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

Asymptotics of solutions of the heat equation in

cones and dihedra under minimal assumptions

on the boundary

Vladimir Kozlov and Jürgen Roßmann

Linköping University Post Print

N.B.: When citing this work, cite the original article.

The original publication is available at www.springerlink.com:

Vladimir Kozlov and Jürgen Roßmann, Asymptotics of solutions of the heat equation in cones

and dihedra under minimal assumptions on the boundary, 2012, Boundary Value Problems,

(2112), 142, .

http://dx.doi.org/10.1186/1687-2770-2012-142

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Postprint available at: Linköping University Electronic Press

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-86257

R E S E A R C H

Open Access

Asymptotics of solutions of the heat equation

in cones and dihedra under minimal

assumptions on the boundary

Vladimir A Kozlov

_{and Jürgen Rossmann}

*_{Correspondence:}

juergen.rossmann@uni-rostock.de 2_{Institute of Mathematics, University} of Rostock, Rostock, D-18051, Germany

Full list of author information is available at the end of the article

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× Rn–m_{. Here the right-hand side f of the heat equation is assumed to}

be an element of a weighted Lp,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

∂u

∂t – u = f inD × R, () u=  on (∂D\M) × R () in the domain

D =x=x, x: x∈ K, x∈ Rn–m,

where K ={x= (x, . . . , xm) : x/|x| ∈ } is a cone in Rm, ≤ m ≤ n,  denotes a

sub-domain of the unit sphere, and M ={x = (x, x) : x= } is the (n – m)-dimensional edge ofD. We are interested in the asymptotics of solutions in the class of the weighted Sobolev spaces W_p,_,q;β(D ×R). Here the space W_pl,l_,q;β(D ×R) is defined for an arbitrary integer l ≥  and real p > , q > , β as the set of all function u(x, t) onD × R with the finite norm

u_Wl,l p,q;β(D×R)=  R  D  |α|+k≤l x p(β–l+k+|α|) ∂_tk∂_xαu(x, t) pdx q/p dt /q . ()

In the case l = , we write W_p,_,q;β= Lp,q;β. If, moreover, β = , then we write Lp,q;= Lp,q.

For the case of smooth boundary ∂ (of class C∞), the asymptotics of solutions was obtained in our previous paper []. For the particular case p = q = , m = n, we refer also to the paper [] by Kozlov and Maz’ya, and for the case p = q= , m = n = , to the paper [] by de Coster and Nicaise. The goal of the present paper is to describe the asymptotics

©2012 Kozlov and Rossmann; licensee Springer. This is an Open Access article distributed under the terms of the Creative Com-mons Attribution License (http://creativecomCom-mons.org/licenses/by/2.0), which permits unrestricted use, distribution, and repro-duction in any medium, provided the original work is properly cited.

of solutions with a remainder in W_p,_,q;β(D × R) under minimal smoothness assumptions on the boundary. Throughout the paper, we assume that ∂∈ C,_.

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

Gx, y, t= λ+_j<σ mj  k= ∂_tkcj(y, t)|x| λ+_j+k φj(ωx) k_k!(σ j+ k)(k) + Rσ  x, y, t

as in [, ] (for the definition of λ+

j, φj, mj, cjand σ(k), see Section .). However, the proof in

[, ] does not work if ∂ is only of the class C,. We give a new proof, which is completely different from that in [, ]. Our tools are estimates for solutions of the Dirichlet problem for the Laplace equation in a cone in weighted LpSobolev spaces and asymptotic

formu-las for solutions of this problem which were obtained in the papers [, ] by Maz’ya and Plamenevski˘ı. Moreover, we use the estimates of the Green function in the recent paper [] by Kozlov and Nazarov. In contrast to the case ∂∈ C∞, the estimates for the second order x- and y-derivatives of the remainder Rσcontain an additional factor (|x|–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 ), we apply the results of Section  in order to obtain the asymptotics of solutions of the problem (), () for f ∈ Lp,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<–β–m/p mj  k= (∂t– x)kHj(x, t) k_k!(σ j+ k)(k) x λ+j+k_φ j(ωx) + w(x, t)

with a remainder w∈ W_p,_,q;β(D × R). Here, Hjis an extension of the function

hj  x, t=  t –∞  Dcj  y, t – τ x, y, t – τf(y, τ ) dy dτ ,

denotes the fundamental solution of the heat equation inRn–m_{. The proof of this result}

(Theorem .) is essentially the same as in []. However, the proofs of some lemmas in [] have to be modified under our weaker assumptions on ∂.

At the end of the paper, we show that the extensions of the functions hjcan be defined as

Hj(x, t) = (Ehj)(x, t) =  ∞   Rn–m T(τ )Rzhj  x– rz, t – rτdzdτ,

where T and R are certain smooth functions onR+andRn–m, respectively (see the

begin-ning of Section  for their definition). This extends the result of [, Corollary .] to the case p= q.

1 The Green function of the heat equation in a cone

We start with the problem

∂u

∂t – xu= f in K× R, ()

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

∂G(x, y, t) ∂t – xG  x, y, t= δx– yδ(t) in K× R, Gx, y, t=  for x∈ ∂K\{}, t ∈ R, Gx, y, t=  for t < . Furthermore, ( – ζ )G(·, y_{,·) ∈ W},

;β(K× R) if λ– <  – β – m/ < λ+ (λ± are defined

be-low), and ζ is a function in C_∞(K × R) equal to one in a neighborhood of the point (x, t) = (y, ). Here W_,β,(K × R) is the space of all functions u = u(x, t) on K× R such that|x|β–+k+|α|_∂k

t∂xαu∈ L(K× R) for k + |α| ≤ . 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{j}∞j=be the nondecreasing sequence of eigenvalues of the Beltrami operator –δ on

(with the Dirichlet boundary condition) counted with their multiplicities, and let{φj}∞j=

be an orthonormal (in L()) sequence of eigenfunctions corresponding to the

eigenval-ues j. Furthermore, we define

λ±_j = – m  ±  ( – m/)_{+ } j and σj= λ+j –  + m .

This means that λ±_j are the solutions of the quadratic equation λ(m–+λ) = j. Obviously,

λ+_j >  and λ– j <  – m for j = , , . . . . By [, Theorem ], ∂_tk∂_xα∂_yγGx, y, t ≤ct–k–(m+|α|+|γ |)/  _|x| |x_{| +}√_t λ+_–|α|–ε _|y| |y_{| +}√_t λ+_–|γ |–ε ×  d(x) |x_| –εα_d(y₎ |y_| –εγ exp  –κ|x _{– y}_| t () for|α| ≤ , |γ | ≤ . Here d(x) denotes the distance of the point xfrom the boundary ∂K . Furthermore, εαis defined as zero for|α| ≤ , while εαis an arbitrarily small positive real

number if|α| = . Actually, the estimate () is proved in [] only for k = , but for a more general class of operators, parabolic operators with discontinuous in time coefficients. If the coefficients in [] do not depend on t, then one can use the same argument as in the proof of [, Theorem ] when treating the derivatives along the edge of the domain

D = K × Rn–m_{. This argument shows that the kth derivative with respect to t will bring}

only an additional factor t–kto the right-hand side of ().

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

Lemma . Let G(x, y, t) be the Green function introduced above, and let Gj(r, ρ, t) denote

the Green function of the initial-boundary value problem ∂tU(r, t) – r–  (r∂r)+ (m – )r∂r– j  U(r, t) =  for r> , t > , U(, t) =  for t> , U(r, ) = (r) for r> .

Then   Gx, y, tφj(ωx) dωx= y –m Gj x , y , t  φj(ωy). ()

Proof The solution of the problem (∂t– x)u



x, t=  for x∈ K, t > , ()

ux, t=  for x∈ ∂K, t > , ux, = φx () is given by the formula

ux, t=  K Gx, y, tφydy. We define Uj(r, t) =   ux, tφj(ωx) dωx.

Then it follows from () and () that

∂tUj(r, t) – r–  (r∂r)+ (m – )r∂r– j  Uj(r, t) =    ∂t– r–  (r∂r)+ (m – )r∂r– j  uxφj(ωx) dωx =   (∂t– x)u  xφj(ωx) dωx= . Furthermore, Uj(r, ) = j(r) def =   φxφj(ωx) dωx. Therefore, Uj(r, t) =  _∞  Gj(r, ρ, t) j(ρ) dρ =  _∞    Gj(r, ρ, t)φj(ωy)φ  ydωydρ =  K Gj  r, y , tφj(ωy)φ  y y –mdy. Comparing this with the formula

Uj(r, t) =   ux, tφj(ωx) dωx=  K   Gx, y, tφj(ωx) dωxφ  ydy, we get (). 

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

We define Gσ(x, y, t) =  for σ < λ+_, while Gσ  x, y, t= λ+_j<σ u(m_j j)x, ∂t  cj  y, t for σ > λ+ , () where u(k)_j x, ∂t  = rλ+j_φ j(ωx) k  μ= rμ_∂μ t μ_μ!(σ j+ μ)(μ) , () cj  y, t=  ( + σj) y λ–j– _|y| t σj+ φj(ωy) exp  –|y _| t , () and mj= [ σ–λ+_j

 ]. Here, we used the notation

σ(μ)= σ (σ – )· · · (σ – μ + ) for μ = , , . . . and σ()= .

We define V_pl_,β(K ) as the weighted Sobolev space with the norm

uV_pl_,β(K )=  K  |α|≤l rp(β–l+|α|) ∂_xαu(x) pdx /p

for  < p <∞ and integer l ≥ .

Lemma . Suppose that σ is a real number such that σ> λ–_ and(σ – λ+_j)/ is not integer

for λ+_j ≤ σ . Furthermore, let  < p < ∞ and β =  – σ – m/p. Then Gx, y, t= Gσ  x, y, t+ Rσ  x, y, t, where ∂k t∂ γ yRσ(·, y, t)∈ Vp,β(K ) for y∈ K, t > , |γ | ≤ .

Proof We prove the lemma by induction in m= [(σ – λ+)/].

First, let λ–

 < σ < λ+. Then it follows from [, Corollary . and Theorem .] (see also

[, Theorem .]) that ∂k t∂

γ

yG(·, y, t)∈ Vp,β(K ) for all y∈ K, t > , |γ | ≤ , where β =

 – σ – m/p. Thus, the assertion of the lemma is true for σ < λ+ .

Suppose the assertion is proved for σ < λ+

 + l. Now let λ++ l < σ < λ+ + (l + ). We

set σ= σ –  if l >  and σ= λ+

– ε if l = , where ε is a sufficiently small positive number.

Then _σ_{– λ}+ j  = _{σ– λ}+ j  –  = mj–  for λ+j < σ.

By the induction hypothesis, we have

Gx, y, t= Gσ



x, y, t+ Rσ



x, y, t,

where Gσ is given by () (with σinstead of σ and mj–  instead of mj), ∂tk∂ γ

yRσ(·, y, t)∈

equation (∂t– y)cj(y, t) = . Therefore,

(∂t– y)Rσ



x, y, t= 

for x, y∈ K, t > . Obviously, Gσ(ax, ay, at) = a–mGσ(x, y, t) for a > . Using the same

equality for the Green function G(x, y, t), we obtain

Rσ  ax, ay, at= a–mRσ  x, y, t for a > . Furthermore, xRσ  x, y, t= xG  x, y, t– xGσ  x, y, t = (∂t– x)Gσ  x, y, t+ ∂tRσ  x, y, t = (∂t– x)  λ+_j<σ mj–  k= ∂_tkcj(y, t) k_k!(σ j+ k)(k) rλ+j+k_φ j(ωx) + ∂tRσ  x, y, t.

Using the formula

xr λ+_j+k φj(ω) = k(σj+ k)r λ+_j+k– φj(ωx), we get xRσ  x, y, t=  λ+_j<σ ∂_tmjcj(y, t)r λ+_j+mj–_φ j(ωx) mj–_(m j– )!(σj+ mj– )(mj–) + ∂tRσ  x, y, t = x+ ∂tRσ  x, y, t, () where =  λ+_j<σ ∂_tmjcj(y, t) mj_m j!(σj+ mj)(mj) rλ+j+mj_φ j(ωx)

(=  for l = ). Let χ be a smooth function with compact support on [,∞) such that

χ(r) =  for r < . Using the notation r =|x|, the function χ can be also considered as a function in K . Since σ< λ+ j + mj< σ for λ+j < σ, we have χ ∂tk∂ γ y(·, y, t)∈ V  p,β(K ) and ( – χ )∂k t∂ γ

y(·, y, t)∈ Vp,β(K ) for all y∈ K, t > . Consequently, ∂tk∂ γ y(Rσ(·, y, t) – χ (·, y, t))∈ V_p_,β(K ) and x∂tk∂ γ y  Rσ  ·, y_{, t}_{– χ }_{·, y}_{, t} = ∂_tk+∂_yγRσ  ·, y_{, t}_{+ } x∂tk∂ γ y( – χ )  ·, y_{, t}_{∈ V} p,β(K ).

Applying [, Theorem .], we obtain

∂_tk∂_yγRσ  x, y, t– χ (r)x, y, t =  σ<λ+μ<σ cμ,k,γ  y, trλ+μ_φ μ(ω) + vk,γ  x, y, t, ()

where vk,γ(·, y, t)∈ Vp,β(K ). The coefficients cμ,k,γ are given by the formula cμ,k,γ  y, t=  K ∂_tk∂_yγ  ∂tRσ  x, y, t+ x( – χ )  x, y, tvμ  xdx, () where vμ(x) = –_σ μr λ–_μφ

μ(ωx). The integral in () is well defined, since

∂_tk∂_yγ∂tRσ  ·, y_{, t}_{+ } x( – χ )  ·, y_{, t}_{∈ V} p,β(K )∩ Vp,β(K )

and vμ∈ V_p_,–β(K ) + V_p_,–β(K ), p= p/(p – ), for σ< λ+μ< σ . The remainder vk,γ and the

coefficients cμ,k,γ in () satisfy the estimate

vk,γ  ·, y_{, t} V_p_,β(K )+  σ<λ+μ<σ cμ,k,γ  y, t ≤ c∂_tk∂_yγ∂tRσ  ·, y_{, t}_{+ } x( – χ )  ·, y_{, t} V_p_,β(K )∩V p,β(K ) . () Obviously, cμ,k,γ(y, t) = ∂tk∂ γ ycμ(y, t) = ∂ k t∂ γ

ycμ,,(y, t). This means that

Rσ  x, y, t– χ (r)x, y, t=  σ<λ+μ<σ cμ  y, trλ+μ_φ μ(ωx) + v  x, y, t, where ∂k t∂ γ yv(·, y, t) = vk,γ(·, y, t)∈ Vp,β(K ). Consequently, Rσ  x, y, t= x, y, t+ Rσ  x, y, t, () where x, y, t= x, y, t+  σ<λ+ μ<σ cμ  y, trλ+μ_φμ(ω x) =  λ+_j<σ ∂_tmjcj(y, t)rλ + j+mj_φ j(ωx) mj_m j!(σj+ mj)(mj) and Rσ(x, y, t) = v(x, y, t) + (χ – )(x, y, t). Obviously, ∂tk∂ γ yRσ(·, y, t) ∈ Vp,β(K ) for

|γ | ≤ . Using () and the equality

Gσ  x, y, t+ x, y, t= Gσ  x, y, t, we conclude that Gx, y, t= Gσ  x, y, t+ Rσ  x, y, t= Gσ  x, y, t+ Rσ  x, y, t. It remains to show that the coefficients

cμ  y, t = –  σμ  ∞     ∂tRσ  x, y, t+ x( – χ )  x, y, tφμ(ωx) dωxrλ – μ+m–_{dr ()}

in () have the form () for σ< λ+

μ< σ . First, note that

(∂t– y)cμ



y, t=  for y∈ K, t > ,

since (∂t– y)Rσ(x, y, t) =  and (∂t– y)(x, y, t) = .

Obviously, the functions ∂tGσ(x, y, t) and

x( – χ )  x, y, t = r–(r∂r)( – χ )  x, y, t+ (m – )∂r( – χ )  x, y, t+ ( – χ )δω 

contain only functions φj(ωx) with λ+j < σ. Thus, the orthogonality of the functions φj

implies    ∂tRσ  x, y, t+ x( – χ )  x, y, tφμ(ωx) dωx =   ∂tG  x, y, tφμ(ωx) dωx () for λ+

μ> σ. Applying Lemma ., we conclude that cμ(y, t) has the form

cμ



y, t= ρ–mφμ(ωy)fμ(ρ, t), ()

where ρ =|y|. Since Rσ(ax, ay, at) = a–mRσ(x, y, t) and (ax, ay, at) = a–m(x, y, t)

for all a > , it follows from () that  σ<λ+ μ<σ  aλ+μ_cμ_ay_{, a}_t_{– a}–m_cμ_y_{, t}_rλ+μ_φμ(ω x) = a–mRσ  x, y, t– Rσ  ax, ay, at.

The function on the right-hand side belongs to V

p,β(K ) for all y∈ K, t > , a > , while the

left-hand side belongs only to V p,β(K ) if

cμ



ay, at= a–m–λμ+_cμ_y_{, t}_.

Combining the last equality with (), we get the representation

cμ  y, t= ρ–m–λ+μ_φμ(ω y)hμ  ρ t = ρλ–μ–_φμ(ω y)hμ  ρ t .

Inserting this into the equation (∂t– y)cμ(y, t) = , we obtain

rhμ(r) + (r – σμ– )rhμ(r) + (σμ+ )hμ(r) = .

The substitution hμ(r) = e–rrσμ+u(r) leads to the differential equation

which has the solution

u(r) = d+ d

  r

s–σμ–_es_ds

with arbitrary constants dand d. Consequently,

cμ  y, t= ρλ–μ–_φμ(ω y)  ρ t σμ+ exp  –ρ  t  d+ d   ρ_/(t)s –σμ–_es_ds . () Using () and (), one gets the estimate

∂_tkcμ



y, t ≤Ck(t)ρλ

+ –ε

with certain functions Ck for ρ =|y| <

√

t. Thus, the constant d in () must be zero.

Integrating (), we get  _∞  cμ  y, tdt= –vμ  y=  σμ ρλ–μ_φ μ(ωy) by means of (). Hence, dρλ – μ–_φ μ(ωy)  _∞   ρ t σμ+ exp  –ρ  t dt=  σμ ρλ–μ_φ μ(ωy).

The integral on the left-hand side is equal to  ρ

_(σ

μ). Thus, we get u(r) = d= /(σμ+ )

and

hμ(r) =



(σμ+ )

rσμ+_e–r_.

This means that the formula () is valid for the coefficients cjif 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 Rσ(x, y, t) in Lemma . in the case

|x_{| <}√_{t. For this, we need the following lemma.}

Lemma . Suppose that u∈ Lp,β(K ) and d∇u ∈ Lp,β(K ), where p > m. Then

sup x∈Kd  xm/pr(x)β ux ≤c  K rpβ dx∇ux p+ ux pdx /p

with a constant c independent of u.

Proof Let x_be a point int K , and let Bbe a ball centered at xwith radius d/ = d(x)/.

We introduce the new coordinates y= d–

 xand set v(y) = u(dy) = u(x). Obviously, the

point y_= d–

 xhas the distance  from ∂K . Hence,

vy_ p≤ c  |y_–y |</  ∇yv  y p+ vy pdy.

This implies ux_ p≤ cd–m_  B  d∇xu  x p+ ux pdx.

Since d/ < d(x) < d/ and r(x)/ < r(x) < r(x)/ for x∈ B, we obtain

d_mrx_pβ ux_ p≤ c  B rpβ dx∇xu  x p+ ux pdx.

The result follows. 

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

Theorem . Suppose that σ is a real number satisfying(). Then

Gx, y, t= Gσ  x, y, t+ Rσ  x, y, t, where ∂_tk∂_xα∂_yγRσ  x, y, t ≤ ct–k–(m+|α|+|γ |)/ _|x| √ t σ–|α| _|y| |y_{| +}√_t λ+_–|γ |–ε ×  d(x) |x_| –εα d(y) |y_| –εγ exp  –κ|y _| t ()

for|x| <√t,|α| ≤ , |γ | ≤ . Here εα=  for|α| ≤ , while εαis an arbitrarily small positive

real number if|α| = .

Proof Since Gσ = Gσ+εfor small positive ε, we may assume, without loss of generality, that

(σ – λ+_j)/ is not integer for λ+_j < σ . We prove the theorem by induction in m= [(σ – λ+)/].

If λ–_ < σ < λ+_, then the assertion of the theorem follows from [, Theorem ]. Suppose that λ+_ + l < σ < λ+_ + (l + ), l≥ , and that the theorem is proved for σ < λ+_ + l. We set σ= σ –  if l > . In the case l = , let σbe an arbitrary real number satisfying the inequalities λ–_ < σ< λ+_ and σ≥ σ – . By the induction hypothesis, we have

Gx, y, t= Gσ



x, y, t+ Rσ



x, y, t,

where Gσis given by () (with σinstead of σ and mj–  instead of mj). Since Gσ= Gσ+δ

for sufficiently small δ, it follows from the induction hypothesis that ∂_tk∂_xα∂_yγRσ  x, y, t ≤ ct–k–(m+|α|+|γ |)/ _|x| √ t σ+δ–|α| _|y| |y_{| +}√_t λ+_–|γ |–ε ×  d(x) |x_| –εα_d(y₎ |y_| –εγ exp  –κ|y _| t () for|x| < √t,|α| ≤ , |γ | ≤ . As was shown in the proof of Lemma ., the remainder

Rσadmits the decomposition

Rσ



x, y, t= x, y, t+ Rσ



where x, y, t= λ+_j<σ rλ+j+mj_φ j(ωx)∂ mj t cj(y, t) mj_m j!(σj+ mj)(mj) and ∂k t∂ γ

yRσ(·, y, t)∈ V_p_,β(K ) for t > , y∈ K, |γ | ≤ . Here β =  – σ – m/p. Furthermore

(cf. ()), xRσ  x, y, t= x  Rσ  x, y, t– x, y, t= x  Rσ  x, y, t– x, y, t = ∂tRσ  x, y, t.

Let χ be a smooth cut-off function on the interval [,∞), χ =  in [, ) and χ =  on (,∞). We define χ(x, t) = χ (t–/|x|) for x∈ K, t > . Then

x  χ  x, t∂_yγ∂tkRσ  x, y, t= fx, y, t, where f = χ∂_yγ∂tk+Rσ+ ∇xχ· ∇x∂_yγ∂tk(Rσ– ) + (xχ)∂_yγ∂tk(Rσ– ).

Thus, by [, Theorem .], there exists a constant c such that χ(·, t)∂_yγ∂tkRσ  ·, y_{, t} V_p_,β(K )≤ cf  ·, y_{, t} V_p_,β(K ) ()

for all y∈ K, t > , |γ | ≤ . We estimate the norm of f . Using (), we get χ∂tk+∂ γ yRσ  ·, y_{, t} V_p_,β(K )≤ ct –k––(m+|γ |+σ_+δ)/ |y| |y_{| +}√_t λ+_–|γ |–ε exp  –κ|y _| t ×  d(y) |y_| –εγ |x_|<√_t|x _|p(β+σ+δ)_dx/p_. Here, p(β + σ+ δ) > –m. Thus, χ∂tk+∂ γ yRσ  ·, y_{, t} V_p_,β(K ) ≤ ct–k–(m+|γ |+σ)/ |y| |y_{| +}√_t λ+_–|γ |–ε_d(y₎ |y_| –εγ exp  –κ|y _| t .

Since∇xχvanishes outside the region

√

t<|x| < √tand|∂_xαχ(x, t)| ≤ ct–|α|/, the

es-timate () also yields ∇xχ· ∇x∂_yγ∂tkRσ  ·, y_{, t} V_p_,β(K )+(xχ)∂ γ y∂ k tRσ  ·, y_{, t} V_p_,β(K ) ≤ ct–k–(m+|γ |+σ)/ |y| |y_{| +}√_t λ+_–|γ |–ε d(y) |y_| –εγ exp  –κ|y _| t .

Finally, it follows from the inequality ∂_yγ∂tkcμ  y, t ≤ct–k–(m+|γ |+λ+μ)/ _|y| √ t λ+_μ–|γ | exp  –|y _| t that ∇xχ· ∇x∂_yγ∂tk  ·, y_{, t} V p,β(K )+ (xχ)∂_yγ∂tk  ·, y_{, t} V p,β(K ) ≤ c λ+_j<σ t–k–(m+|γ |+σ)/ _|y| √ t λ+_j–|γ | exp  –|y _| t ≤ ct–k–(m+|γ |+σ)/ |y| |y_{| +}√_t λ+_–|γ | exp  –|y _| t . Consequently, by (), χ(·, t)∂_yγ∂tkRσ  ·, y_{, t} V p,β(K )≤ ct –k–(m+|γ |+σ)/ |y| |y_{| +}√_t λ+_–|γ |–ε ×  d(y) |y_| –εγ exp  –κ|y _| t ()

with a positive constant κ. Applying the estimate  |α|≤ x β–+|α|+m/p ∂_xαχ  x, t∂_yγ∂tkRσ  x, y, t ≤cχ∂_yγ∂tkRσ  ·, y_{, t} V p,β(K )

for p > m (cf. [, Lemma ..]), we obtain () for|α| ≤ .

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

cd



x≤ ρx≤ cd



x

with positive constants c and c (cf. [, Chapter VI, § .]). Moreover, ρ satisfies the

inequality

∂_xαρx ≤crx–|α|. ()

We consider the function

vx, y, t= χ  x, tρx∂xj∂ γ y∂ k tRσ  x, y, t

for ≤ j ≤ m. It follows from the equation xRσ = ∂tRσ that

where f= χρ∂xj∂ γ y∂tk+Rσ, f= (x(χρ))∂xj∂ γ y∂tkRσ and f= ∇x(χρ)· ∇x∂xj∂ γ y∂tkRσ.

Using () and (), we obtain f  ·, y_{, t} V_p_,β(K )≤ ct –k–(m+|γ |+σ)/ |y| |y_{| +}√_t λ+_–|γ |–ε d(y) |y_| –εγ exp  –κ|y _| t . Let χ(x, t) = χ (|x|/( √

t)). The inequalities|x(χρ)| ≤ cr–and|∇x(χρ)| ≤ c yield

_f ·, y_{, t} V_p_,β(K )+f  ·, y_{, t} V_p_,β(K ) ≤ cχ(·, t)∂ γ y∂ k tRσ  ·, y_{, t} V p,β(K ) ≤ ct–k–(m+|γ |+σ)/ |y| |y_{| +}√_t λ+_–|γ |–ε d(y) |y_| –εγ exp  –κ|y _| t

(see ()). Consequently by [, Theorem .], the function v = χρ∂xj∂

γ y∂tkRσ satisfies the estimate v·, y, t_V p,β(K )≤ cf+ f+ fV  p,β(K ) ≤ ct–k–(m+|γ |+σ)/ |y| |y_{| +}√_t λ+_–|γ |–ε d(y) |y_| –εγ exp  –κ|y _| t .

Applying Lemma . to the function u(x, y, t) = χ(x, t)∂_xα∂_yγ∂tkRσ(x, y, t) with an

arbi-trary multi-index α with length|α| = , we get sup x∈K dxm/p x β χ  x, t∂_xα∂_yγ∂tkRσ  x, y, t ≤ c  K rpβ ρ∇xχ∂_xα∂_yγ∂tkRσ  x, y, t p+ χ∂_xα∂_yγ∂tkRσ  x, y, t pdx /p ≤ cχρ∇x∂_yγ∂tkRσ  ·, y_{, t} V p,β(K )+ χ∂_yγ∂tkRσ  ·, y_{, t} V p,β(K )  ≤ ct–k–(m+|γ |+σ)/ |y| |y_{| +}√_t λ+_–|γ |–ε d(y) |y_| –εγ exp  –κ|y _| t

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

2 Asymptotics of solutions of the problem inD

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

 – β – m/p > λ–_ =  – m – λ+_ and  – β – m/p= λ+_j for j = , , . . . , () and q is an arbitrary real number > . Let G(x, y, t) be the Green function of the problem (), (). Furthermore, let x, y, t= (π t)(m–n)/exp  –|x _{– y}_| t

be the fundamental solution of the heat equation inRn–m_{. Then}

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

is the Green function of the problem (), (). We consider the solution

u(x, t) =  t –∞  DG(x, y, t – τ)f (y, τ) dy dτ () of the problem (), ().

We again denote by Gσ(x, y, t) the function () introduced in Section . In the sequel,

σ is an arbitrary real number such that

σ>  – β – m/p, λ+_j ∈ [ – β – m/p, σ ] for all j/ () and mj= _{σ– λ}+ j  = _{ – β – λ}+ j – m/p  for λ+_j <  – β – m/p. () Then Gσ(x, y, t) = G–β–m/p(x, y, t). Let χ be an infinitely differentiable function onR+=

(,∞) equal to one on the interval (, ) and vanishing on (, ∞). We define

χ  x, y= χ _|x| |y_| , χ  x, t, τ= χ  _|x| √ t– τ . Obviously, u=  + v, where (x, t) =  t –∞  DχχGσ  x, y, t – τ x, y, t – τf(y, τ ) dy dτ , () v(x, t) =  t –∞  D  Gx, y, t – τ– χχGσ  x, y, t – τ × x, y, t – τf(y, τ ) dy dτ . () We also consider the decomposition

u= + w, where =  λ+_j<–β–m/p u(m_j j)x, ∂t– x  Hj(x, t) () and Hj(x, t) =  t –∞  D χ  x, yχ  x, t, τcj  y, t – τ x, y, t – τf(y, τ ) dy dτ ()

is an extension of the function hj  x, t=  t –∞  Dcj  y, t – τ x, y, t – τf(y, τ ) dy dτ () with cjdefined by (). Our goal is to show that both remainders v and w are elements of

the space W_p,_,q;β(D × R). We start with the case p = q.

2.1 Estimates in weighted LpSobolev spaces

Let W_pl,l_,q;β(D × R) be the weighted Sobolev space with the norm (). Furthermore, let

W_pl,l_;β(D × R) = W_pl,l_,p;β(D × R), Lp;β(D × R) = Wp,;β(D × R).

In this subsection, we assume that f ∈ Lp;β(D × R), where p and β satisfy (). First, we

prove that  – ∈ W_p,_;β(D × R). This was shown in [, Corollary .] for the case ∂ ∈

C∞. In the case ∂∈ C,_{, we must keep in mind that the second-order derivatives of the}

eigenfunctions φjmust not be bounded. Then we have the estimate

∂_xαφj(ωx) ≤c x

–|α|d(x) |x_|

–εα

() for|α| ≤ , where εα=  for|α| ≤  and εαis an arbitrarily small positive real number if

|α| ≤ . However, this requires only a small modification of the proof in [].

Lemma . Suppose that f ∈ Lp,β(D × R). Then ∂xα∂tk( – )∈ Lp;β–+|α|+k(D × R) and

∂_xα∂_tk– _L

p;β–+|α|+k(D×R)≤ cf Lp,β(D×R)

for|α| ≤  and all k.

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

– = – λ+_j<σ  t –∞  Dχ  x, yu(m_j j)x, ∂t  , χ  cj  y, t – τ × x, y, t – τf(y, τ ) dy dτ , where [u(m_j j)(x, ∂t), χ] = u (mj) j (x, ∂t)χ – χu (mj)

j (x, ∂t) denotes the commutator of

u(m_j j)(x, ∂t) and χ. Obviously, the inequalities

x ≤ y and √t– τ≤ x ≤√t– τ are satisfied on the support of the kernel

Kj(x, y, t, τ ) = χ  x, yu(m_j j)x, ∂t  , χ  cj  y, t – τ x, y, t – τ. () Since, moreover, the eigenfunctions φjsatisfy the inequality () for|α| ≤ , we obtain

∂_xα∂_tkKj(x, y, t, τ ) ≤c(t – τ )–n/  d(x) |x_| –ε x –|α|–k–σ y σexp  –|y _|_+|x_{– y}_| (t – τ )

for|α| ≤ . Using Hölder’s inequality, we obtain ∂_xα∂_tk– (x, t) ≤c  d(x) |x_| –ε x –|α|–k–σA/pB/p, where A=  t–|x|/ t–|x|  D(t – τ ) –n/ _y pβ _{f(y, τ )} p_exp  –|y _|_+|x_{– y}_| (t – τ ) dy dτ and B=  t–|x_|_/ t–|x|  D |y|>|x|/ (t – τ )–n/ y p(σ –β)exp  –|y _|_+|x_{– y}_| (t – τ ) dy dτ.

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

B≤ c  t–|x_|_/ t–|x| (t – τ )p(σ –β)/dτ  |z_|>/ z  p(σ –β) exp  –|z _|  dz ×  Rn–m exp  –|z _|  dz, i.e., B≤ c|x|p(σ –β)+_{. Consequently,}  R  D x p(β–+|α|+k) ∂_xα∂_tk– (x, t) pdx dt ≤ c  R  D x –  d(x) |x_| –pε A(x, t) dx dt ≤ c  R  D y pβ f(y, τ ) pD(y, τ ) dy dτ , where D(y, τ ) =  τ+|y| τ  D √ t–τ <|x|<√t–τ x –  d(x) |x_| –pε (t – τ )–n/ × exp  –|y _|_+|x_{– y}_| (t – τ ) dx dt.

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

D(y, τ ) =  τ+|y| τ (t – τ )–exp  – |y _| (t – τ ) dt  K <|z_|< z –  d(z) |z_| –pε dz.

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. [, Lemma A.]).

Lemma . LetK be the integral operator (Kf )(x, t) =  t –∞  Rn K(x, y, t, τ )f (y, τ ) dy dτ ()

with a kernel K(x, y, t, τ ) satisfying the estimate |K| ≤ c(t – τ)–(n+–r)/  _|x| |x_{| +}√_{t– τ} a+r _|y_| |y_{| +}√_{t– τ} b _|x_|μ–r |y_|μ exp _{–κ|x – y|} t– τ ,

where κ > ,  < r ≤ , a + b > –m, –m_p – a < μ < m – m_p + b. ThenK is bounded on

Lp(Rn× R).

In the proof of the following assertion, we use another decomposition of the remainder

vas in [, Lemma .]. This allows us to apply directly the estimate in Theorem ..

Lemma . Let p and β satisfy the condition(). Furthermore, let v be the function (),

where f ∈ Lp;β(D × R),  < p < ∞. Then ∂xαv∈ Lp;β–+|α|(D × R) for |α| ≤  and



|α|≤

∂_xαv_L

p;β–+|α|(D×R)≤ cf Lp;β(D×R)

with a constant c independent of f. The same is true for the function w.

Proof Obviously, v=   j=  t –∞  Dj(x, y, t, τ )f (y, τ ) dy dτ , where (x, y, t, τ ) = χ  x, t, τ(G – Gσ)  x, y, t – τ x, y, t – τ, (x, y, t, τ ) =   – χ  x, t, τGx, y, t – τ x, y, t – τ and (x, y, t, τ ) =   – χ  x, yχ  x, t, τGσ  x, y, t – τ x, y, t – τ. We show that the integral operators with the kernels

K_j(α)(x, y, t, τ ) = x β–+|α| y –β∂_xαj(x, y, t, τ )

are bounded in Lp(D × R) for j = , ,  and |α| ≤ . Using Theorem ., we get

K_(α)(x, y, t, τ ) ≤c|x _|β–+|α| |y_|β (t – τ ) –(n+|α|)/_√|x| t– τ σ–|α| _|y| |y_{| +}√_{t– τ} λ+_–ε × exp  –κ|x – y|  t– τ ,

where ε is an arbitrarily small positive number. Applying Lemma . with r =  –|α|, μ = β,

a= σ – , b = λ+

 – ε, we conclude that the integral operator with the kernel K (α)

 (x, y, t, τ )

is bounded in Lp(D × R) for |α| ≤ .

Since|x| ≤ |x| +√t– τ≤ |x| on the support of K_(α), the estimate () implies K_(α)(x, y, t, τ ) ≤c|x _|β–+|α| |y_|β (t – τ ) –(n+|α|)/ |x| |x_{| +}√_{t– τ} a _|y_| |y_{| +}√_{t– τ} λ+_–ε × exp  –κ|x – y|  t– τ

with arbitrary real a. Thus, by Lemma ., the integral operator with the kernel K(x, y, t, τ )

is bounded in Lp(D × R) for |α| ≤ .

We consider the kernel K_(α). Since Gσ(x, y, t) has the form

Gσ  x, y, t= λ+_j<σ mj  k= cj,k x λ+_j+k y λ+j_φ j(ωx)φj(ωy)∂tkt –λ+_j–m/ exp  –|y _| t ,

we get the representation

K_(α)x, y, t, τ= λ+_j<σ mj  k= Kj,k(x, y, t, τ ), where Kj,k(x, y, t, τ ) ≤c|x _|β–+|α| |y_|β x  λ+_j+k–|α| y λ+j_{(t – τ )}–k–λ+j–n/_exp  –κ|x – y|  t– τ .

Here we used the fact that|y| ≤ |x| ≤ √t– τ on the support of the function ( – χ)χ.

The inequalities|y| ≤ |x| ≤ √t– τ and λ+

j + k≤ σ imply Kj,k(x, y, t, τ ) ≤c|x _|β–+|α| |y_|β (t – τ ) –(n+|α|)/_√|x| t– τ σ–|α| _|y| √ t– τ λ+_–σ × exp  –κ|x – y|  t– τ . It is no restriction to assume that σ < λ+

 + m – β – m/p in addition to () and ().

Therefore, we can apply Lemma . with r =  –|α|, a = σ –  and b = λ+_ – σ to the integral operator with the kernel Kj,k. It follows that the integral operator with the kernel

K_(α)(x, y, t, τ ) is bounded in Lp(D ×R) for |α| ≤ . Consequently, the integral operator with

the kernel K(α)(x, y, t, τ ) =   j= K_j(α)(x, y, t, τ ) =|x _|β–+|α| |y_|β   j= ∂_xαj(x, y, t, τ )

Furthermore, the assertions of [, Lemmas ., ., Theorem .] are also valid if ∂ is only of the class C,_{. The proof under this weaker assumption on  does not require}

any modifications of the method in []. We give here only the formulation of [, Theo-rem .].

Theorem . Let f ∈ Lp;β(D × R), where p and β satisfy the condition (). Then there

exists a solution of the problem(), () which has the form

u= 

λ+_j<–β–m/p

u(m_j j)x, ∂t– x



Hj(x, t) + w,

where w∈ W_p,_;β(D × R) and u(k)_j , mj, Hjare given by(), () and (), respectively. The

functions Hjdepend only on|x|, xand t and satisfy the estimates

∂_tk∂_xγHj_L p;β+λ+_j+k+|γ |–(D×R)≤ ck,γf Lp;β(D×R) () fork +|γ | >  – β – λ+_j – m/p and ∂_tk∂_xα∂_xγHj_L p;β+λ+_j+k+|α|+|γ |–(D×R)≤ ck,α,γf Lp;β(D×R) () for all k, α, γ ,|α| ≥ .

2.2 Weighted Lp,qestimates for the remainder

We assume now that f ∈ Lp,q;β(D × R) and consider the decomposition

u= + w

of the solution (), where is defined by (). Our goal is to show that w∈ W_p,_,q;β(D ×R) if p and β satisfy the condition (). For the proof, we will use the next lemma which follows directly from [, Theorem .].

Lemma . Suppose thatK is a linear operator on Lp(Rn× R) satisfying the following

conditions:

(i) KhLp(Rn×R)≤ chLp(Rn×R)for all h∈ Lp(Rn× R),

(ii) _|t–t

|>δ(Kh)(·, t)Lp(Rn)dt≤ c



Rh(·, t)Lp(Rn)dtfor all δ >  and for all functions

hwith support in the layer|t – t| < δ such that



Rh(x, t) dt≡  .

Then the inequality

KhLp,q(Rn×R)≤ chLp,q(Rn×R)

holds for arbitrary q,  < q < p. Here the constant c depends only on c, c, p and q.

The condition (ii) of the last lemma can be verified in some cases by means of the fol-lowing lemma (cf. [, Lemma ]).

Lemma . Suppose that the kernel of the integral operator() satisfies the estimate K(x, y, t, τ ) ≤ c δ (t – τ )(n+–r)/  |x_| |x_{| +}√_{t– τ} a+r _|y| |y_{| +}√_{t– τ} b d(x) |x_| –ε_d(y₎ |y_| –ε ×|x_|y|_μ_|μ–r exp _{–κ|x – y|} t– τ for t> t+ δ,|τ – t| ≤ δ, where κ > ,  ≤ r ≤ , a + b > –m, –m_p – a < μ < m –m_p + b, ≤ ε< /p, ≤ ε<  – /p. Then  ∞ t+δ (Kh)(·, t)_L p(D)dt≤ chLp,(D×R)

for all h∈ Lp,(D × R) with support in the layer |t – t| ≤ δ. Here, the constant c is

indepen-dent of tand δ.

It is more easy to estimate the remainder v = u – , where  is defined by (). For this reason, we estimate the difference  – first.

Lemma . Let  and be the functions() and (), respectively. If f ∈ Lp,q;β(D × R),

then ∂_tk∂α

x( – )∈ Lp,q;β–+k+|α|(D × R) and

∂_tk∂_xα– _L

p,q;β–+k+|α|(D×R)≤ ck,αf Lp;β(D×R)

for all k and α,|α| ≤ . Here, the constants ck,αare independent of f. In particular,  – ∈

W_p,_,q;β(D × R). Proof We have – = – λ+_j<σ  t –∞  DKj(x, y, t, τ )f (y, τ ) dy dτ ,

where Kjis given by (). LetKj,k,αbe the integral operator with the kernel

Kj,k,α(x, y, t, τ ) = x

β–+k+|α|

y –β∂_xα∂_tkKj(x, y, t, τ ),

where |α| ≤ . As was shown in the proof of Lemma ., this operator is bounded in

Lp(D × R). Now let h be a function in Lp,(D × R) with support in the layer |t – t| ≤ δ

satisfying the condition_Rh(x, t) dt≡ . Then

(Kj,k,αh)(x, t) =  t –∞  D  τ t ∂ ∂sKj,k,α(x, y, t, s) ds h(y, τ ) dy dτ . Analogously to the proof of Lemma ., we obtain

_∂s∂ Kj,k,α(x, y, t, s) ≤ c(t – s)––n/  d(x) |x_| –ε_|x_|β–σ – |y_|β–σ exp  –|x – y|  (t – s) ()

for|α| ≤ . Since |x| ≤ |x| +√t– s≤ |x| and |y| ≤ |y| +√t– s≤ |y| on the support of

Kj,k,α(x, y, t, s), we can append the factors

 _|x| |x_{| +}√_{t– s} a and  _|y| |y_{| +}√_{t– s} b

with arbitrary exponents a and b on the right-hand side of (). For t > t+ δ and|τ – s| <

|τ – t| < δ, we obviously have (t – τ)/ < t – s < (t – τ). Consequently,

 τ t ∂ ∂sKj,k,α(x, y, t, s) ds ≤ c δ (t – τ )+n/  d(x) |x_| –ε_|x_|β–σ – |y_|β–σ ×  |x_| |x_{| +}√_{t– τ} a _|y_| |y_{| +}√_{t– τ} b exp  –|x – y|  (t – s)

for t > t+δ and|τ –t| < δ, where a and b are arbitrary real numbers and ε is an arbitrarily

small positive real number. Hence, by Lemmas . and ., the operatorKj,k,αis bounded

in Lp,q(D × R) for  < q ≤ p.

We consider the operator ˜Kj,k,αwith the kernel

˜Kj,k,α(x, y, t, τ ) = Kj,k,α(y, x, –τ , –t) = (–)k|y _|β–+k+|α| |x_|β ∂ k τ∂ α yKj(y, x, –τ , –t).

It follows from the boundedness of the operator Kj,k,α in Lp that ˜Kj,k,α is bounded in

Lp(D × R), p= p/(p – ). Furthermore, one can check that

_tτ  ∂ ∂s˜Kj,k,α(x, y, t, s) ds ≤ c_{(t – τ )}δ+n/  d(y) |y_| –ε _|x_|σ–β |y_|σ–β+ ×  _|x| |x_{| +}√_{t– τ} a _|y_| |y_{| +}√_{t– τ} b exp  –|x – y|  (t – s)

with arbitrary a and b. Thus, as in the first part of the proof, we conclude that ˜Kj,k,α(and

therefore also the adjoint operator ofKj,k,α) is bounded in Lp,q(D × R) for  < q< p. This

means thatKj,k,αis bounded in Lp,q(D × R) for all p, q > . The lemma is proved. 

By means of Lemma ., it is also possible to prove the assertion of [, Theorem .] under the weaker assumption on  of the present paper.

Theorem . Let f ∈ Lp,q;β(D × R), where p and β satisfy the condition () and q is an

arbitrary real number,  < q <∞. Then there exists a solution of the problem (), () which

has the form

u= 

λ+_j<–β–m/p

u(m_j j)x, ∂t– x



Hj(x, t) + w,

where u(m_j j), Hjare given by() and (), respectively, and w∈ Wp,,q;β(D × R). The

estimate

∂_tk∂_xα∂_xγHj_L

p,q;β+λ+_j+k+|α|+|γ |–(D×R)≤ ck,α,γf Lp,q;β(D×R) ()

for all k, α, γ such that|α| ≥  or k + |γ | >  – β – λ+ j – m/p.

Proof We have to show that the integral operatorK(k,α)_{with the kernel}

K(k,α)(x, y, t, τ ) =|x _|β–+k+|α| |y_|β ∂ k t∂ α x(G – χχGσ)  x, y, t – τ x, y, t – τ

is bounded in Lp,q(D × R) for k + |α| ≤ . For p = q this is true by Theorem .. Let ,

, and be the same functions as in the proof of Lemma . and let

K_j(k,α)(x, y, t, τ ) = x β–+k+|α| y –β∂_xα∂_tkj(x, y, t, τ ). Then K(k,α)_{= K}(k,α)  + K (k,α)  + K (k,α)

 . We show that the operators K (k,α)

j satisfy the

condi-tion (ii) of Lemma .. Let h be a funccondi-tion in Lp,(D×R) with support in the layer |t–t| ≤ δ

satisfying the condition_Rh(x, t) dt =  for all x. Then  K(k,α) j h  (x, t) =  t –∞  D  τ t ∂ ∂sK (k,α) j (x, y, t, s) ds h(y, τ ) dy dτ .

Using Theorem ., we get ∂sK(k,α)(x, y, t, s) ≤ c(t – s)–k––(n+|α|)/ |x| |x_{| +}√_{t– s} σ–|α| _|y| |y_{| +}√_{t– s} λ+_–ε ×  d(x) |x_| –ε_|x_|β–+k+|α| |y_|β exp  –κ|x – y|  t– s . Thus, _tτ  ∂ ∂sK (k,α)  (x, y, t, s) ds ≤ c δ (t – τ )(n+k+|α|+)/  _|x| |x_{| +}√_{t– τ} σ–|α| _|y| |y_{| +}√_{t– τ} λ+_–ε ×  d(x) |x_| –ε_|x_|β–+k+|α| |y_|β exp  –κ|x – y|  (t – τ )

for t > t+ δ and|τ – t| < δ. Applying Lemma . with r =  – k – |α|, a = σ + k –  and

b= λ+ – ε, we conclude that  ∞ t+δ K(k,α) j h  (·, t)_L p(D)dt≤ chLp,(D×R) ()