Pointwise estimates and Lp

Pointwise estimates and L _p convergence rates to diffusion waves for p-system with damping

Weike Wang

and Tong Yang

*

aDepartment of Mathematics, Wuhan University, Wuhan, China

bDepartment of Mathematics, City University of Hong Kong, 83 Tat Chee Ace, Hong Kong, Hong Kong Received March 10, 2001; revised April 17, 2002

Abstract

By introducing a new approximate Green function, we obtain the pointwise estimates on the solutions of Euler equations with linear frictional damping, from which we can deduce the optimalL_p ð1pppþNÞconvergence rates to the nonlinear diffusion waves. The pointwise estimates andL_p ð1ppo2Þconvergence rates given in this paper are new.

r2002 Elsevier Science (USA). All rights reserved.

MSC:35K; 76N; 35L

Keywords: p-System with damping; Nonlinear diffusion waves; Approximate Green function; Pointwise estimates

1. Introduction

In this paper, we study the time-asymptotic behavior of solutions to thep-system with frictional damping, which in Lagrangian coordinates can be written as

vtux¼0;

utþpðvÞ_x¼ au; a>0; p⁰o0;

(

ð1:1Þ with the initial data

ðv;uÞðx;0Þ ¼ ðv0ðxÞ;u0ðxÞÞ:

*Corresponding author.

E-mail address:matyang@math.cityu.edu.hk (T. Yang).

1Supported in part by the RGC Competitive Earmarked Research Grant 9040534.

0022-0396/03/$ - see front matterr2002 Elsevier Science (USA). All rights reserved.

PII: S 0 0 2 2 - 0 3 9 6 ( 0 2 ) 0 0 0 5 6 - 6

Herevðx;tÞ>0 anduðx;tÞrepresent the specific volume and velocity, respectively, anda>0 is the frictional coefficient. The pressurepðvÞis assumed to be a smooth function ofvwithpðvÞ>0; p⁰ðvÞoC0o0 forvunder consideration.

It was proved in [3] that the solutions to (1.1) time asymptotically behave like those governed by the Darcy’s law inL2 andLNnorms if the solution is away from vacuum. That is, as t tends to infinity, the solution ðvðx;tÞ;uðx;tÞÞ of (1.1) approaches to the solutionðvðx;% tÞ;uðx;% tÞÞgoverned by the following system with the same end states asvðx;0Þat infinity:

%

vt¼ ¹_apðvÞ%_xx; pðvÞ% _x¼ au:% (

ð1:2Þ

The convergence rates inL_p ð2pppNÞnorms were studied in [9,10] by using the energy method and an approximate Green function.

However, the pointwise estimates and Lp ð1ppo2Þ convergence rate cannot be obtained by the method used in [10]. The reason can be explained as follows.

Since the large time behavior of the solution to the Euler equations with linear damping is governed by a parabolic equation derived by using Darcy’s law, the main idea in [10] is to consider a linear parabolic equation by putting the second-order derivative with respect to time variable to the right-hand side of the equation and treating it as a source. Notice that this second-order derivative term is linear. The advantage of this method is that the obtained linear equation becomes a heat equation with variable coefficient which depends on the diffusion wave.

Therefore, the approximate Green function can be defined easily which is a variation of the heat kernel. However, the second-order derivative with respect to time in the source requires higher-order derivatives on the solutions in the analysis.

Consequently, to close the decay estimates of the solution needs a crude decay estimates on the highest order derivatives of the solution with respect to time obtained by energy method. Since the energy method is crucial there, to our knowledge, this will not give any detailed pointwise estimates. But the estimates for L_p withpX2 can be obtained, cf. [10].

How to overcome the above difficulty is the main purpose of this paper. Here, we find a new way to construct an approximate Green function for the purpose of obtaining the pointwise estimates on the solutions so that a sharper result on the convergence rates in Lp ð1pppNÞ can be established. In particular, the L₁ estimate is obtained. Our idea is to keep the second-order derivative with respect to time variable on the left-hand side of the equation. Therefore, the linear equation now is of the second order, and of hyperbolic type with linear damping and variable coefficient. Notice that here the source term is nonlinear where the highest order derivative has a factor of lower-order derivatives. In the following analysis, we first give a detailed pointwise estimates of the Green function to the above linear hyperbolic equation by treating the variable coefficient as a parameter through Fourier analysis. Then we introduce a new method to define an approximate Green function to the original linearized equation. Based on

these analysis, we can estimate the error due to the approximate Green function and the integral of the product of the approximate Green function and the nonlinear source to obtain the desired estimates. We should mention that some of the techniques used here in the Fourier analysis on the Green function come from the study on Navier–Stokes equations and hyperbolic systems with relaxation in [6,12].

Furthermore, as a preparation for the study of planar diffusion wave in multi- dimensional space, we give estimates on higher-order derivatives on the solutions in this one-dimensional setting. For this, we make some interesting improvement on the analysis in closing the a priori estimates in the last section.

There are other interesting problems concerning (1.1) with partial results obtained so far. One of them is the behavior of the solutions when vacuum appears. For this problem, the equivalence between (1.1) and (1.2) is known for some special families of solutions by construction and the LN solutions by compensated compactness.

Another problem is the existence of global weak solutions with finite total variation.

This was proved for the case when the end states of the initial data at infinities coincide. But the problem without this restriction is still open. Interested readers please refer to [1,2,4,5,13] and reference therein. Some interesting works on the convergence to the Darcy’s law from the compressible Euler flow and the study of limiting behavior of nonhomogeneous hyperbolic systems when the relaxed equilibrium is described by parabolic equations are done in [7,8]. There are also some works on the case with boundary and the case for nonisentropic gas which we do not refer here because it is irrelevant to this paper.

The rest of the paper is arranged as follows: The main theorems are stated in Section 2. In Section 3, we will first study the Green function with a parameter in details by Fourier analysis. In Section 4, the new approximate Green function is introduced and the proof of the main theorems is given in the last section.

Throughout this paper,Candbwill be used to denote a generic positive large and small constants, respectively.

2. The main result

As in [3,9,10], we are interested in the large time behavior of the solution of (1.1) with initial data satisfying

ðv;uÞðx;0Þ-ðv₇;u₇Þ as x-7N ð2:1Þ withv_þ not necessarily being equal to v: Denote the self-similar solution, i.e. the diffusion wave, of (1.2) in the form ofjð^pffiffiffiffiffiffi_tþ1^x Þbyvðx;% tÞwith the same end states at infinities asvðx;0Þ;i.e.,

%

vðx;tÞ ¼f x ffiffiffiffiffiffiffiffiffiffi tþ1 p

; vð7% N;tÞ ¼v₇: ð2:2Þ

Also set

%

uðx;tÞ 1

apðvÞ% _x: ð2:3Þ

From [3,10], we know thatvðx;% tÞhas the following exponential decay property in_tþ1^x2: jvðx;% tÞ vþj_x2o1þtþ jvðx;tÞ vj_x2>1þt

pCjvþvje^{Ca x}

2

1þt;

j@^k_xvðx;% tÞjpCjvþvjð1þtÞ^k=2e^{Ca x}

2

1þt: ð2:4Þ

Since the u component of the solution is expected to decay exponentially in t at x¼7N;an auxiliary functions ð˜u;vÞ* was introduced in [3] as follows:

uðx;˜ tÞ ¼e^at uþ ðuþuÞ Z x

N

m₀ðyÞdy

ð2:5Þ and

*

vðx;tÞ ¼uþu

a e^atm0ðxÞ; ð2:6Þ

wherem0ðxÞis a smooth function with compact support satisfying

Z N

N

m₀ðxÞdx¼1:

It is easy to check thatð˜u;vÞ* satisfies

*

v_tu˜_x¼0;

˜

ut¼ au:˜ (

Let the initial datav0ðxÞbe a small perturbation of a diffusion wavevðx;% 0Þ:We are going to study how the solution behaves pointwise asttends to infinity. As in [3], there exists a shiftx0 such that the initial data satisfies:

Z N

N

ðv0ðyÞ vðy% þx₀;0Þ vðy;* 0ÞÞdy¼0:

By the first equation of (1.1), this implies that the functionVðx;tÞdefined by Vðx;tÞ ¼

Z x N

ðvðy;tÞ vðy% þx₀;tÞ vðy;* tÞÞdy; ð2:7Þ

satisfiesVð7N;tÞ ¼0:Herex0 is a constant uniquely determined by

Z N

N

ðvðx;0Þ vðx% þx₀;0ÞÞdx¼uþu

a : ð2:8Þ

For later use, denote

Uðx;tÞ ¼uðx;tÞ uðx% þx₀;tÞ uðx;˜ tÞ; ð2:9Þ V0ðxÞ ¼Vðx;0Þ and U0ðxÞ ¼Uðx;0Þ ¼Vtðx;0Þ: From (1.1), (1.2), (2.5)–(2.9), we have

VtU¼0;

U_tþ ðpðVxþv%þvÞ * pðvÞÞ% _xþaU¼¹_apðvÞ% _xt; ðV;UÞj_t¼0 ðV0;U0ÞðxÞ-0 as x-7N:

8>

<

>: ð2:10Þ

By linearizing the second equation of (2.10) about v;% we have the following system:

V_tU ¼0;

Utþ ðp⁰ðvÞV% xÞ_xþaU ¼F1þF2;

ðV;UÞj_t¼0 ðV0;U₀ÞðxÞ-0 as x-7N; 8>

<

>: ð2:11Þ

whereFjðx;tÞ ¼ ðF˜jðx;tÞÞ_x ðj¼1;2Þ;and F˜₁ðx;tÞ ¼1

apðvÞ%_t;

F˜₂ðx;tÞ ¼ ðpðVxþv%þvÞ * pðvÞ % p⁰ðvÞV% xÞ: ð2:12Þ From now on, we will study system (2.11). First, the following theorem is a direct consequence of the a priori estimates obtained in [3,9], and we omit its proof for brevity.

Theorem 2.1. For sufficiently smalle0;if

juþuj þ jvþvj þ jjV0jj_Hnþ3þ jjU0jj_Hnþ2pe0

then there exists a global in time solutionðVðx;tÞ;Uðx;tÞÞof(2.10) satisfying jjVðtÞjj_Hnþ3þ jjUðtÞjj_Hnþ2pCe₀; ð2:13Þ where n is a positive integer.

Our main results of this paper are stated in the following.

Theorem 2.2. For mX3;if V0ðxÞand U0ðxÞsatisfy the condition of Theorem2.1for n¼2m;and for some sufficiently smalle₀ and any positive number N>¹₂;

juþuj þ jvþvjpe0; X

hpm

j@_x^hV0ðxÞj þ X

hpm2

j@^h_xU0ðxÞjpe0ð1þx²Þ^N; ð2:14Þ then there exists a global in time solutionðVðx;tÞ;Uðx;tÞÞof(2.10), which satisfies

j@_x^kVðx;tÞjpCe₀ð1þtÞ^ðkþ1Þ=2B_Nðx;tÞ;

j@_x^lUðx;tÞj j@t@_x^lVðx;tÞjpCe₀ð1þtÞ^ðlþ3Þ=2B_Nðx;tÞ; ð2:15Þ where kpm1;lpm3and

BNðx;tÞ ¼ 1þ x² 1þt

^N

:

That is,in the original functionðu;vÞ; we have

j@^k1_x ðvðx;tÞ vðx% þx0;tÞ vðx;* tÞÞjpCe0ð1þtÞ^ðkþ1Þ=2BNðx;tÞ;

j@^l_xðuðx;tÞ uðx% þx0;tÞ uðx;˜ tÞÞjpCe0ð1þtÞ^ðlþ3Þ=2BNðx;tÞ:

Remark 2.1. If the initial perturbation decays exponentially in x; then the perturbation decays exponentially inxfor any time. In fact, if

juþuj þ jvþvjpe₀; X

hpm

j@_x^hV₀ðxÞj þ X

hpm2

j@^h_xU₀ðxÞjpe₀e^Cx2; ð2:16Þ

then, forkpm1; lpm3;ðVðx;tÞ;Uðx;tÞÞsatisfies:

j@_x^kVðx;tÞjpCe0ð1þtÞ^ðkþ1Þ=2e^bx

2

1þt;

j@_x^lUðx;tÞjpCe0ð1þtÞ^ðlþ3Þ=2e^bx

2

1þt: ð2:17Þ

As a corollary of Theorem 2.2, we have the following theorem.

Theorem 2.3. For pA½1;N and mX2; if V0ðxÞand U0ðxÞ satisfy the conditions of Theorem 2.1 for n¼2m and (2.14), then there exists a global in time solution

ðVðx;tÞ;Uðx;tÞÞof(2.10),which satisfies:

jj@_x^kVðtÞjj_LppCe0ð1þtÞ^12ð1þk1pÞ; jj@_x^lUðtÞjj_LppCe₀ð1þtÞ

1 2ð3þl1

pÞ

; ð2:18Þ

where kpm1; lpm3:That is,for the functions ðu;vÞ;we have jj@_x^k1ðvðx;tÞ vðx% þx0;tÞ vðx;* tÞÞjj_LppCe0ð1þtÞ^12ð1þk1pÞ; jj@_x^lðuðx;tÞ uðx% þx₀;tÞ uðx;˜ tÞÞjj_LppCe₀ð1þtÞ

1 2ð3þl1

pÞ

:

3. Green function with a parameter

In this section, we first consider a linear partial differential equation with constant coefficient and parameterlas follows:

V_ttlV_xxþaV_t¼0: ð3:1Þ IfGⁿðl;x;tÞis the Green function of (3.1), then it satisfies:

G_ttⁿðl;x;tÞ lðGⁿðl;x;tÞÞ_xxþaGⁿ_tðl;x;tÞ ¼0;

Gⁿðl;x;0Þ ¼0; Gⁿ_tðl;x;0Þ ¼dðxÞ; ð3:2Þ wherelis a parameter satisfyingC₁>l>C₀>0 with constantsC₁andC₀;anddis the Dirac function.

We now establish estimates of Green function of (3.2) which will be used to obtain the pointwise estimates for the approximate Green function. This can be done by Fourier analysis as in [6] for the Navier–Stokes equations and [12] for hyperbolic system with relaxation.

The Fourier transform offðxÞis fðx;ˆ tÞ ¼

Z

R

fðx;tÞe^ixxdx;

and the inverse Fourier transform is

fðx;tÞ ðF¹fÞðx;ˆ tÞ ¼ ð2pÞ¹ Z

R

fðx;ˆ tÞe^ixxdx:

The symbol of the operator for Eq. (3.1) is

t²þlx²þat: ð3:3Þ

Here,tandxcorrespond to_@t^@ andD_xrespectively, andD_x¼¹_i@^@_x:It is easy to see that the eigenvalues of (3.3) fortare

t¼m₇ðxÞ ¹₂ða7 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a²4lx² q

Þ: ð3:4Þ

From (3.2), we have by direct calculation that

Gˆⁿðl;x;tÞ ¼ ða²4lx²Þ¹⁼²ðe^mþðxÞte^mðxÞtÞ: ð3:5Þ For convenience, we sometimes decompose Gˆⁿðl;x;tÞ ¼Gˆ^þðl;x;tÞ þGˆðl;x;tÞ;

where

Gˆ⁷ðl;x;tÞ ¼7m¹₀ e^m7ðxÞt; m₀¼ ða²4lx²Þ¹⁼²: Notice that

@_lðGˆ⁷Þðl;x;tÞ ¼ ð2x²m²₀ 8x²tm¹₀ ÞGˆ⁷ðl;x;tÞ: ð3:6Þ For Gˆⁿðl;x;tÞ and Gˆ⁷ðl;x;tÞ; we first have the following lemma from standard consideration.

Lemma 3.1. Gˆⁿðl;x;tÞis a holomorphic function ofx:Gˆ⁷ðl;x;tÞalso is a holomorphic function ofxexcept isolated singularitiesx¼7₂^paffiffi_l:

In the following, we are going to obtain some detailed properties of the Green functionGⁿðl;x;tÞ:Denote

Gⁿðl;x;tÞ ¼ Z e

e

Gˆⁿðl;x;tÞe^ixxdxþ Z

epjxjpR

Gˆⁿðl;x;tÞe^ixxdx þ

Z

RpjxjpN

Gˆⁿðl;x;tÞe^ixxdx

¼:Gⁿ₁ðl;x;tÞ þGⁿ₂ðl;x;tÞ þG₃ⁿðl;x;tÞ:

ForG⁷;the correspondingG⁷_j ðj¼1;2;3Þcan be defined in the same way.

Lemma 3.2. For sufficiently smalle;there exist positive constants C and b;such that j@_t^l@^h_lD^k_xG₁^þðl;x;tÞjpCð1þtÞ^{ð1þ2lþkÞ=2}e^bx2=ð1þtÞ; ð3:7Þ

j@^l_t@_l^hD^k_xG₁ðl;x;tÞjpCe^at=2e^bx2=ð1þtÞ ð3:8Þ for any non-negative integers l;h and k:

Proof. SinceGˆ₇is a holomorphic function ofxat the origin. Thus, we can move the path of integration from½e;etoSðe;e;cÞ;where

Sða;b;cÞ ¼ fxjRex¼a;Imx: 0-cg [fxjImx¼c;Rex:a-bg [fxjRex¼b;Imx:c-0g:

Letx¼zþiZ;zandZare real numbers. Sincejxjpeande>0 is sufficiently small, we have

ReðmþtþixxÞ ¼ ða¹lZ²txZÞ a¹lz²tþOð1ÞðZ⁴þz⁴Þt;

ReðmtþixxÞ ¼ atþ ða¹lZ²txZÞ þa¹lz²tþOð1ÞðZ⁴þz⁴Þt:

We only prove (3.7) in the following since the corresponding estimate onG₁can be obtained in the same way. In fact, from the above identities, we have

@_t^l@_l^hD^k_x Z e

e

Gˆ^þðl;x;tÞe^ixxdx

¼Oð1Þ Z e

e

e^ða1lZ2txZÞe^a1lz2te^{Oð1ÞðZ4þz4Þt}ðZ²þz²Þ^k=2þl ððZ²þz²Þð1þtÞÞ^hdz

þOð1Þ Z c

0

e^ða1lZ2txZÞe^a1lz2te^{Oð1ÞðZ4þz4Þt}ðZ²þz²Þ^k=2þl ððZ²þz²Þð1þtÞÞ^hdZ:

For ^ajxj_2lto^e₂; we let c¼_2lt^ax; then ða¹lZ²txZÞj_Z¼c¼^ax_2lt²: Denote the left-hand of (3.7) byR^þ_l;h;k;then

R^þ_l;h;kpCð1þtÞ^k=2le^ax

2

16lt

Z e e

e^4a1lz2te^{Oð1ÞðZ4þz4Þt} ððZ²þz²Þð1þtÞÞ^k=2þhþldz

þCð1þtÞ^k=2le^4a1le2t Z _2lt^ax

0

e^{ð4a1lZ2tjxjZÞ}e^{Oð1ÞðZ4þz4Þt} ððZ²þz²Þð1þtÞÞ^k=2þlþhdZ

pCð1þtÞ^ðkþ1Þ=2lðe^bx2=ð1þtÞþe^btÞ:

For^ajxj_2ltX^e

2;we letc¼^e₂signx;then R^þ_l;h;kpCð1þtÞ^k=2le^ejxj=4

Z e e

e^4a1lz2te^{Oð1ÞðZ4þz4Þt} ððZ²þz²Þð1þtÞÞ^k=2þhþldz

þCð1þtÞ^k=2le^3a1le2t Z e=2

0

e^jxjZe^{Oð1ÞðZ4þz4Þt} ððZ²þz²Þð1þtÞÞ^k=2þlþhdZ

pCð1þtÞ^ðkþ1Þ=2lðe^btþe^b1jxjÞ pCð1þtÞ^ðkþ1Þ=2le^bt:

This completes the proof of the lemma. &

Lemma 3.3. For fixedeand R;there exist positive constants b and C;such that j@^l_t@_l^hD^k_xGⁿ₂ðl;x;tÞjpCe^bt: ð3:9Þ

Proof. For any fixede andR; we can choose b1>0 sufficiently small such that if epjxjpR

je^mþðxÞtjpCe^b1t: As fore^mðxÞt;since RemðxÞp^a₂;we have

je^mðxÞtjpCe^at=2: Thus ifjm₀ðxÞjXe1>0;one has

jGˆⁿ₂ðl;x;tÞjpCe^b2t:

Notice that e^mþðxÞte^mðxÞt¼Oð1Þe^at=2m₀ðxÞt if jm₀ðxÞjoe₁; and e₁ is sufficiently small, we also get above inequality. Thus, for any fixedeandR;we deduce that

jGⁿ₂ðl;x;tÞjpCe^b2t Z

epjxjpR dxpCe^b2t:

By applying@_t^l@_l^hD^k_x on both sides of (3.1), we can prove similarly that j@_t^l@^h_lD^k_xG₂ⁿðl;x;tÞjpCe^bt

Z

epjxjpRð1þ jxjÞ^kþlþhdxpCe^bt for someb>0;which implies (3.9). &

Now we considerGⁿ₃ðl;x;tÞ:By letting

m_Z¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z4l=a²

p ;

and then taking the Taylor expansion ofm_Z inZ;we have m_Z¼i2 ffiffiffi

pl

a þiZ a ffiffiffil p

þOð1ÞðZ²Þ: ð3:10Þ

Since

m₇ðxÞ ¼¹₂ða7xa ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x²4l=a² q

Þ;

whenjxjis sufficiently large, we arrive at m₇ðxÞ ¼1

2 a72i ffiffiffi pl

x7i a² ffiffiffil

p xþOðjxj³Þ

:

This implies that

e^m7ðxÞt ¼e^at=2e^7ipffiffi_lxt e⁷ⁱ

a²t 2^pffiffi_lx

ð1þOð1Þðjxj³ÞtÞ: ð3:11Þ Eq. (3.11) will be used in the proofs of the following lemmas.

Lemma 3.4. For R sufficiently large,there exists a constant C>0;such that jGⁿ₃ðl;x;tÞjpCe^at=4: ð3:12Þ Moreover,there exists a distribution

K_l;k;hⁿ ðl;x;tÞ ¼e^at=2lþkþh1X

j¼0

ðq^þ_jþhðtÞd^ðlþkþhjÞðxþ ffiffiffi pl

tÞ þq_jþhðtÞd^ðlþkþhjÞðx ffiffiffi

pl tÞÞ;

such that when lþkX1;we have

j@_t^l@_l^hD^k_xGⁿ₃ðl;x;tÞ K_l;k;hⁿ ðl;x;tÞjpCe^at=4; ð3:13Þ where q⁷_jþhðtÞ; j¼0;y;lþk are polynomials of t with degrees not greater than jþh correspondingly.

Proof. Since forjxjlarge enough,

m¹₀ ðxÞ ¼ i 2 ffiffiffi

pl

xþOðjxj²Þ;

Eq. (3.11) gives

Gˆⁿðl;x;tÞ ¼ i 2 ffiffiffi pl

xe^at=2ðe^ipffiffi_lxt

e^ipffiffi_lxt Þ

þOðjxj²Þð1þtþOðjxj¹ÞtÞe^at=2: Thus

jGⁿ₃ðl;x;tÞjpCe^at=2 Z

RpjxjpN

i 2 ffiffiffi pl

xðe^ipffiffi_lxt

e^ipffiffi_lxt Þe^ixxdx

þCe^at=4 pCe^at=2

Z N

R

sinðx ffiffiffi pl

tÞxsinðxþ ffiffiffi pl

tÞx

x dx

þCe^at=4

pCe^at=4;

which implies (3.12). For the general case, since

@_t^l@_l^hx^kGˆⁿðl;x;tÞ

¼Oð1Þe^at=2 e^ipffiffi_lx ^lþkþhX

j¼0

q^þ_jþhðtÞjxj^jþq^þ_lþkþ2hþ1ðtÞjxj^lkh1

!

þe^ipffiffi_lx ^lþkþhX

j¼0

q_jþhðtÞjxj^jþq_lþkþ2hþ1ðtÞjxj^lkh1

!!

jxj^1þkþlþh if we set

Kˆⁿ_l;k;hðl;x;tÞ ¼Oð1Þe^at=2

Z N

N

e^ipffiffi_lx ^lþkþh1X

j¼0

q^þ_jþhðtÞjxj^j

!

þe^ipffiffi_lx ^lþkþh1X

j¼0

q_jþhðtÞjxj^j

!!

jxj^1þkþhþle^ixxdx;

where q⁷_mðtÞ is polynomial of t with degree not greater than m; and jq^þ_jþhðtÞj ¼ jq_jþhðtÞjthen we can deduce that

j@_t^lD^k_x@_l^hG₃ⁿðl;x;tÞ K_l;k;hⁿ ðl;x;tÞj pZ

jxjXR

ð@_t^lx^k@^h_lGˆⁿðl;x;tÞ Kˆⁿ_l;k;hðl;x;tÞÞe^ixxdx

þ

Z

jxjpR

Kˆⁿ_l;k;hðl;x;tÞdx

pCe^at=3 Z

jxjXR

ðe^ipffiffi_lx

q^þ_lþkþ2hðtÞx¹þe^ipffiffi_lx

q_lþkþ2hðtÞx¹Þe^ixxdx

þe^at=3

Z

jxjXR

e^Oðjxj1ÞtOðjxj²Þe^ixxdx

þCe^at=4: Similar to the proof of (3.12), we get (3.13). &

Lemma 3.5. For M>0 sufficiently large, there exists a constant C>0; such that whenjxj=ð1þtÞXM;we have

j@_t^lD^k_x@_l^hGⁿðl;x;tÞ K_l;k;hⁿ ðl;x;tÞjpCe^at=4e^{jxj2=ð1þtÞ}: ð3:14Þ

Proof. By Lemma 3.1, we can move the path of integration toSðN;N;2x=ð1þ tÞÞ:Then

Gⁿðl;x;tÞ ¼C

Z N

N

Gˆⁿðl;zþ2ix=ð1þtÞ;tÞe^ixze^2x2=ð1þtÞdz:

Sincejxj=ð1þtÞ>M andM is sufficiently large, we know thatjxj ¼ jzþ2ix=ð1þ tÞj is large enough and then (3.11) can be applied to variable zþ2ix=ð1þtÞ:

Combining this observation with the above identity imply that Gⁿðl;x;tÞ

¼Oð1Þe^at=2 e^ð2x2þpffiffi_{lxtÞ=ð1þtÞ}Z N N

ðzþ2ix=ð1þtÞÞ¹e^iðxþpffiffi_ltÞz dz

þe^ð2x2pffiffi_{lxtÞ=ð1þtÞ}Z N N

ðzþ2ix=ð1þtÞÞ¹e^iðxpffiffi_ltÞz dz

þOð1Þe^at=2e^2x2=ð1þtÞ

Z N

N

Oððz²þ ð2x=ð1þtÞÞ²Þ¹ð1þte^Oðz2þð2x=ð1þtÞÞ²ÞtÞÞdz:

Since

1

zþ2ix=ð1þtÞ¼1

z 2ix=ð1þtÞ

z²þ ð2x=ð1þtÞÞ² ð2x=ð1þtÞÞ² z²þ ð2x=ð1þtÞÞ²

1 z; we have

Z N

N

ðz72ix=ð1þtÞÞ¹e^iðxþpffiffi_ltÞz dz

pC jx7 ffiffiffi pl

tj þ jxj

1þtþ jxj² ð1þtÞjx7 ffiffiffi

pl tj

! :

In summary, by noticing thatjxj=ð1þtÞ>M;we have jGⁿðl;x;tÞjpCe^at=2e^x2=ð1þtÞ

jx7 ffiffiffi pl

tj þ jxj

1þtþ jxj² ð1þtÞjx7 ffiffiffi

pl

tjþ1þe^t=M

!

pCe^at=2e^bx2=ð1þtÞ:

Thus, we have (3.14) forl¼k¼h¼0:The general case can be proved similarly as in Lemma 3.5. &

Combining Lemmas 3.2 with 3.5, we have

Theorem 3.1. For non-negative integers l; k and h; there exists a distribution K_l;k;hⁿ ðl;x;tÞ;such that

j@_t^lD^k_x@_l^hGⁿðl;x;tÞ K_l;k;hⁿ ðl;x;tÞjpCð1þtÞ^{ð1þkþ2lÞ=2}e^bx2=t: ð3:15Þ

Proof. Ifjxj=ð1þtÞpM;we have tpM²x²

1þt; jx7 ffiffiffi pl

tjpðMþ ffiffiffi pl

Þð1þtÞ:

Thus, (3.15) holds for jxj=ð1þtÞpM by Lemmas 3.2–3.4. For jxj=ð1þtÞ>M;

Lemma 3.5 implies (3.15), and this completes the proof. &

4. Approximate Green function

In this section, we construction the approximate Green function for the unknown functionVðx;tÞ:

V_tt ðaðx;tÞVxÞ_xþaV_t¼F; ð4:1Þ where aðx;tÞ ¼ p⁰ðvðx;% tÞÞ>C₀ >0; F ¼F₁þF₂: The constructed approximate Green functionGðx;t;y;sÞfor (4.1) satisfies the basic requirement

Gðx;t;y;tÞ ¼0; G_sðx;t;y;tÞ ¼dðyxÞ: ð4:2Þ By multiplying (4.1) whose variables are now changed toðy;sÞbyGand integrating over the regionðy;sÞAR ð0;tÞ;we have

Vðx;tÞ

¼

Z N

N

Gsðx;t;y;0ÞV0ðyÞdyþ

Z N

N

Gðx;t;y;0ÞðaV0ðyÞ þVtðy;0ÞÞdy þ

Z t 0

Z N

N

Gðx;t;y;sÞFðy;sÞdy ds þ

Z t 0

Z N

N

ððGssaG_sÞðx;t;y;sÞ

ðaðy;sÞGyðx;t;y;sÞÞ_yÞVðy;sÞdy ds: ð4:3Þ Ifaðy;sÞis a constant andGis a Green function of (4.1), then we know that the last integral of (4.3) is equal to zero. However, it now depends on the profile of the diffusion wave. Therefore, we can only try to minimize the termGssaGs ðaGyÞ_y: For this purpose, we compare it with the linear partial differential equation with constant coefficient and a parameter l; i.e. (3.1). From the discussion of the last section, the Green function Gⁿðl;x;tÞ satisfies (3.2) and Theorem 3.1. Define an approximate Green function for (4.1) as follows:

Gðx;t;y;sÞ ¼Gⁿðaðy;sðt;sÞÞ;xy;tsÞ; ð4:4Þ wheresðt;sÞAC³ðR²Þand

sðt;sÞ ¼ s; s>t=2þ1;

t=2; spt=2:

(

Moreover, we can choosesðt;sÞto be smooth whensAðt=2;t=2þ1Þ;such that X

1pl1þl2p3

j@_t^l1@_s^l2sðt;sÞjpC: ð4:5Þ

Whent>1;we also obtain

sðt;sÞ¹p C

1þt: ð4:6Þ

It is clear that the approximate Green function defined in (4.4) satisfies the condition (4.2), and

G₃₃ⁿðaðy;sÞ;xy;tsÞ aðy;sÞG₂₂ⁿðaðy;sÞ;xy;tsÞ

þaGⁿ₃ðaðy;sÞ;xy;tsÞ ¼0; ð4:7Þ whereGⁿ_j ¼Gⁿ_jðaðy;sÞ;xy;tsÞdenotes the partial derivative with respect to the jth ðj¼1;2;3;4Þvariable of Gⁿ:

The functionGðx;t;y;sÞis not symmetric with respect to the variablesx;tandy;s:

Instead, we have the following relations:

@_xG¼ @yGþ@_aðGⁿÞay;

@_tG¼ @sGþ@_aðGⁿÞðasþa_tÞ; ð4:8Þ whereay;at andasrepresent the derivatives of awith respect toy;t;s;respectively.

Sinceaðy;sðt;sÞÞ ¼ p⁰ðvðy;% sðt;sÞÞÞ;it follows from [10] that

j@_t^l1@_s^l2@_y^kaðy;sðt;sÞÞjpCe₀ð1þtÞ^{ðk=2þl1þl2Þ} ð4:9Þ for kþl1þl2X1: Now we give a pointwise estimate to the approximate Green function Gðx;t;y;sÞ: In fact, from Theorem 3.1 and the above discussion, it is straightforward to have the following theorem.

Theorem 4.1. For l¼0;1and a positive integer k;there exists a distribution K_l;kðx;y;t;sÞ

¼e^{aðtsÞ=2 lþk1}X

j¼0

ð˜q^þ_j ðtsÞd^ðlþkjÞðxyþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðy;sðt;sÞÞ

p Þ

þq˜_jðtsÞd^ðlþkjÞðxy ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðy;sðt;sÞÞ

p Þ; ð4:10Þ

such that

X

l1þl2¼l;k1þk2¼k

ð@_t^l1@^l_s²@_x^k1@^k_y²Gðx;t;y;sÞ K_l;kðx;y;t;sÞÞ

pCðð1þtÞ^{ð1þkþ2lÞ=2}þ ð1þtsÞ^{ð1þkþ2lÞ=2}Þe^{bðxyÞ2=ðtsÞ}; ð4:11Þ

where l¼0;1 and ˜q⁷_j ðj¼0;y;lþk1Þ are polynomials of t with degrees not greater than j correspondingly.

Given a functiongðy;sÞ;andkX1;from (4.8) and (4.9), we have Z b

a

Z N

N

@_x^hþk@^l_tGðx;t;y;sÞgðy;sÞdy ds

¼ Z b

a

Z N

N

@^h_x@_t^lGðx;t;y;sÞ@_y^kgðy;sÞdy ds

þOð1Þe0

X

k⁰ok

Z b a

Z N

N

ðð1þtsÞ^{ð1þ2lþhÞ=2} ð1þtÞ^ðkk0ÞÞe^{bðxyÞ2=ðtsÞ}

þKl;hðx;y;t;sÞÞ@_y^k0gðy;sÞdy ds: ð4:12Þ Set the error due to the approximate Green function byR_G;

R_GG_ssðx;t;y;sÞ aG_sðx;t;y;sÞ ðaðy;sÞGyðx;t;y;sÞÞ_y: Then (4.7) implies that

R_G¼ ðaðy;sÞ aðy;sÞÞGⁿ₂₂þ2G₁₃ⁿa_sðy;sÞ þGⁿ₁₁ðasðy;sÞÞ² þG₁ⁿassðy;sÞ aðy;sÞð2G₁₂ⁿayðy;sÞ þGⁿ₁₁ðayðy;sÞ²Þ

þG₁ⁿa_yyðy;sÞ aGⁿ₁a_sðy;sÞ: ð4:13Þ Notice that

jaðy;sÞ aðy;sÞj

¼ Oð1Þe0ð1þtsÞ¹⁼²ð1þsÞ¹⁼²e

by²

1þt; sot=2þ1;

0; sXt=2þ1:

8<

: ð4:14Þ

It follows from Theorem 4.1, (4.9) and (4.13) that there exists a distribution KGðx;y;t;sÞ

¼ ðaðy;sÞ aðy;sÞÞK_2;0;0ⁿ þ2K_0;1;1ⁿ asðy;sÞ þK_0;0;2ⁿ ðasðy;sÞÞ² þK_0;0;1ⁿ a_ssðy;sÞ aðy;sÞð2K_1;0;1ⁿ a_yðy;sÞ þK_0;0;2ⁿ ðayðy;sÞ²Þ þK_0;0;1ⁿ ayyðy;sÞ aK_0;0;1ⁿ asðy;sÞ;

such that

R_G¼Oð1Þe0Yðy;t;sÞðð1þtsÞ¹⁼²e^{bðxyÞ2=ðtsÞ}þK_Gðx;y;t;sÞÞ; ð4:15Þ where

Yðy;t;sÞ ¼ ðð1þtÞ¹þ ð1þtsÞ¹⁼²ð1þsÞ¹⁼²Þe

by² 1þt:

Furthermore, it can be shown by straightforward calculation that there exists a distribution

KG;l;kðx;y;t;sÞ

¼e^{aðtsÞ=2 lþkþ1}X

j¼0

ðQ^þ_j ðtsÞd^ðlþkþ2jÞðxyþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðy;sðt;sÞÞ

p Þ

þQ_j ðtsÞd^ðlþkþ2jÞðxy ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi aðy;sðt;sÞÞ

p Þ;

such that

X

l1þl2¼l;k1þk2¼k

@^l_t¹@_s^l2@_x^k1@_y^k2R_Gðx;t;y;sÞ

¼Oð1Þe0Yðy;t;sÞðð1þtsÞ^1þkþ2l2 e^{bðxyÞ2=ðtsÞ}

þKG;l;kðx;y;t;sÞÞ: ð4:16Þ

Here l¼0;1 and Q⁷_j ðj¼0;y;lþkþ1Þ are polynomials of t with degrees not greater thanj correspondingly.

In deducing the pointwise estimates, we need to perform integrations by parts for R_G and a given functiongðy;sÞ:For later use, we list the following two identities:

Z b a

Z N

N

@_x^kR_Gðx;t;y;sÞgðy;sÞdy ds

¼ Z b

a

Z N

N

RGðx;t;y;sÞ@^k_ygðy;sÞdy ds þOð1Þe0

X

lok

Z b a

Z N

N

Yðy;t;sÞ

ðð1þtÞ^ðklÞ=2Þð1þtsÞ¹⁼²e^{bðxyÞ2=ðtsÞ}

þKGðx;y;t;sÞÞ@_y^lgðy;sÞdy ds ð4:17Þ

and Z b

a

Z N

N

@_t@_x^kR_Gðx;t;y;sÞgðy;sÞdy ds

¼ Z b

a

Z N

N

RGðx;t;y;sÞ@s@_y^kgðy;sÞdy ds

Z N

N

@^k_xRGðx;t;y;sÞgðy;sÞdyj^s¼b_s¼a

þOð1Þe0

X

lok

Z b a

Z N

N

Yðy;t;sÞðð1þtÞ^{ð2þklÞ=2Þ}ð1þtsÞ¹⁼²

e^{bðxyÞ2=ðtsÞ}þKGðx;y;t;sÞÞ@_y^lgðy;sÞdy ds: ð4:18Þ All the above pointwise estimates and identities will be used in the next section to obtain pointwise estimates and theLp convergence rates on the solutions to (4.1).

5. The Proof of Theorem 2.2

From (4.3), we have

@_t^l@^k_xVðx;tÞ ¼I₁^l;kþI₂^l;kþI₃^l;k; ð5:1Þ where

I₁^l;k¼

Z N

N

@_t^l@_x^kG_sðx;t;y;0ÞVðy;0Þdy

Z N

N

@_t^l@_x^kGðx;t;y;0ÞðaVðy;0Þ þV_tðy;0ÞÞdy;

I₂^l;k¼@_t^l Z t

0

Z N

N

@_x^kGðx;t;y;sÞFðy;sÞdy ds;

I₃^l;k¼@_t^l Z t

0

Z N

N

@_x^kR_Gðx;t;y;sÞVðy;sÞdy ds:

We will derive the estimates on I_j^l;k ðj¼1;2;3Þ in the following. First, we give a lemma which will be used in the analysis.

Lemma 5.1. If0oC0plpC1;then for a positive number N>¹₂;we have B_Nðx7 ffiffiffi

pl

t;tÞpCð1þtÞ^2NB_Nðx;tÞ; ð5:2Þ

Z N

N

e^bðxyÞ2=tð1þy²Þ^NdypCBNðx;tÞ ð5:3Þ

and

Z N

N

e^{bðxyÞ2=ðtsÞ}BNðy;sÞdypC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þtsÞð1þsÞ

1þt r

BNðx;tÞ: ð5:4Þ

Proof. Sincee^x2=tpCB_Nðx;tÞ;(5.2) and(5.3) are direct consequences of Lemmas 3.4 and 5.1 of [11]. For (5.4), we only need to prove that

J_N ¼:

Z N

N

B_Nðxy;tsÞBNðy;sÞdypC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þtsÞð1þsÞ

1þt r

B_Nðx;tÞ:

Ifx²pt;then 1p2^NBNðx;tÞ:Thus,

J_NpZ N N

B_Nðxy;tsÞdypCe₀ð1þtsÞ¹⁼²:

Also we can obtain I_NpCe₀ð1þsÞ¹⁼² similarly. Thus, (5.4) hold when x²pt: For case whenx²>t;since

1þðxyÞ² 1þts

!N

1þ y² 1þs

^N

p Cð1þ_1þts^x2 Þ^Nð1þ_1þs^y2Þ^N; jxjXjyj=2;

Cð1þ^ðxyÞ_1þts²Þ^Nð1þ_1þs^x2Þ^N; jxjpjyj=2;

8<

:

we have

J_NpC

Z N

N

1þ x² 1þts

^N

1þ y² 1þs

^N

dy

þC

Z N

N

1þðxyÞ² 1þts

!N

1þ x² 1þs

^N

dy

pC 1þts 1þt

N

ð1þsÞ¹⁼²þ 1þs 1þt

N

ð1þtsÞ¹⁼²

! BNðx;tÞ

pC

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þtsÞð1þsÞ

1þt r

BNðx;tÞ:

Thus the lemma is proved. &

ForI₁^l;k;we know from Theorem 4.1 and the conditions in Theorem 2.2 that

I₁^l;k¼

Z N

N

ð@^l_t@_x^kG_sðx;t;y;0Þ Klþ1;kðxy;tÞÞVðy;0Þdy

þ

Z N

N

ð@_t^l@_x^kGðx;t;y;0Þ K_l;kðxy;tÞÞðaVþV_tÞðy;0Þdy

þ

Z N

N

ðKlþ1;kðxy;tÞVðy;0Þ þK_l;kðxy;tÞðaVþV_tÞðy;0ÞÞdy

pCe₀ð1þtÞ^1þkþ2l2

RN

Ne^ðxyÞ2=tð1þy²Þ^Ndy

þe^at=3ð1þx²Þ^N; 2lþkpm1;

1; 2lþk>m1:

8>

><

>>

:

By Lemma 5.1, we know that

jI₁^l;kjpCe₀ð1þtÞ^{ð1þkþ2lÞ=2} BNðx;tÞ; 2lþkpm1;

1; 2lþk>m1:

(

ð5:5Þ

ForI₂^l;k;we first set

F^k;lðx;tÞ ¼ ð1þtÞ^{nð1þkþ2lÞ} ð1þ_1þt^x2Þ^N; 2lþkpm1;

1; 2lþk>m1;

(

and forlp1 and a even integer m;set

MðtÞ ¼ sup

0pspt;xAR;2lþkp2m F^k;lðx;sÞj@_t^l@_x^kVðx;sÞj; ð5:6Þ where

nðhÞ ¼1 2

h; hpm;

m2; h¼mþ1;mþ2;

m4; h¼mþ3;mþ4;

^ ^

0; h¼2m1;2m:

8>

>>

>>

><

>>

>>

>>

:

It follows easily from [3,10] that

@^l_t@_x^kvðx;% tÞpCe0ð1þtÞ^ðkþ2lÞ=2e^bx

2

1þt: ð5:7Þ

Thus,

j@^l_t@_x^kF˜₁ðx;tÞjpCe₀ð1þtÞ^{ðkþ2lþ2Þ=2}e^bx

2

1þt: ð5:8Þ

Since

F˜2¼ ðp⁰ðvÞ*% vþp⁰⁰ðv%þyðVxþvÞÞððV* xþvÞ* ²=2Þ;

with 0oyo1;we have

j@_t^l@^k_xvðx;* tÞjpCjuþuje^at@_x^km0ðxÞ:

Noticing thatm₀ðxÞis a smooth function with compact support, we have

j@_t^l@_x^kvðx;* tÞjpCe0e^ate^x2: ð5:9Þ

By (5.6) and Theorem 2.1, we have forlp1;1þ2lþkp2m j@_t^l@^k_x@_xðV_x²Þj

p

CM²ðtÞð1þtÞ^3þkþ2l2 B2Nðx;tÞ; 1þ2lþkpm2;

CM²ðtÞð1þtÞ^m2B_Nðx;tÞ; 1þ2lþk

¼m1;m;

CM²ðtÞð1þtÞ^m22 BNðx;tÞ; 1þ2lþk

¼mþ1;mþ2;

^ ^

CM²ðtÞð1þtÞ¹BNðx;tÞ; 1þ2lþk

¼2m3;2m2;

Ce0MðtÞð1þtÞ¹BNðx;tÞ; 1þ2lþk

¼2m1;2m:

8>

>>

>>

>>

>>

>>

>>

>>

>>

>>

>>

><

>>

>>

>>

>>

>>

>>

>>

>>

>>

>>

>>

:

ð5:10Þ

Also, (5.7), (5.9) and (5.10) yield

j@_t^l@_x^kF2jpCðe0þM²ðtÞÞ

ð1þtÞ^3þkþ2l2 B2Nðx;tÞ; 1þ2lþkpm2;

ð1þtÞ^m2B_Nðx;tÞ; 1þ2lþk

¼m1;m;

ð1þtÞ^m22 BNðx;tÞ; 1þ2lþk

¼mþ1;mþ2;

^ ^

ð1þtÞ¹B_Nðx;tÞ; 1þ2lþk

¼2m1;2m:

8>

>>

>>

>>

>>

>>

>>

>>

>>

<

>>

>>

>>

>>

>>

>>

>>

>>

>:

ð5:11Þ

By (4.12), for eachk;we can choose ak%according to (5.10) and (5.11) so that I₂^0;k¼

Z t t=2

Z N

N

ð@^k_x ^k˜GK_0;kk˜Þðx;y;t;sÞ@_y^k˜Fðy;sÞdy ds

þ Z t

t=2

Z N

N

K_0;kk˜ðx;y;t;sÞ@_y^k˜Fðy;sÞdy ds

þOð1Þe0

X

hok˜

Z t t=2

Z N

N

ðð1þtsÞ^{ð1þkkÞ=2˜} ð1þtÞ^ðkhÞ=2˜