Open Access Research

Asymptotics of solutions of the heat equation in cones and dihedra under minimal assumptions on the boundary

Vladimir A Kozlov1 and Jürgen Rossmann2*

Author Affiliations

1 Institute of Mathematics, Linköping University, Linköping, SE-58183, Sweden

2 Institute of Mathematics, University of Rostock, Rostock, D-18051, Germany

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Boundary Value Problems 2012, 2012:142  doi:10.1186/1687-2770-2012-142


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/142


Received:25 June 2012
Accepted:14 November 2012
Published:3 December 2012

© 2012 Kozlov and Rossmann; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the first part of the paper, the authors obtain the asymptotics of Green’s function of the first boundary value problem for the heat equation in an m-dimensional cone K. The second part deals with the first boundary value problem for the heat equation in the domain K × R n m . Here the right-hand side f of the heat equation is assumed to be an element of a weighted L p , q -space. The authors describe the behavior of the solution near the ( n m ) -dimensional edge of the domain.

Introduction

The paper is concerned with the first boundary value problem for the heat equation

(1)

(2)

in the domain

D = { x = ( x , x ) : x K , x R n m } ,

where K = { x = ( x 1 , , x m ) : x / | x | Ω } is a cone in R m , 2 m n , Ω denotes a subdomain of the unit sphere, and M = { x = ( x , x ) : x = 0 } is the ( n m ) -dimensional edge of  D . We are interested in the asymptotics of solutions in the class of the weighted Sobolev spaces W p , q ; β 2 , 1 ( D × R ) . Here the space W p , q ; β 2 l , l ( D × R ) is defined for an arbitrary integer l 0 and real p > 1 , q > 1 , β as the set of all function u ( x , t ) on D × R with the finite norm

u W p , q ; β 2 l , l ( D × R ) = ( R ( D | α | + 2 k 2 l | x | p ( β 2 l + 2 k + | α | ) | t k x α u ( x , t ) | p d x ) q / p d t ) 1 / q . (3)

In the case l = 0 , we write W p , q ; β 0 , 0 = L p , q ; β . If, moreover, β = 0 , then we write L p , q ; 0 = L p , q .

For the case of smooth boundary Ω (of class C ), the asymptotics of solutions was obtained in our previous paper [1]. For the particular case p = q = 2 , m = n , we refer also to the paper [2] by Kozlov and Maz’ya, and for the case p = q 2 , m = n = 2 , to the paper [3] by de Coster and Nicaise. The goal of the present paper is to describe the asymptotics of solutions with a remainder in W p , q ; β 2 , 1 ( D × R ) under minimal smoothness assumptions on the boundary. Throughout the paper, we assume that Ω C 1 , 1 .

The paper consists of two parts. The first part (Section 1) deals with the asymptotics of the Green function for the heat equation in the cone K. We obtain the same decomposition

G ( x , y , t ) = λ j + < σ k = 0 m j t k c j ( y , t ) | x | λ j + + 2 k ϕ j ( ω x ) 4 k k ! ( σ j + k ) ( k ) + R σ ( x , y , t )

as in [4,5] (for the definition of λ j + , ϕ j , m j , c j and σ ( k ) , see Section 1.1). However, the proof in [4,5] does not work if Ω is only of the class C 1 , 1 . We give a new proof, which is completely different from that in [4,5]. Our tools are estimates for solutions of the Dirichlet problem for the Laplace equation in a cone in weighted L p Sobolev spaces and asymptotic formulas for solutions of this problem which were obtained in the papers [6,7] by Maz’ya and Plamenevskiĭ. Moreover, we use the estimates of the Green function in the recent paper [8] by Kozlov and Nazarov. In contrast to the case Ω C , the estimates for the second order x - and y -derivatives of the remainder R σ contain an additional factor ( | x | 1 d ( x ) ) ε with a negative exponent −ε. Here, d ( x ) is the distance from the boundary of ∂K.

In the second part of the paper (Section 2), we apply the results of Section 2 in order to obtain the asymptotics of solutions of the problem (1), (2) for f L p , q ; β ( D × R ) . We show that, under a certain condition on β, there exists a solution of the form

u ( x , t ) = λ j + < 2 β m / p k = 0 m j ( t Δ x ) k H j ( x , t ) 4 k k ! ( σ j + k ) ( k ) | x | λ j + + 2 k ϕ j ( ω x ) + w ( x , t )

with a remainder w W p , q ; β 2 , 1 ( D × R ) . Here, H j is an extension of the function

h j ( x , t ) = t D c j ( y , t τ ) Φ ( x , y , t τ ) f ( y , τ ) d y d τ ,

Φ denotes the fundamental solution of the heat equation in R n m . The proof of this result (Theorem 2.2) is essentially the same as in [1]. However, the proofs of some lemmas in [1] have to be modified under our weaker assumptions on Ω.

At the end of the paper, we show that the extensions of the functions h j can be defined as

H j ( x , t ) = ( E h j ) ( x , t ) = 0 R n m T ( τ ) R ( z ) h j ( x r z , t r 2 τ ) d z d τ ,

where T and R are certain smooth functions on R + and R n m , respectively (see the beginning of Section 3 for their definition). This extends the result of [[1], Corollary 4.5] to the case p q .

1 The Green function of the heat equation in a cone

We start with the problem

(4)

(5)

Let G ( x , y , t ) be the Green function for the problem (4), (5). It is defined for every y K as the solution of the problem

Furthermore, ( 1 ζ ) G ( , y , ) W 2 ; β 2 , 1 ( K × R ) if λ 1 < 2 β m / 2 < λ 1 + ( λ 1 ± are defined below), and ζ is a function in C 0 ( K × R ) equal to one in a neighborhood of the point ( x , t ) = ( y , 0 ) . Here W 2 , β 2 , 1 ( K × R ) is the space of all functions u = u ( x , t ) on K × R such that | x | β 2 + 2 k + | α | t k x α u L 2 ( K × R ) for 2 k + | α | 2 . The goal of this section is to describe the behavior of the Green function for | x | < t .

1.1 Asymptotics of Green’s function

Let { Λ j } j = 1 be the nondecreasing sequence of eigenvalues of the Beltrami operator −δ on Ω (with the Dirichlet boundary condition) counted with their multiplicities, and let { ϕ j } j = 1 be an orthonormal (in L 2 ( Ω ) ) sequence of eigenfunctions corresponding to the eigenvalues  Λ j . Furthermore, we define

λ j ± = 2 m 2 ± ( 1 m / 2 ) 2 + Λ j and σ j = λ j + 1 + m 2 .

This means that λ j ± are the solutions of the quadratic equation λ ( m 2 + λ ) = Λ j . Obviously, λ j + > 0 and λ j < 2 m for j = 1 , 2 ,  .

By [[8], Theorem 3],

| t k x α y γ G ( x , y , t ) | c t k ( m + | α | + | γ | ) / 2 ( | x | | x | + t ) λ 1 + | α | ε ( | y | | y | + t ) λ 1 + | γ | ε × ( d ( x ) | x | ) ε α ( d ( y ) | y | ) ε γ exp ( κ | x y | 2 t ) (6)

for | α | 2 , | γ | 2 . Here d ( x ) denotes the distance of the point x from the boundary ∂K. Furthermore, ε α is defined as zero for | α | 1 , while ε α is an arbitrarily small positive real number if | α | = 2 . Actually, the estimate (6) is proved in [8] only for k = 0 , but for a more general class of operators, parabolic operators with discontinuous in time coefficients. If the coefficients in [8] do not depend on t, then one can use the same argument as in the proof of [[8], Theorem 3] when treating the derivatives along the edge of the domain D = K × R n m . This argument shows that the kth derivative with respect to t will bring only an additional factor t k to the right-hand side of (6).

The following lemma will be applied in the proof of Lemma 1.2. Here and in the sequel, we use the notation r = | x | and ω x = x / | x | .

Lemma 1.1Let G ( x , y , t ) be the Green function introduced above, and let G j ( r , ρ , t ) denote the Green function of the initial-boundary value problem

Then

Ω G ( x , y , t ) ϕ j ( ω x ) d ω x = | y | 1 m G j ( | x | , | y | , t ) ϕ j ( ω y ) . (7)

Proof The solution of the problem

(8)

(9)

is given by the formula

u ( x , t ) = K G ( x , y , t ) ϕ ( y ) d y .

We define

U j ( r , t ) = Ω u ( x , t ) ϕ j ( ω x ) d ω x .

Then it follows from (8) and (9) that

Furthermore,

U j ( r , 0 ) = Φ j ( r ) = def Ω ϕ ( x ) ϕ j ( ω x ) d ω x .

Therefore,

U j ( r , t ) = 0 G j ( r , ρ , t ) Φ j ( ρ ) d ρ = 0 Ω G j ( r , ρ , t ) ϕ j ( ω y ) ϕ ( y ) d ω y d ρ = K G j ( r , | y | , t ) ϕ j ( ω y ) ϕ ( y ) | y | 1 m d y .

Comparing this with the formula

U j ( r , t ) = Ω u ( x , t ) ϕ j ( ω x ) d ω x = K Ω G ( x , y , t ) ϕ j ( ω x ) d ω x ϕ ( y ) d y ,

we get (7). □

In the sequel, σ is an arbitrary real number satisfying the conditions

σ > λ 1 , σ λ j + for all  j . (10)

We define G σ ( x , y , t ) = 0 for σ < λ 1 + , while

G σ ( x , y , t ) = λ j + < σ u j ( m j ) ( x , t ) c j ( y , t ) for  σ > λ 1 + , (11)

where

(12)

(13)

and m j = [ σ λ j + 2 ] . Here, we used the notation

σ ( μ ) = σ ( σ 1 ) ( σ μ + 1 ) for  μ = 1 , 2 , and σ ( 0 ) = 1 .

We define V p , β l ( K ) as the weighted Sobolev space with the norm

u V p , β l ( K ) = ( K | α | l r p ( β l + | α | ) | x α u ( x ) | p d x ) 1 / p

for 1 < p < and integer l 0 .

Lemma 1.2Suppose thatσis a real number such that σ > λ 1 and ( σ λ j + ) / 2 is not integer for λ j + σ . Furthermore, let 1 < p < and β = 2 σ m / p . Then

G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) ,

where t k y γ R σ ( , y , t ) V p , β 2 ( K ) for y K , t > 0 , | γ | 2 .

Proof We prove the lemma by induction in m 1 = [ ( σ λ 1 + ) / 2 ] .

First, let λ 1 < σ < λ 1 + . Then it follows from [[7], Corollary 4.1 and Theorem 4.2] (see also [[6], Theorem 3.2]) that t k y γ G ( , y , t ) V p , β 2 ( K ) for all y K , t > 0 , | γ | 2 , where β = 2 σ m / p . Thus, the assertion of the lemma is true for σ < λ 1 + .

Suppose the assertion is proved for σ < λ 1 + + 2 l . Now let λ 1 + + 2 l < σ < λ 1 + + 2 ( l + 1 ) . We set σ = σ 2 if l > 0 and σ = λ 1 + ε if l = 0 , where ε is a sufficiently small positive number. Then

[ σ λ j + 2 ] = [ σ λ j + 2 ] 1 = m j 1 for  λ j + < σ .

By the induction hypothesis, we have

G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) ,

where G σ is given by (11) (with σ instead of σ and m j 1 instead of m j ), t k y γ R σ ( , y , t ) V p , β 2 ( K ) , β = 2 σ m / p . The coefficients c j ( y , t ) in G σ are given by (13) and satisfy the equation ( t Δ y ) c j ( y , t ) = 0 . Therefore,

( t Δ y ) R σ ( x , y , t ) = 0

for x , y K , t > 0 . Obviously, G σ ( a x , a y , a 2 t ) = a m G σ ( x , y , t ) for a > 0 . Using the same equality for the Green function G ( x , y , t ) , we obtain

R σ ( a x , a y , a 2 t ) = a m R σ ( x , y , t ) for  a > 0 .

Furthermore,

Δ x R σ ( x , y , t ) = Δ x G ( x , y , t ) Δ x G σ ( x , y , t ) = ( t Δ x ) G σ ( x , y , t ) + t R σ ( x , y , t ) = ( t Δ x ) λ j + < σ k = 0 m j 1 t k c j ( y , t ) 4 k k ! ( σ j + k ) ( k ) r λ j + + 2 k ϕ j ( ω x ) + t R σ ( x , y , t ) .

Using the formula

Δ x r λ j + + 2 k ϕ j ( ω ) = 4 k ( σ j + k ) r λ j + + 2 k 2 ϕ j ( ω x ) ,

we get

Δ x R σ ( x , y , t ) = λ j + < σ t m j c j ( y , t ) r λ j + + 2 m j 2 ϕ j ( ω x ) 4 m j 1 ( m j 1 ) ! ( σ j + m j 1 ) ( m j 1 ) + t R σ ( x , y , t ) = Δ x Σ + t R σ ( x , y , t ) , (14)

where

Σ = λ j + < σ t m j c j ( y , t ) 4 m j m j ! ( σ j + m j ) ( m j ) r λ j + + 2 m j ϕ j ( ω x )

( Σ = 0 for l = 0 ). Let χ be a smooth function with compact support on [ 0 , ) such that χ ( r ) = 1 for r < 1 . Using the notation r = | x | , the function χ can be also considered as a function in K. Since σ < λ j + + 2 m j < σ for λ j + < σ , we have χ t k y γ Σ ( , y , t ) V p , β 2 ( K ) and ( 1 χ ) t k y γ Σ ( , y , t ) V p , β 2 ( K ) for all y K , t > 0 . Consequently, t k y γ ( R σ ( , y , t ) χ Σ ( , y , t ) ) V p , β 2 ( K ) and

Applying [[7], Theorem 4.2], we obtain

(15)

where v k , γ ( , y , t ) V p , β 2 ( K ) . The coefficients c μ , k , γ are given by the formula

c μ , k , γ ( y , t ) = K t k y γ ( t R σ ( x , y , t ) + Δ x ( 1 χ ) Σ ( x , y , t ) ) v μ ( x ) d x , (16)

where v μ ( x ) = 1 2 σ μ r λ μ ϕ μ ( ω x ) . The integral in (16) is well defined, since

t k y γ ( t R σ ( , y , t ) + Δ x ( 1 χ ) Σ ( , y , t ) ) V p , β 0 ( K ) V p , β 0 ( K )

and v μ V p , β 0 ( K ) + V p , β 0 ( K ) , p = p / ( p 1 ) , for σ < λ μ + < σ . The remainder v k , γ and the coefficients c μ , k , γ in (15) satisfy the estimate

(17)

Obviously, c μ , k , γ ( y , t ) = t k y γ c μ ( y , t ) = t k y γ c μ , 0 , 0 ( y , t ) . This means that

R σ ( x , y , t ) χ ( r ) Σ ( x , y , t ) = σ < λ μ + < σ c μ ( y , t ) r λ μ + ϕ μ ( ω x ) + v ( x , y , t ) ,

where t k y γ v ( , y , t ) = v k , γ ( , y , t ) V p , β 2 ( K ) . Consequently,

R σ ( x , y , t ) = Σ ( x , y , t ) + R σ ( x , y , t ) , (18)

where

Σ ( x , y , t ) = Σ ( x , y , t ) + σ < λ μ + < σ c μ ( y , t ) r λ μ + ϕ μ ( ω x ) = λ j + < σ t m j c j ( y , t ) r λ j + + 2 m j ϕ j ( ω x ) 4 m j m j ! ( σ j + m j ) ( m j )

and R σ ( x , y , t ) = v ( x , y , t ) + ( χ 1 ) Σ ( x , y , t ) . Obviously, t k y γ R σ ( , y , t ) V p , β 2 ( K ) for | γ | 2 . Using (18) and the equality

G σ ( x , y , t ) + Σ ( x , y , t ) = G σ ( x , y , t ) ,

we conclude that

G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) .

It remains to show that the coefficients

(19)

in (15) have the form (13) for σ < λ μ + < σ . First, note that

( t Δ y ) c μ ( y , t ) = 0 for  y K , t > 0 ,

since ( t Δ y ) R σ ( x , y , t ) = 0 and ( t Δ y ) Σ ( x , y , t ) = 0 .

Obviously, the functions t G σ ( x , y , t ) and

contain only functions ϕ j ( ω x ) with λ j + < σ . Thus, the orthogonality of the functions ϕ j implies

(20)

for λ μ + > σ . Applying Lemma 1.1, we conclude that c μ ( y , t ) has the form

c μ ( y , t ) = ρ 1 m ϕ μ ( ω y ) f μ ( ρ , t ) , (21)

where ρ = | y | . Since R σ ( a x , a y , a 2 t ) = a m R σ ( x , y , t ) and Σ ( a x , a y , a 2 t ) = a m Σ ( x , y , t ) for all a > 0 , it follows from (18) that

σ < λ μ + < σ ( a λ μ + c μ ( a y , a 2 t ) a m c μ ( y , t ) ) r λ μ + ϕ μ ( ω x ) = a m R σ ( x , y , t ) R σ ( a x , a y , a 2 t ) .

The function on the right-hand side belongs to V p , β 2 ( K ) for all y K , t > 0 , a > 0 , while the left-hand side belongs only to V p , β 2 ( K ) if

c μ ( a y , a 2 t ) = a m λ μ + c μ ( y , t ) .

Combining the last equality with (21), we get the representation

c μ ( y , t ) = ρ m λ μ + ϕ μ ( ω y ) h μ ( ρ 2 4 t ) = ρ λ μ 2 ϕ μ ( ω y ) h μ ( ρ 2 4 t ) .

Inserting this into the equation ( t Δ y ) c μ ( y , t ) = 0 , we obtain

r 2 h μ ( r ) + ( r σ μ 1 ) r h μ ( r ) + ( σ μ + 1 ) h μ ( r ) = 0 .

The substitution h μ ( r ) = e r r σ μ + 1 u ( r ) leads to the differential equation

r 2 u ( r ) + ( σ μ + 1 r ) r u ( r ) = 0

which has the solution

u ( r ) = d 1 + d 2 r 1 s σ μ 1 e s d s

with arbitrary constants d 1 and d 2 . Consequently,

c μ ( y , t ) = ρ λ μ 2 ϕ μ ( ω y ) ( ρ 2 4 t ) σ μ + 1 exp ( ρ 2 4 t ) ( d 1 + d 2 ρ 2 / ( 4 t ) 1 s σ μ 1 e s d s ) . (22)

Using (6) and (17), one gets the estimate

| t k c μ ( y , t ) | C k ( t ) ρ λ 1 + ε

with certain functions C k for ρ = | y | < t . Thus, the constant d 2 in (22) must be zero. Integrating (19), we get

0 c μ ( y , t ) d t = v μ ( y ) = 1 2 σ μ ρ λ μ ϕ μ ( ω y )

by means of (20). Hence,

d 1 ρ λ μ 2 ϕ μ ( ω y ) 0 ( ρ 2 4 t ) σ μ + 1 exp ( ρ 2 4 t ) d t = 1 2 σ μ ρ λ μ ϕ μ ( ω y ) .

The integral on the left-hand side is equal to 1 4 ρ 2 Γ ( σ μ ) . Thus, we get u ( r ) = d 1 = 2 / Γ ( σ μ + 1 ) and

h μ ( r ) = 2 Γ ( σ μ + 1 ) r σ μ + 1 e r .

This means that the formula (13) is valid for the coefficients c j if σ < λ j + < σ . The proof of the lemma is complete. □

1.2 Point estimates for the remainder in the asymptotics of Green’s function

We are interested in point estimates for the remainder R σ ( x , y , t ) in Lemma 1.2 in the case | x | < t . For this, we need the following lemma.

Lemma 1.3Suppose that u L p , β ( K ) and d u L p , β ( K ) , where p > m . Then

sup x K d ( x ) m / p r ( x ) β | u ( x ) | c ( K r p β ( | d ( x ) u ( x ) | p + | u ( x ) | p ) d x ) 1 / p

with a constantcindependent ofu.

Proof Let x 0 be a point int K, and let B 0 be a ball centered at x 0 with radius d 0 / 2 = d ( x 0 ) / 2 . We introduce the new coordinates y = d 0 1 x and set v ( y ) = u ( d 0 y ) = u ( x ) . Obviously, the point y 0 = d 0 1 x 0 has the distance 1 from ∂K. Hence,

| v ( y 0 ) | p c | y y 0 | < 1 / 2 ( | y v ( y ) | p + | v ( y ) | p ) d y .

This implies

| u ( x 0 ) | p c d 0 m B 0 ( | d 0 x u ( x ) | p + | u ( x ) | p ) d x .

Since d 0 / 2 < d ( x ) < 3 d 0 / 2 and r ( x 0 ) / 2 < r ( x ) < 3 r ( x 0 ) / 2 for x B 0 , we obtain

d 0 m r ( x 0 ) p β | u ( x 0 ) | p c B 0 r p β ( | d ( x ) x u ( x ) | p + | u ( x ) | p ) d x .

The result follows. □

Using the last two lemmas, we can prove the following theorem.

Theorem 1.1Suppose thatσis a real number satisfying (10). Then

G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) ,

where

| t k x α y γ R σ ( x , y , t ) | c t k ( m + | α | + | γ | ) / 2 ( | x | t ) σ | α | ( | y | | y | + t ) λ 1 + | γ | ε × ( d ( x ) | x | ) ε α ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) (23)

for | x | < 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 .

Proof Since G σ = G σ + ε for small positive ε, we may assume, without loss of generality, that ( σ λ j + ) / 2 is not integer for λ j + < σ . We prove the theorem by induction in m 1 = [ ( σ λ 1 + ) / 2 ] .

If λ 1 < σ < λ 1 + , then the assertion of the theorem follows from [[8], Theorem 3]. Suppose that λ 1 + + 2 l < σ < λ 1 + + 2 ( l + 1 ) , l 0 , and that the theorem is proved for σ < λ 1 + + 2 l . We set σ = σ 2 if l > 0 . In the case l = 0 , let σ be an arbitrary real number satisfying the inequalities λ 1 < σ < λ 1 + and σ σ 2 . By the induction hypothesis, we have

G ( x , y , t ) = G σ ( x , y , t ) + R σ ( x , y , t ) ,

where G σ is given by (11) (with σ instead of σ and m j 1 instead of m j ). Since G σ = G σ + δ for sufficiently small δ, it follows from the induction hypothesis that

| t k x α y γ R σ ( x , y , t ) | c t k ( m + | α | + | γ | ) / 2 ( | x | t ) σ + δ | α | ( | y | | y | + t ) λ 1 + | γ | ε × ( d ( x ) | x | ) ε α ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) (24)

for | x | < 2 t , | α | 2 , | γ | 2 . As was shown in the proof of Lemma 1.2, the remainder R σ admits the decomposition

R σ ( x , y , t ) = Σ ( x , y , t ) + R σ ( x , y , t ) ,

where

Σ ( x , y , t ) = λ j + < σ r λ j + + 2 m j ϕ j ( ω x ) t m j c j ( y , t ) 4 m j m j ! ( σ j + m j ) ( m j )

and t k y γ R σ ( , y , t ) V p , β 2 ( K ) for t > 0 , y K , | γ | 2 . Here β = 2 σ m / p . Furthermore (cf. (14)),

Δ x R σ ( x , y , t ) = Δ x ( R σ ( x , y , t ) Σ ( x , y , t ) ) = Δ x ( R σ ( x , y , t ) Σ ( x , y , t ) ) = t R σ ( x , y , t ) .

Let χ be a smooth cut-off function on the interval [ 0 , ) , χ = 1 in [ 0 , 1 ) and χ = 0 on ( 2 , ) . We define χ 1 ( x , t ) = χ ( t 1 / 2 | x | ) for x K , t > 0 . Then

Δ x ( χ 1 ( x , t ) y γ t k R σ ( x , y , t ) ) = f ( x , y , t ) ,

where

f = χ 1 y γ t k + 1 R σ + 2 x χ 1 x y γ t k ( R σ Σ ) + ( Δ x χ 1 ) y γ t k ( R σ Σ ) .

Thus, by [[7], Theorem 4.1], there exists a constant c such that

χ 1 ( , t ) y γ t k R σ ( , y , t ) V p , β 2 ( K ) c f ( , y , t ) V p , β 0 ( K ) (25)

for all y K , t > 0 , | γ | 2 . We estimate the norm of f. Using (24), we get

χ 1 t k + 1 y γ R σ ( , y , t ) V p , β 0 ( K ) c t k 1 ( m + | γ | + σ + δ ) / 2 ( | y | | y | + t ) λ 1 + | γ | ε exp ( κ | y | 2 t ) × ( d ( y ) | y | ) ε γ ( | x | < 2 t | x | p ( β + σ + δ ) d x ) 1 / p .

Here, p ( β + σ + δ ) > m . Thus,

Since x χ 1 vanishes outside the region t < | x | < 2 t and | x α χ 1 ( x , t ) | c t | α | / 2 , the estimate (24) also yields

Finally, it follows from the inequality

| y γ t k c μ ( y , t ) | c t k ( m + | γ | + λ μ + ) / 2 ( | y | t ) λ μ + | γ | exp ( | y | 2 6 t )

that

Consequently, by (25),

χ 1 ( , t ) y γ t k R σ ( , y , t ) V p , β 2 ( K ) c t k ( m + | γ | + σ ) / 2 ( | y | | y | + t ) λ 1 + | γ | ε × ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) (26)

with a positive constant κ. Applying the estimate

| α | 1 | x | β 2 + | α | + m / p | x α χ 1 ( x , t ) y γ t k R σ ( x , y , t ) | c χ 1 y γ t k R σ ( , y , t ) V p , β 2 ( K )

for p > m (cf. [[9], Lemma 1.2.3]), we obtain (23) for | α | 1 .

It remains to prove the estimate (23) for | α | = 2 . Let ρ ( x ) be the “regularized distance” of the point x to the boundary ∂K, i.e., ρ is a smooth function in K satisfying the inequalities

c 1 d ( x ) ρ ( x ) c 2 d ( x )

with positive constants c 1 and c 2 (cf. [[10], Chapter VI, § 2.1]). Moreover, ρ satisfies the inequality

| x α ρ ( x ) | c r ( x ) 1 | α | . (27)

We consider the function

v ( x , y , t ) = χ 1 ( x , t ) ρ ( x ) x j y γ t k R σ ( x , y , t )

for 1 j m . It follows from the equation Δ x R σ = t R σ that

Δ x v = f 1 + f 2 + f 3 ,

where f 1 = χ 1 ρ x j y γ t k + 1 R σ , f 2 = ( Δ x ( χ 1 ρ ) ) x j y γ t k R σ and f 3 = 2 x ( χ 1 ρ ) x x j y γ t k R σ . Using (24) and (27), we obtain

f 1 ( , y , t ) V p , β 0 ( K ) c t k ( m + | γ | + σ ) / 2 ( | y | | y | + t ) λ 1 + | γ | ε ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) .

Let χ 2 ( x , t ) = χ ( | x | / ( 2 t ) ) . The inequalities | Δ x ( χ 1 ρ ) | c r 1 and | x ( χ 1 ρ ) | c yield

(see (26)). Consequently by [[7], Theorem 4.1], the function v = χ 1 ρ x j y γ t k R σ satisfies the estimate

v ( , y , t ) V p , β 2 ( K ) c f 1 + f 2 + f 3 V p , β 0 ( K ) c t k ( m + | γ | + σ ) / 2 ( | y | | y | + t ) λ 1 + | γ | ε ( d ( y ) | y | ) ε γ exp ( κ | y | 2 t ) .

Applying Lemma 1.3 to the function u ( x , y , t ) = χ 1 ( x , t ) x α y γ t k R σ ( x , y , t ) with an arbitrary multi-index α with length | α | = 2 , we get

for | α | = 2 , | γ | 2 , p > m . Since p can be chosen arbitrarily large, the estimate (23) holds in the case | α | = 2 . The proof is complete. □

2 Asymptotics of solutions of the problem in D

Now we consider the problem (1), (2) in the domain D . Throughout this section, it is assumed that f L p , q ; β ( D × R ) , where p and β satisfy the inequalities

2 β m / p > λ 1 = 2 m λ 1 + and 2 β m / p λ j + for  j = 1 , 2 , , (28)

and q is an arbitrary real number >1. Let G ( x , y , t ) be the Green function of the problem (4), (5). Furthermore, let

Φ ( x , y , t ) = ( 4 π t ) ( m n ) / 2 exp ( | x y | 2 4 t )

be the fundamental solution of the heat equation in R n m . Then

G ( x , y , t ) = G ( x , y , t ) Φ ( x , y , t )

is the Green function of the problem (1), (2). We consider the solution

u ( x , t ) = t D G ( x , y , t τ ) f ( y , τ ) d y d τ (29)

of the problem (1), (2).

We again denote by G σ ( x , y , t ) the function (11) introduced in Section 1. In the sequel, σ is an arbitrary real number such that

σ > 2 β m / p , λ j + [ 2 β m / p , σ ] for all  j (30)

and

m j = [ σ λ j + 2 ] = [ 2 β λ j + m / p 2 ] for  λ j + < 2 β m / p . (31)

Then G σ ( x , y , t ) = G 2 β m / p ( x , y , t ) . Let χ be an infinitely differentiable function on R + = ( 0 , ) equal to one on the interval ( 0 , 1 ) and vanishing on ( 2 , ) . We define

χ 1 ( x , y ) = χ ( | x | | y | ) , χ 2 ( x , t , τ ) = χ ( | x | t τ ) .

Obviously,

u = Σ + v ,

where

(32)

(33)

We also consider the decomposition

u = Σ + w ,

where

Σ = λ j + < 2 β m / p u j ( m j ) ( x , t Δ x ) H j ( x , t ) (34)

and

H j ( x , t ) = t D χ 1 ( x , y ) χ 2 ( x , t , τ ) c j ( y , t τ ) Φ ( x , y , t τ ) f ( y , τ ) d y d τ (35)

is an extension of the function

h j ( x , t ) = t D c j ( y , t τ ) Φ ( x , y , t τ ) f ( y , τ ) d y d τ (36)

with c j defined by (13). Our goal is to show that both remainders v and w are elements of the space W p , q ; β 2 , 1 ( D × R ) . We start with the case p = q .

2.1 Estimates in weighted L p Sobolev spaces

Let W p , q ; β 2 l , l ( D × R ) be the weighted Sobolev space with the norm (3). Furthermore, let

W p ; β 2 l , l ( D × R ) = W p , p ; β 2 l , l ( D × R ) , L p ; β ( D × R ) = W p ; β 0 , 0 ( D × R ) .

In this subsection, we assume that f L p ; β ( D × R ) , where p and β satisfy (28). First, we prove that Σ Σ W p ; β 2 , 1 ( D × R ) . This was shown in [[1], Corollary 2.3] for the case Ω C . In the case Ω C 1 , 1 , we must keep in mind that the second-order derivatives of the eigenfunctions ϕ j must not be bounded. Then we have the estimate

| x α ϕ j ( ω x ) | c | x | | α | ( d ( x ) | x | ) ε α (37)

for | α | 2 , where ε α = 0 for | α | 1 and ε α is an arbitrarily small positive real number if | α | 1 . However, this requires only a small modification of the proof in [1].

Lemma 2.1Suppose that f L p , β ( D × R ) . Then x α t k ( Σ Σ ) L p ; β 2 + | α | + 2 k ( D × R ) and

x α t k ( Σ Σ ) L p ; β 2 + | α | + 2 k ( D × R ) c f L p , β ( D × R )

for | α | 2 and allk.

Proof A simple calculation (see the proof of [[1], Corollary 1]) yields

Σ Σ = λ j + < σ t D χ 1 ( x , y ) ( [ u j ( m j ) ( x , t ) , χ 2 ] c j ( y , t τ ) ) × Φ ( x , y , t τ ) f ( y , τ ) d y d τ ,

where [ u j ( m j ) ( x , t ) , χ 2 ] = u j ( m j ) ( x , t ) χ 2 χ 2 u j ( m j ) ( x , t ) denotes the commutator of u j ( m j ) ( x , t ) and χ 2 . Obviously, the inequalities

| x | 2 | y | and t τ | x | 2 t τ

are satisfied on the support of the kernel

K j ( x , y , t , τ ) = χ 1 ( x , y ) ( [ u j ( m j ) ( x , t ) , χ 2 ] c j ( y , t τ ) ) Φ ( x , y , t τ ) . (38)

Since, moreover, the eigenfunctions ϕ j satisfy the inequality (37) for | α | 2 , we obtain

| x α t k K j ( x , y , t , τ ) | c ( t τ ) n / 2 ( d ( x ) | x | ) ε | x | | α | 2 k σ | y | σ exp ( | y | 2 + | x y | 2 8 ( t τ ) )

for | α | 2 . Using Hölder’s inequality, we obtain

| x α t k ( Σ Σ ) ( x , t ) | c ( d ( x ) | x | ) ε | x | | α | 2 k σ A 1 / p B 1 / p ,

where

A = t | x | 2 t | x | 2 / 4 D ( t τ ) n / 2 | y | p β | f ( y , τ ) | p exp ( | y | 2 + | x y | 2 8 ( t τ ) ) d y d τ

and

B = t | x | 2 t | x | 2 / 4 D | y | > | x | / 2 ( t τ ) n / 2 | y | p ( σ β ) exp ( | y | 2 + | x y | 2 8 ( t τ ) ) d y d τ .

The substitution y = z t τ , y = x + z t τ yields

B c t | x | 2 t | x | 2 / 4 ( t τ ) p ( σ β ) / 2 d τ | z | > 1 / 2 | z | p ( σ β ) exp ( | z | 2 8 ) d z × R n m exp ( | z | 2 8 ) d z ,

i.e., B c | x | p ( σ β ) + 2 . Consequently,

where

D ( y , τ ) = τ τ + | y | 2 D t τ < | x | < 2 t τ | x | 2 ( d ( x ) | x | ) p ε ( t τ ) n / 2 × exp ( | y | 2 + | x y | 2 8 ( t τ ) ) d x d t .

Substituting x = z t τ and x = y + z t τ , we obtain

D ( y , τ ) = τ τ + | y | 2 ( t τ ) 1 exp ( | y | 2 8 ( t τ ) ) d t K 1 < | z | < 2 | z | 2 ( d ( z ) | z | ) p ε d z .

This means that D ( y , τ ) is a constant. This proves the lemma. □

Next, we estimate the first-order x-derivatives of the remainder v. For this, we employ the following lemma (cf. [[11], Lemma A.1]).

Lemma 2.2Let K be the integral operator

( K f ) ( x , t ) = t R n K ( x , y , t , τ ) f ( y , τ ) d y d τ (39)

with a kernel K ( x , y , t , τ ) satisfying the estimate

| K | c ( t τ ) ( n + 2 r ) / 2 ( | x | | x | + t τ ) a + r ( | y | | y | + t τ ) b | x | μ r | y | μ exp ( κ | x y | 2 t τ ) ,

where κ > 0 , 0 < r 2 , a + b > m , m p a < μ < m m p + b . Then K is bounded on L p ( R n × R ) .

In the proof of the following assertion, we use another decomposition of the remainder v as in [[1], Lemma 2.4]. This allows us to apply directly the estimate in Theorem 1.1.

Lemma 2.3Letpandβsatisfy the condition (28). Furthermore, letvbe the function (33), where f L p ; β ( D × R ) , 1 < p < . Then x α v L p ; β 2 + | α | ( D × R ) for | α | 1 and

| α | 1 x α v L p ; β 2 + | α | ( D × R ) c f L p ; β ( D × R )

with a constantcindependent off. The same is true for the functionw.

Proof Obviously,

v = j = 1 3 t D Ψ j ( x , y , t , τ ) f ( y , τ ) d y d τ ,

where

and

Ψ 3 ( x , y , t , τ ) = ( 1 χ 1 ( x , y ) ) χ 2 ( x , t , τ ) G σ ( x , y , t τ ) Φ ( x , y , t τ ) .

We show that the integral operators with the kernels

K j ( α ) ( x , y , t , τ ) = | x | β 2 + | α | | y | β x α Ψ j ( x , y , t , τ )

are bounded in L p ( D × R ) for j = 1 , 2 , 3 and | α | 1 . Using Theorem 1.1, we get

| K 1 ( α ) ( x , y , t , τ ) | c | x | β 2 + | α | | y | β ( t τ ) ( n + | α | ) / 2 ( | x | t τ ) σ | α | ( | y | | y | + t τ ) λ 1 + ε × exp ( κ | x y | 2 t τ ) ,

where ε is an arbitrarily small positive number. Applying Lemma 2.2 with r = 2 | α | , μ = β , a = σ 2 , b = λ 1 + ε , we conclude that the integral operator with the kernel K 1 ( α ) ( x , y , t , τ ) is bounded in L p ( D × R ) for | α | 1 .

Since | x | | x | + t τ 2 | x | on the support of K 2 ( α ) , the estimate (6) implies

| K 2 ( α ) ( x , y , t , τ ) | c | x | β 2 + | α | | y | β ( t τ ) ( n + | α | ) / 2 ( | x | | x | + t τ ) a ( | y | | y | + t τ ) λ 1 + ε × exp ( κ | x y | 2 t τ )

with arbitrary real a. Thus, by Lemma 2.2, the integral operator with the kernel K 2 ( x , y , t , τ ) is bounded in L p ( D × R ) for | α | 1 .

We consider the kernel K 3 ( α ) . Since G σ ( x , y , t ) has the form

G σ ( x , y , t ) = λ j + < σ k = 0 m j c j , k | x | λ j + + 2 k | y | λ j + ϕ j ( ω x ) ϕ j ( ω y ) t k t λ j + m / 2 exp ( | y | 2 4 t ) ,

we get the representation

K 3 ( α ) ( x , y , t , τ ) = λ j + < σ k = 0 m j K j , k ( x , y , t , τ ) ,

where

| K j , k ( x , y , t , τ ) | c | x | β 2 + | α | | y | β | x | λ j + + 2 k | α | | y | λ j + ( t τ ) k λ j + n / 2 exp ( κ | x y | 2 t τ ) .

Here we used the fact that | y | | x | 2 t τ on the support of the function ( 1 χ 1 ) χ 2 . The inequalities | y | | x | 2 t τ and λ j + + 2 k σ imply

| K j , k ( x , y , t , τ ) | c | x | β 2 + | α | | y | β ( t τ ) ( n + | α | ) / 2 ( | x | t τ ) σ | α | ( | y | t τ ) 2 λ 1 + σ × exp ( κ | x y | 2 t τ ) .

It is no restriction to assume that σ < 2 λ 1 + + m β m / p in addition to (30) and (31). Therefore, we can apply Lemma 2.2 with r = 2 | α | , a = σ 2 and b = 2 λ 1 + σ to the integral operator with the kernel K j , k . It follows that the integral operator with the kernel K 3 ( α ) ( x , y , t , τ ) is bounded in L p ( D × R ) for | α | 1 . Consequently, the integral operator with the kernel

K ( α ) ( x , y , t , τ ) = j = 1 3 K j ( α ) ( x , y , t , τ ) = | x | β 2 + | α | | y | β j = 1 3 x α Ψ j ( x , y , t , τ )

is bounded in L p ( D × R ) for | α | 1 . This proves the lemma. □

Furthermore, the assertions of [[1], Lemmas 2.5, 2.6, Theorem 2.7] are also valid if Ω is only of the class C 1 , 1 . The proof under this weaker assumption on Ω does not require any modifications of the method in [1]. We give here only the formulation of [[1], Theorem 2.7].

Theorem 2.1Let f L p ; β ( D × R ) , wherepandβsatisfy the condition (28). Then there exists a solution of the problem (1), (2) which has the form

u = λ j + < 2 β m / p u j ( m j ) ( x , t Δ x ) H j ( x , t ) + w ,

where w W p ; β 2 , 1 ( D × R ) and u j ( k ) , m j , H j are given by (12), (31) and (35), respectively. The functions H j depend only on | x | , x andtand satisfy the estimates