1013–1019 DOI: 10.18514/MMN.2019.3040 INTERMEDIATE REGULARITY RESULTS FOR THE SOLUTION OF A HIGH ORDER PARABOLIC EQUATION AREZKI KHELOUFI Received 19 September, 2019 Abstract

Vol. 20 (2019), No. 2, pp. 1013–1019 DOI: 10.18514/MMN.2019.3040

INTERMEDIATE REGULARITY RESULTS FOR THE SOLUTION OF A HIGH ORDER PARABOLIC EQUATION

AREZKI KHELOUFI Received 19 September, 2019

Abstract. In this work we give new intermediate regularity results for the solution of the follow- ing2m-th order parabolic equation

@tuC. 1/^m

n

X

iD1

@^2m_xi uD0;

wheremis a positive integer, subject to Dirichlet condition on the lateral boundary of a cyl- indrical domain and to a non-homogeneous initial Cauchy data.

2010Mathematics Subject Classification: 35K05; 35K55 Keywords: regularity results, parabolic equations, Sobolev spaces

1. INTRODUCTION

Let˝be an open bounded set ofRⁿwith boundary andQthe cylinderR_C˝ with lateral boundary˙ DR_C :We assume that˝ is of classC^2m:Consider in Qthe following boundary value problem:

8 ˆˆ ˆˆ

<

ˆˆ ˆˆ :

@tuC. 1/^mPn

iD1@^2m_xi uD0 inQ;

@^juD0 on˙,j D0; 1; :::; m 1;

u .0; x/Du0.x/,x2˝;

(1.1)

wheremis a positive integer and@^jis the derivative of orderj throughout the normal vectoron˙:

Classical results on the resolution of Problem (1.1) when the initial Cauchy data u0belongs toL².˝/or to the usual Sobolev spaceH₀^m.˝/can be found in [6], see also [1], [2], [8] and [9] and the references therein.

Our interest in this work is the regularity of the solutionuof (1.1) in terms of the regularity of the initial datau0. More precisely, we are interested on the question of the regularity of the solution of (1.1) whenu0is ”between”H₀^m.˝/andL².˝/ :The

c 2019 Miskolc University Press

results obtained here complement those obtained for the heat equation, i.e.mD1, in [4].

The organization of this paper is as follows. In Section 2, we begin by preliminar- ies where we define the basic functional spaces, in which we will work and we give some of their properties needed for our study. Then, we recall a classical result for Problem (1.1) and we prove a fundamental lemma which will allow us to prove our main result in Section 3.

2. PRELIMINARIES

2.1. Function spaces

In this subsection, we recall the definitions of the basic functional spaces, in which we will work. We will need some anisotropic Sobolev spaces (see [6]), which we recall in the following definitions

H^r;s Rⁿ Dn

u2L² Rⁿ Wh

1C^2r=2

C 1C^2s=2i

bu2L² Rⁿo wherebuis the Fourier transform ofuandr; sare two non-negative numbers. We put

H^r;s.˝/D˚

uj^˝ Wu2H^r;s Rⁿ ;

with˝ is an open subset ofRⁿ. For0r1 and a positive integerm;we recall that the spaceH^r;2mr.Q/, whereQDR_C˝;can be defined by

H^r;2mr.Q/DL² R_C; H^2mr.˝/

\H^r R_C; L².˝/

and the spaceH₀^r.˝/is defined by

H₀^r.˝/D fu2H^r.˝/IuD0on g for ¹₂< r1, is the boundary of˝:

H₀^r.˝/DH

1 2

00.˝/

forrD¹₂;

H₀^r.˝/DH^r.˝/

for0r < ¹₂.

We will need also some interpolation spaces in Hilbert spaces (see [7] and [3]), which we recall in the following definition.

LetX,Y be two Hilbert spaces with

X Y continuously.

Here, we give one of the usual methods, namely, that of Lions-Peetre [7] which allow us to build spaces

ŒX; Y  0 < < 1;

”intermediate” betweenX andY:

Definition 1. The space ŒX; Y  0 < < 1; is a sub-space of Y consisting of elementsawhich can be written in the form

aD Z ₁

0

u .t /dt

t (2.1)

with

tu .t /2L².X /,t ¹u .t /2L².Y / : (2.2) This space is endowed with the norm

a7 !inf

"

Z 1 0

t²ju .t /jX²

dt t

¹₂ C

Z 1 0

t^{2. 1/}ju .t /j²Y

dt t

¹₂# I the inf is taken with respect touverifying (2.1) and (2.2).

HereL².V / denotes the space of functions fromt > 0 with values inV, which are square integrable for the Haar measuredt =t:

Example 1. H^r;s.˝/can also be defined as a real interpolation space between Hr=.1 /;s=.1 /.˝/andL².˝/ ; 20; 1Œ ;(see [10])

H^r;s.˝/Dh

Hr=.1 /;s=.1 /.˝/ ; L².˝/i

: (2.3)

In this work, we consider the casesD2mr,D1 r;

H^r;2mr.˝/D

H^1;2m.˝/ ; L².˝/

1 r 8r20; 1Œ : (2.4) The spaceH^r;2mr.˝/is well defined by Relationship (2.4) because the right hand side term of (2.4) is well defined as an interpolation space between two well defined spacesH^1;2m.˝/andL².˝/ :

Example2. The usual Sobolev spacesH^s.˝/(s0) can be defined by interpol- ation

H^s.˝/Dh

H^{s=.1 /}.˝/ ; L².˝/

i

; 20; 1Œ :

Hereafter some interpolation theory properties on spacesŒ:; : (see Triebel [10]), needed for proving our main result in the following section.

Theorem 1([10]). LetA0,A1be two Hilbert spaces with A0A1continuously.

Then

L².A0/ ; L².A1/

DL².ŒA0; A1/ ; 0 < < 1.

A direct consequence of Theorem1is

Corollary 1. For each0r1and any positive integerm;we have L² R_C; H^2m.˝/

; L².R_C; H^m.˝//

1 r D L² R_C;

H^2m.˝/ ; H^m.˝/

1 r

D L²

R_C; H^m.1Cr/.˝/

: with˝an open bounded set ofRⁿ:

2.2. Lemmas

The next result is well known ( cf. Lions and Magenes [6]).

Lemma 1. 1) For givenu0inH₀^m.˝/ ;Problem (1.1) has a unique solutionuin H^1;2m.Q/defined by

H^1;2m.Q/DL² R_C; H^2m.˝/

\H¹ R_C; L².˝/

: 2) For givenu0inL².˝/, Problem (1.1) has a unique weak solutionuin

L² R_C; H₀^m.˝/

\H¹ R_C; H ^m.˝/

\L¹ R_C; L².˝/

:

We will need the following lemma for proving our main result in the following section.

Lemma 2. Letu02L².˝/. Then the solutionuof Problem (1.1) associated to u0is inH¹² R_C; L².˝/

:Moreover, there exists a positive constantC (independent ofu0) such that

kuk_H¹₂.R_C;L².˝//Cku0kL².˝/:

Proof. Consider a sequence of spectral elements._k; '_k/ ; k2Nof the Dirichlet problem for the operator. 1/^mPn

iD1@^2m_x

i

8 ˆˆ ˆˆ

<

ˆˆ ˆˆ :

. 1/^mPn

iD1@^2m_xi 'k Dk'k

'_k2H₀^m.˝/

k'_kkL².˝/D1:

The sequence.'_k/_k2Nis a basis ofL².˝/. Ifu02L².˝/we may write u0.x/D X

k2N

ak'k.x/

withku0k_L²²_.˝/DP

k2Na_k²:The solution associated tou0is u .t; x/DX

k2N

a_kexp. _kt / '_k.x/ : Noteeuthe extension ofutoR, i.e.,

eu .t; x/DX

akexpj ktj'k.x/ :

By the Fourier transform

beu .; x/DCX a_kk

²C²_k'_k.x/ ; whereC is a constant, from which

beu .; :/

2

DC²X

a²_k ²_k ²C²_k2

and by elementary calculations, we check easily that Z

Rjj beu .; :/

2

d DC⁰X a_k²; whereC⁰is a constant. Consequently

eu2H¹² R; L².˝/

; then

u2H¹² R_C; L².˝/

by restriction ofeutot > 0:

3. MAIN RESULT

In the sequel, we will assume thatu02H₀^rm.˝/ ; 0r1:ThusH₀^rm.˝/is the interpolation space of order1 r betweenH₀^m.˝/andL².˝/ :Indeed, it suffices to takeD1 randsDrmin Example2. We look for the regularity ofuin term ofr:Our main result in this work is

Theorem 2. For given u0 in H₀^rm.˝/ 0r 1; Problem (1.1) has a unique weak solutionuinH^1C2r;m.1Cr/.Q/ :

Proof. Letu02H₀^rm.˝/ 0r1thenu02L².˝/and consequently (1.1) ad- mits a unique weak solution (see Lemma1)uinL² R_C; H₀^m.˝/

:In order to show that this solution is inL²

R_C; H^m.1Cr/.˝/

it suffices to interpolate the operator S which associatesutou0. IndeedS Wu07!uis linear continuous fromH₀^m.˝/

toL² R_C; H^2m.˝/

and fromL².˝/toL².R_C; H^m.˝// :By interpolation, it is linear continuous from

H₀^m.˝/ ; L².˝/

1 r into

L² R_C; H^2m.˝/

; L² R_C; H^m.˝/

1 r: Thanks to Corollary1,S is linear continuous from

H₀^rm.˝/ intoL²

R_C; H^m.1Cr/.˝/

:

We can interpolate againSfor proving thatu2H^rC21 R_C; L².˝/

:Indeed,Sis lin- ear continuous fromL².˝/intoH¹² R_C; L².˝/

(see Lemma2) and fromH₀^m.˝/

intoH¹ R_C; L².˝/

(see Theorem1). By interpolation, it is linear continuous from H₀^m.˝/ ; L².˝/

1 r into h

H¹² R_C; L².˝/

; H¹ R_C; L².˝/i

1 r: But, (see Triebel [10])

h

H¹² R_C; L².˝/

; H¹ R_C; L².˝/i

1 rDH^rC21 R_C; L².˝/

: Then,Sis linear continuous fromH₀^rm.˝/intoH^rC21 R_C; L².˝/

:This ends the

proof of Theorem2.

Remark1. Note that the anisotropic Hilbert spaces used in this work may be ex- tended to the case of parabolic Sobolev spaces built onL^p; p¤2:An idea for this extension can be found in [5].

ACKNOWLEDGEMENT

The author is grateful to anonymous reviewers for carefully reading this paper and for their valuable comments and suggestions, which have improved the paper.

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Author’s address

Arezki Kheloufi

Bejaia University, Department of Technology, Lab. of Applied Mathematics, 06000 Bejaia, Algeria E-mail address:arezkinet2000@yahoo.fr