V p ; β 2 0 ( K ) )

(see, e.g., [18], Theorem 3.3.5]) implies (28). □

Now we are able to prove the main result of this section.

Theorem 3.2

Suppose that λ < λ 1 + ( 0 ) C ϵ and
λ 1 + ( 0 ) < μ < λ 1 + ( 0 ) + 1 C ϵ , λ j + ( x , t ) [ μ ϵ , μ + ϵ ] for all  j , x , t ,
(29)
where C is the same constant as in Theorem  3.1. Then the Green function G ( x , y , t , τ ) admits the decomposition
G ( x , y , t , τ ) = λ j + < μ ψ j ( x , y , t , τ ) u j ( x , t ; x ) + R ( x , y , t , τ ) ,
(30)
where u j is defined by (24), and
| t k τ l x α y γ R ( x , y , t , τ ) | c ( t τ ) k l ( n + | α | + | γ | ) / 2 ( | x | t τ ) μ | α | ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ )
(31)

for 0 < t τ < T , | x | < t τ , | α | 1 , | α | , | γ | , | γ | 2 , k , l 1 . Here, ε γ is the same constant as in Theorem  2.1. The coefficients ψ j ( x , y , t , τ ) satisfy the estimate (27).

Proof

Let ζ be a smooth function on the interval ( 0 , ) , ζ ( r ) = 1 for r < 1 and ζ ( r ) = 0 for r > 2 . Furthermore, let χ ( x , t ) = ζ ( t 1 / 2 | x | ) for x = ( x , x ) D and t > 0 . It follows from the equality
L 0 ( 0 , x , t , x , 0 ) R ( x , y , t , τ ) = L 0 ( 0 , x , t , x , 0 ) G ( x , y , t , τ ) = ( L 0 ( 0 , x , t , x , 0 ) L ( x , t , x ) ) G ( x , y , t , τ ) + t G ( x , y , t , τ )
that
L 0 ( 0 , x , t , x , 0 ) χ ( x , t τ ) τ l y γ R ( x , y , t , τ ) = f ( x , y , t , τ )
for t > τ , where
f = χ ( x , t τ ) τ l y γ ( ( L 0 ( 0 , x , t , x , 0 ) L ( x , t , x ) ) G ( x , y , t , τ ) + t G ( x , y , t , τ ) ) + [ L 0 ( 0 , x , t , x , 0 ) , χ ( x , t τ ) ] τ l y γ ( G ( x , y , t , τ ) ψ j ( x , y , t , τ ) u j ( x , t ; x ) ) .
Here, [ L 0 , χ ] = L 0 χ χ L 0 denotes the commutator of L 0 and χ. Furthermore, τ l y γ R ( x , y , t , τ ) = 0 for x K . We estimate the V p ; β 2 ( K ) -norm of the function χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) for 2 β m / p = μ . By [15], Theorem 4.1],
χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K ) c f ( , x , y , t , τ ) V p ; β 0 ( K ) .
(32)
Here, the constant c is independent of x , y, t, τ. Indeed, by Lemma 3.2, we have
χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K ) c ϵ L 0 ( 0 , 0 , x , 0 ) χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) V p ; β 0 ( K ) ,
with a constant c independent of x , y, t, τ. Furthermore, under the condition (3), the inequality
( L 0 ( 0 , x , t , x , 0 ) L 0 ( 0 , 0 , x , 0 ) ) χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) V p ; β 0 ( K ) c ϵ χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K )
holds. Thus,
χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K ) c 1 ϵ χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K ) + c 2 ϵ f ( , x , y , t , τ ) V p ; β 0 ( K ) ,
which implies (32) if ϵ is sufficiently small. Next, we estimate the V p ; β 0 -norm of f ( , x , y , t ) . By Theorem 3.1,
| χ ( x , t τ ) τ l y γ ( L 0 ( 0 , x , t , x , 0 ) L ( x , t , x ) ) G ( x , y , t , τ ) | c ( t τ ) l ( n + | γ | + λ + 1 ) / 2 | x | λ 1 ( d ( x ) | x | ) ε ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ )
for 0 < t τ < T , where λ < λ 1 + ( 0 ) C ϵ . Here, λ can be chosen such that p ( β + λ 1 ) > m . Therefore,
χ ( , t τ ) τ l y γ ( L 0 ( 0 , x , t , x , 0 ) L ( , x , t , x ) ) G ( , x , y , t , τ ) V p ; β 0 ( K ) c ( t τ ) l + ( β n | γ | 2 + m / p ) / 2 ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ ) .
Analogously, we obtain
χ ( , t τ ) t τ l y γ G ( , x , y , t ) V p ; β 0 ( K ) c ( t τ ) l + ( β n | γ | 2 + m / p ) / 2 ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ ) .
Since [ L 0 ( 0 , x , t , x ) , χ ( x , t τ ) ] G ( x , y , t , τ ) vanishes for | x | < t τ and | x | > 2 t τ , we obtain the estimate
[ L 0 ( 0 , x , t , x , 0 ) , χ ( x , t τ ) ] y γ G ( , x , y , t , τ ) V p ; β 0 ( K ) c ( t τ ) l + ( β n | γ | 2 + m / p ) / 2 ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ )
by means of Theorem 3.1. Using Lemma 3.1, we get the same estimate for the V p ; β 0 ( K ) -norm of the functions [ L 0 ( 0 , x , t , x , 0 ) , χ ( , t τ ) ] τ l y γ ψ j ( x , y , t , τ ) u j ( x , t ; ) . Consequently, (32) implies
χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K ) c ( t τ ) l + ( β n | γ | 2 + m / p ) / 2 ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ )
(33)
for 0 < t τ < T . We prove an analogous estimate for the x - and t-derivatives of τ l y γ R . Obviously,
L 0 ( 0 , x , t , x , 0 ) ( χ ( x , t τ ) x j τ l y γ R ( x , y , t , τ ) ) = x j f ( x , y , t , τ ) ( x j L 0 ( 0 , x , t , x , 0 ) ) ( χ ( x , t τ ) τ l y γ R ( x , y , t , τ ) )
for j m + 1 , where f is the same function as above. Since, moreover, x j τ l y γ R ( x , y , t , τ ) = 0 for x K and j m + 1 , we get
χ ( , t τ ) x j τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K ) c ( x j f V p ; β 0 ( K ) + χ ( , t τ ) τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K ) )
for j m + 1 . The V p ; β 0 ( K ) -norms of x j f can be estimated in the same way as f. This together with (33) leads to the estimate
χ ( , t τ ) x j τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K ) c ( t τ ) l + ( β n | γ | 3 + m / p ) / 2 ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ )
for j m + 1 . Analogously, the inequality
χ ( , t τ ) t k x α τ l y γ R ( , x , y , t , τ ) V p ; β 2 ( K ) c ( t τ ) k l + ( β n | α | | γ | 2 + m / p ) / 2 ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ )
holds for | α | 2 and k 1 . Applying the estimate
| α | 1 | x | β 2 + | α | + m / p | x α v ( x , y , t , τ ) | c v ( , x , y , t , τ ) V p ; β 2 ( K )
for v ( x , y , t , τ ) = χ ( x , t τ ) t k x α τ l y γ R ( x , y , t , τ ) , p > m (cf.[18], Lemma 1.2.3]), we get
| t k τ l x α y γ R ( x , y , t , τ ) | c ( t τ ) k l ( n + | α | + | γ | ) / 2 ( | x | t τ ) μ | α | ( | y | t τ ) λ | γ | × ( d ( y ) | y | ) ε γ exp ( κ ( | y | 2 + | x y | 2 ) t τ )

for | x | < t τ , 0 < t τ < T , | α | 1 , | α | , | γ | , | γ | 2 . This proves (31). □

Comparing the representation (30) with the estimate (23), we conclude that
inf x , t λ 1 + ( x , t ) λ 1 + ( 0 , 0 ) C ϵ ,
(34)

where C is the same constant as in Theorem 3.1.

3.3 Asymptotics of the coefficients ψ j ( x , y , t , τ )

Let G 0 ( x , t ; z , y , s ) be the Green function of the first boundary value problem for the operator
s L 0 ( 0 , x , t , z ) = s i , j = 1 n a i , j ( 0 , x , t ) 2 z i z j
with constant coefficients a i , j ( 0 , x , t ) depending on the parameters x and t. This means that
( s L 0 ( 0 , x , t , z ) ) G 0 ( x , t ; z , y , s ) = δ ( z y ) δ ( s ) for  z , y D , s R , G 0 ( x , t ; z , y , s ) = 0 for  z D , y D , s R , G 0 ( x , t ; z , y , s ) | s < 0 = 0 .
We write the operator L 0 ( 0 , x , t , z ) in the form
L 0 ( 0 , x , t , z ) = z T ( A ( x , t ) z + B ( x , t ) z ) + z T ( B T ( x , t ) z + A ( x , t ) z ) ,
where z and z denote the nabla operators in the z - and z -variables, respectively. As in Section 2, let U = U ( x , t ) and W = W ( x , t ) be square and continuously differentiable (with respect to x and t) matrices such that U A U T = I m and W ( A B T A 1 B ) W T = I n m . By Theorem 2.1, the function G 0 admits the decomposition
G 0 ( x , t ; z , y , s ) = λ j + ( x , t ) < μ ψ j , 0 ( x , t ; z , y , s ) u j ( x , t ; z ) + R 0 ( x , t ; z , y , s )
if λ 1 + ( x , t ) < μ < λ 1 + ( x , t ) + 1 and μ λ j + ( x , t ) for all j. Here,
ψ j , 0 ( x , t ; z , y , s ) = K v j ( x , t ; x ) L 0 ( 0 , x , t , x , 0 ) G 0 ( x , t ; x , z , y , s ) d x
(35)
(cf. (19)), the functions u j ( x , t ; ) and v j ( x , t ; ) are defined by (24), and R 0 satisfies the estimate in Theorem 2.1. A more explicit formula for the function ψ j , 0 is
ψ j , 0 ( x , t ; z , y , s ) = 2 π ( m n ) / 2 ( 4 s ) λ j + ( x , t ) n / 2 | det A | 1 / 2 Γ ( λ j + + m / 2 ) u j ( x , t ; y ) × exp ( q ( x , t ; y , z y ) 4 s )
(36)
(cf. (13)), where A ( x , t ) is the coefficients matrix of the operator L 0 ( 0 , x , t , z ) , and
q ( x , t ; y , y ) = | U y | 2 + | W ( y + B T A 1 y ) | 2
is a quadratic form with respect to y and y satisfying the inequality (14). We define Ψ j , 0 ( x , y , t , τ ) = ψ j , 0 ( x , t ; x , y , t τ ) , i.e.,
Ψ j , 0 ( x , y , t , τ ) = 2 π ( m n ) / 2 ( 4 t 4 τ ) λ j + ( x , t ) n / 2 | det A | 1 / 2 Γ ( λ j + + m / 2 ) u j ( x , t ; y ) × exp ( q ( x , t ; y , x y ) 4 ( t τ ) )
(37)

for x R n m , y D , τ < t .

Theorem 3.3

The coefficients ψ j ( x , y , t , τ ) in Theorem  3.2admit the decomposition
ψ j ( x , y , t , τ ) = Ψ j , 0 ( x , y , t , τ ) + r j ( x , y , t , τ ) ,
where r j satisfies the estimate
| r j ( x , y , t , τ ) | c ( t τ ) ( n 1 + λ j + ( x , t ) ) / 2 ( | y | t τ ) λ exp ( κ ( | y | 2 + | x y | 2 ) t τ )

for 0 < t τ < T , λ < λ 1 + ( 0 ) C ϵ .

Proof

For shortness, we write λ j + instead of λ j + ( x , t ) in the proof of this theorem. Since ( s L 0 ( 0 , x , t , y ) ) G 0 ( x , t ; x , y , s ) = 0 for x , y D , s > 0 , we have
( s L 0 ( 0 , x , t , y ) ) ψ j , 0 ( x , t ; z , y , s ) = 0
for y D , s > 0 , x , z R n m , t R . This means that the function Ψ j , 0 satisfies the equation
( τ L 0 ( 0 , x , t , y ) ) Ψ j , 0 ( x , y , t , τ ) = 0 for  y D , τ < t .
On the other hand, it follows from (26) that
( τ L ( y , τ , y ) ) ψ j ( x , y , t , τ ) = 0 for  y D , τ < t .
Here L denotes the formally adjoint differential operator to L. Consequently,
( τ L ( y , τ , y ) ) r j ( x , y , t , τ ) = ( L ( y , τ , y ) L 0 ( 0 , x , t , y ) ) Ψ j , 0 ( x , y , t , τ )
for y D and τ < t . Furthermore, r j ( x , y , t , t ) = 0 for x R n m , y D . This follows from the representation
r j ( x , y , t , τ ) = K v j ( x , t ; x ) L 0 ( 0 , x , t , x , 0 ) ( G ( x , y , t , τ ) G 0 ( x , t ; x , y , t τ ) ) d x
of the function r j = ψ j Ψ j , 0 (cf. (26) and (35)) and from the equality G ( x , y , t , t ) = G 0 ( x , t ; x , y , 0 ) = δ ( x y ) . Thus,
r j ( x , y , t , τ ) = τ t D G ( z , y , s , τ ) ( L ( z , s , z ) L 0 ( 0 , x , t , z ) ) Ψ j , 0 ( x , z , t , s ) d z d s .
Since Ψ j , 0 has the form (37), we get
| ( L ( z , s , z ) L 0 ( 0 , x , t , z ) ) Ψ j , 0 ( x , z , t , s ) | c ( | α | 1 z α Ψ j , 0 ( x , z , t , s ) + ( | z | + | z x | + t s ) | α | = 2 z α Ψ j , 0 ( x , z , t , s ) ) c ( t s ) λ j + + ( 1 n