PARABOLIC INEQUALITIES IN L AS LIMITS OF RENORMALIZED EQUATIONS

PARABOLIC INEQUALITIES IN L AS LIMITS OF RENORMALIZED EQUATIONS

K. AZELMAT, M. KBIRI ALAOUI, D. MESKINE, AND A. SOUISSI Received 21 April 2006; Revised 10 August 2006; Accepted 21 August 2006

The paper deals with the existence of solutions of some parabolic bilateral problems ap- proximated by the renormalized solutions of some parabolic equations.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

1. Introduction

LetΩbe a bounded domain inR^NandT >0. We denote byQthe cylinderΩ×(0,T) and Γ=∂Q.

Let

A(u)= −diva(x,t,u,∇u) (1.1) be a Leray-Lions operator acting onL^p(0,T;W0^1,p(Ω)), 1< p <∞, into its dualL^p(0,T;

W^−1,p(Ω))(1/ p+ 1/ p=1). Consider the following parabolic problem:

u∈᏷=

v∈L^p0,T;W0^1,p(Ω):v(t)∈Ka.e., _T

0

∂v

∂t,u−v

dt+

Qa(x,t,u,∇u)(∇u− ∇v)dx dt≤ _T

0 f,u−vdt,

∀v∈᏷∩ v∈L^p0,T;W0^1,p(Ω):∂v

∂t ^∈L^p0,T;W^−1,p(Ω);v(0)=0

, (P)

whereKis a given convex inW0^1,p(Ω) and f ∈L^p(0,T;W^−1,p(Ω)).

It is well known that (P) admits at least one solution via a classical penalty method (see Lions [5] forp≥2 and Landes-Mustonen [4] for 1< p <2). Recently in [6], the authors

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 46265, Pages1–18

DOI 10.1155/IJMMS/2006/46265

L

approximated (P) by the following sequence of parabolic equations:

∂un

∂t +Au_n+hx,u_nⁿ⁻¹hx,u_nGx,t,u_n,∇u_n= f inQ, u_n(x,t)=0 on∂Q,

un(x, 0)=0 inΩ,

(P_n)

wherehandGare two Carath´eodory functions satisfying some natural growth condi- tions. The obtained convexKdepends on two obstacles constructed fromh.

In theL¹ case, that is, f ∈L¹(Ω×]0,T[), the formulations (P) and (Pn) are not ap- propriate. So, we introduce the renormalized problem (Rn) associated to (Pn) (see the definition below). The study of the asymptotic behavior of (R_n) asn→ ∞leads to some bilateral parabolic problem. Our approach allows us also to prove the existence of solu- tions for general parabolic inequalities of type

T_k(u)∈᏷, _T

0

∂v

∂t,Tk(u−v)

dt+

Qa(x,t,u,∇u)∇Tk(u−v)dx dt +

QH(x,t,u,∇u)Tk(u−v)dx dt≤

Qf Tk(u−v)dx dt, ∀v∈᏷∩D∩L^∞(Q), (1.2) whereD= {v∈L^p(0,T;W0^1,p(Ω)), ∂v/∂t∈L^p(0,T,W0^−1,p(Ω)) +L¹(Q), v(0)=0}and whereHis a given Carath´eodory function satisfying some natural growth assumption.

For some recent and classical results for some parabolic inequalities problems, the reader can refer to [2,7,9,10].

2. Main result

LetΩbe an open bounded subset ofR^N,N≥2 and 1< p <+∞. We denote byQthe cylinderΩ×(0,T) andΓ=∂Q.

LetA(u)= −div(a(x,t,∇u)) be a Leray-Lions operator defined onL^p(0,T;W0^1,p(Ω)) into its dualL^p(0,T;W^−1,p(Ω)), wherea:Ω×R×R^N→R^Nis a Carath´eodory function satisfying for a.e.x∈Ω, for allt∈Rand for allζ,ζ∈R^N, (ζ=ζ) the following hold:

a(x,t,ζ)≤βk(x,t) +|ζ|^p−1 , a(x,t,ζ)−a(x,t,ζ)(ζ−ζ)>0,

a(x,t,ζ)ζ≥α|ζ|^p,

(2.1)

withα >0,β >0,k∈L^p(Q).

Furthermore, leth:Ω×R→Rbe a Carath´eodory function such that

h(x, 0)=0, h(x,s) is nondecreasing with respect tos. (2.2)

Gis a Carath´eodory function satisfying the following assumptions:

G(x,t,s,ξ)≤b|s|

c(x,t) +|ξ|^p

, G(x,t,s, 0)=0, (2.3) v∈L^p0,T;W0^1,p(Ω):G(x,t,v,∇v)=0 a.e. inQ

⊂

v∈L^p0,T;W0^1,p(Ω):h(x,v)≤1 a.e. inQ.

(2.4) Let us suppose

for almostx∈Ω\Ω^∞+ there exists=(x)>0 such that h(x,s)>1, ∀s∈

q+(x),q+(x) + , for almostx∈Ω\Ω^∞−there exists=(x)>0 such that

h(x,s)<−1, ∀s∈

q−(x)−,q−(x),

(2.5)

wherebis a continuous nondecreasing function andc(x,t)∈L¹(Q),c≥0, and q+(x)=infs >0,h(x,s)≥1,

q−(x)=sups >0,h(x,s)≤ −1, Ω^∞+ =

x∈Ω:q+(x)=+∞ , Ω^∞₋=

x∈Ω:q−(x)= −∞

.

(2.6)

We define for allsandkinR,k≥0,T_k(s)=max(−k, min(k,s)).

We will say thatunis a renormalized solution of (Pn) if Tk

un

∈L^p0,T;W0^1,p(Ω), ∀k >0,

hlim→∞

{h≤|un|≤h+1}ax,t,∇u_n∇u_ndx dt=0, unsatisfies in the distributional sense Aun

t−divax,t,∇un Aun

+ax,t,∇un

∇unAun +hx,u_nⁿ⁻¹hx,u_nGx,t,u_n,∇u_nAu_n=f Au_n,

∀A∈C¹(R), A,A∈L^∞(Ω), Ahas a compact support andunsatisfies the initial condition in the sense thatAun

∈C[0,T],L¹(Ω).

(Rn)

Thanks to [8, Theorem 3.2, page 164], there exists at least one solutionunof (Rn).

Theorem 2.1. Under the hypotheses (2.1)–(2.5), f ∈L¹(Q), the problem (Pn) has at least one renormalized solution (un) such that

T_ku_n−→T_k(u) strongly inL^p0,T;W0^1,p(Ω), (2.7)

L

whereuis a solution of the following obstacle problem:

q−(x)≤u(x,t)≤q+(x) a.e. (x,t)∈Q, T_k(u)∈L^p0,T;W0^1,p(Ω), _T

0

∂v

∂t,Tk(u−v)

dt+

Qa(x,t,∇u)∇Tk(u−v)dx dt

≤

Qf Tk(u−v)dx dt, ∀v∈᏷∩D∩L^∞(Q),

(R)

where

D= v∈L^p0,T;W0^1,p(Ω), ∂v

∂t ^∈L^p0,T;W^−1,p(Ω)+L¹(Q),v(0)=0

,

᏷=

v∈L^p0,T;W0^1,p(Ω),v(t)∈K, K=

v∈W0^1,p(Ω), q−≤v≤q+

. (2.8)

Moreover, ifq−,q+∈L^∞(Ω), thenu∈L^p(0,T;W0^1,p(Ω))∩L^∞(Q).

Remark 2.2. The same result can be obtained when dealing with general operator of Leray-Lions type depending also onu, that is,A(u)= −div(a(x,t,u,∇u)).

Proof ofTheorem 2.1.

Step 1. LetA(t)=Hm(t),Hm(t)=_t

0hm(s)ds, where

hm(s)=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

1 if|s| ≤m,

affine ifm≤ |s| ≤m+ 1, 0 ifm+ 1≤ |s|.

(2.9)

Taking nowTk(Hm(un)) as test function in (Rn), we obtain _T

0

∂Hm

un

∂t ,Tk

Hm

un

dt+

|Hm(un)|<kax,t,∇un

∇unh²_mun

dx dt +

Q

hx,unⁿ⁻¹hx,unGx,t,un,∇unhm un

Tk Hm

un dx dt +

Qa·,t,∇un

∇unh_mun Tk

Hm un

dx dt=

Qf hm un

Tk Hm

un dx dt.

(2.10) Since

_T

0

∂Hm

un

∂t ,Tk

Hm

un

dt

=

Ω

_Hm(un(x,T))

0 Tk(s)ds

dx−

Ω

_Hm(un(x,0))

0 Tk(s)ds

dx

(2.11)

and by using the fact that_Ω(₀^Hm(un(x,T))Tk(s)ds)≥0, we obtain

{|Hm(un)_|<k}ax,t,∇un

∇unh²_mun

dx dt≤Ck+

{m≤|un|≤m+1_}ax,t,∇un

∇undx dt,

Q

hx,u_nⁿ⁻¹hx,u_nGx,t,u_n,∇u_nh_mu_nT_kH_mu_ndx dt

≤Ck+

{m≤|un|≤m+1}ax,t,∇un

∇undx dt.

(2.12) We haveHm(s) (resp.,hm(s)) tends tos(resp., to 1) asmgoes to +∞.

Using Fatou’s lemma and the definition of the renormalized solution leads to

Q

∇T_ku_n^pdx dt≤Ck, (2.13)

Q

hx,u_nⁿ⁻¹hx,u_nGx,t,u_n,∇u_nT_ku_ndx dt≤Ck, (2.14)

which gives

Q

hx,unⁿGx,t,un,∇unTk

un

k dx dt≤C, (2.15) and ask→0 we obtain

Q

hx,u_nⁿGx,t,u_n,∇u_ndx dt≤C. (2.16)

Choosing now aC²functionρk, such thatρk(s)=sfor|s| ≤kand 2ksign(s) for|s|>2k, we get

ρ_ku_nt−divax,t,∇u_nρ_ku_n+ax,t,∇u_n∇u_nρ_ku_n +hx,unⁿ⁻¹hx,unGx,t,un,∇unρ_kun

= f ρk

un

.

(2.17)

We deduce that (ρ_k(u_n))_tis bounded inL¹(Q) +L^p(0,T;W^−1,p(Ω)).

Now thanks to the following result.

Lemma 2.3 [11]. Let p >1. If (un) is a bounded sequence ofL^p(0,T;W0^1,p(Ω)) such that

∂un/∂tis bounded inL¹+L^p(0,T;W^−1,p(Ω)), thenunis relatively compact inL^p(Q).

L

We deduce thatρk(un) is relatively compact inL^p(Q) and so there exists a measurable functionusuch thatun→ua.e. inQ.

Finally, we deduce from (2.13) thatTk(un)Tk(u) weakly inL^p(0,T;W0^1,p(Ω)), and strongly inL^p(Q).

Step 2. We are dealing now with the almost convergence of the gradient.

We have to prove that, for 0< θ <1,

nlim→∞

Q

ax,t,∇Tk

un

−ax,t,∇Tk(u)∇Tk

un

− ∇Tk(u)^θdx dt=0. (2.18)

Letω∈L^p(0,T;W0^1,p(Ω)), we define for anyμ >0,ωμthe time regularization ofω,

ωμ(x,t)=μ _t

−∞ω(x,s) expμ(s−t)ds, (2.19)

whereωis the zero extension ofωfors > T. Furthermore,ωμsatisfies the following prop- erties (see [3]):

ωμ−→ω strongly inL^p0,T;W0^1,p(Ω),

∂ωμ

∂t ⁼μω−ωμ

in the distributional sense.

(2.20)

Lettingη >0, we obtain

Q

ax,t,∇T_ku_n−ax,t,∇T_k(u)∇T_ku_n− ∇T_k(u)^θ

≤CmeasTk

un

−Tk(u)μ≥η^1−θ

+C

{|Tk(un)₋Tk(u)_μ|<η}

ax,t,∇Tk

un

−ax,t,∇Tk(u)∇Tk

un

− ∇Tk(u)

θ

.

(2.21)

On the other hand, we have

{|Tk(un)₋Tk(u)_μ|<η}

ax,t,∇T_ku_n−ax,t,∇T_k(u)∇T_ku_n− ∇T_k(u)dx dt

≤

{|Tk(un)−Tk(u)μ|<η}

ax,t,∇T_ku_n

−ax,t,∇Tk(u)μ

∇Tk un

− ∇Tk(u)μ dx dt +

{|Tk(un)₋Tk(u)_μ|<η}ax,t,∇T_ku_n∇T_k(u)_μ− ∇T_k(u)dx dt +

{|Tk(un)−Tk(u)μ|<η}

ax,t,∇Tk(u)μ

−ax,t,∇Tk(u)∇Tk un

dx dt

−

{|Tk(un)−Tk(u)μ|<η}ax,t,∇Tk(u)μ

∇Tk(u)μdx dt +

{|Tk(un)₋Tk(u)_μ|<η}ax,t,∇Tk(u)∇Tk(u)dx dt

≤I1+I2+I3+I4+I5.

(2.22)

TakeTη(Hm(un)−Tk(u)μ) as test function in (Rn) withA(t)=Hm(t). We obtain _T

0

∂Hm

un

∂t ,Tη

Hm

un

−Tk(u)μ

dt +

{|Hm(un)₋Tk(u)_μ|<η}ax,t,∇u_n∇u_nh²_mu_n− ∇T_k(u)_μh_mu_ndx dt +

Q

hx,u_nⁿ⁻¹hx,u_nGx,t,u_n,∇u_nh_mu_nT_ηH_mu_n−T_k(u)_μdx dt

+

Qax,t,∇u_n∇u_nh_m(u_n)T_ηH_mu_n−T_k(u)_μdx dt

=

Qf h_mu_nT_ηH_mu_n−T_k(u)_μdx dt.

(2.23) We have

_T

0

∂Hm

un

∂t ,Tη

Hm

un

−Tk(u)μ

dt

= _T

0

∂Hm

un

−Tk(u)μ

∂t ,Tk

Hm

un

−Tk(u)μ

dt

+ _T

0

∂Tk(u)μ

∂t ,Tk

Hm

un

−Tk(u)μ

dt.

(2.24)

L Using the fact that

_T

0

∂Hm

un

−Tk(u)μ

∂t ,Tη

Hm

un

−Tk(u)μ

dt≥0, _T

0

∂Tk(u)μ

∂t ,Tk

Hm

un

−Tk(u)μ

dt

=μ

Q

Tk(u)−Tk(u)μ

Tη

Hm

un

−Tk(u)μ

dx dt,

(2.25)

consequently, lim sup

n→∞ lim sup

m→∞

_T

0

∂Hm un

∂t ,Tη Hm

un

−Tk(u)μ dt

≥μ

Q

T_k(u)−T_k(u)_μT_ηu−T_k(u)_μdx dt=(m,n)≥0.

(2.26)

This implies that

{|Hm(un)−Tk(u)_μ|<η}ax,t,∇u_n∇u_nh²_mu_n− ∇T_k(u)_μh_mu_ndx dt +

Q

hx,unⁿ⁻¹hx,unGx,t,un,∇unhm un

Tη Hm

un

−Tk(u)μ dx dt +

Qax,t,∇un

∇unh_mun Tη

Hm un

−Tk(u)μ dx dt

≤

Qf h_mu_nT_ηH_mu_n−T_k(u)_μdx dt+(m,n),

(2.27) which gives by using the fact that

Q

hx,unⁿ⁻¹hx,unGx,t,un,∇unhm un

Tη Hm

un

−Tk(u)μ

dx dt≤Cη,

{|Hm(un)−Tk(u)μ|<η}ax,t,∇un

∇unh²_mun

− ∇Tk(u)μhm un

dx dt

≤Cη+(m,n) +η

{m≤|un|≤m+1_}ax,t,∇un

∇undx dt,

(2.28) which gives asm→ ∞,

{|un−Tk(u)μ|<η}ax,t,∇un

∇un− ∇Tk(u)μdx dt≤Cη+(n). (2.29)

Finally from (2.22), I1≤Cη+(n)−

{|Tk(un)−Tk(u)μ|<η}ax,t,∇Tk(u)μ

∇Tk un

− ∇Tk(u). (2.30) Sincea(x,t,∇T_k(u)_μ)χ{|Tk(un)−Tk(u)μ|<η}→a(x,t,∇T_k(u)_μ)χ{|Tk(u)−Tk(u)μ|<η}inL^p(Q) and Tk(un)→Tk(u) weakly inL^p(0,T;W0^1,p(Ω)), then

−

{|Tk(un)₋Tk(u)_μ|<η}ax,t,∇Tk(u)μ

∇Tk

un

− ∇Tk(u)dx dt

= −

{|Tk(u)₋Tk(u)_μ|<η}ax,t,∇T_k(u)_μ∇T_k(u)− ∇T_k(u)dx dt+(n).

(2.31)

So

I1≤Cη+(n). (2.32)

For what concerns the termI2, one has

I2=(n,μ), (2.33)

since

ax,t,∇Tk un

χ{|Tk(un)−Tk(u)μ|<η}−→ax,t,∇Tk(u)χ{|Tk(u)−Tk(u)μ|<η} inL^p(Q)^N, ∇T_k(u)_μ− ∇T_k(u)χ{|Tk(un)−Tk(u)μ|<η}−→

∇T_k(u)_μ− ∇T_k(u)χ{|Tk(u)−Tk(u)μ|<η}. (2.34) In the same way, we show that

I3=(n,μ), I4=(n,μ), I5=(n,μ). (2.35) Combining the above estimates, we get

nlim→∞

Q

ax,t,∇T_ku_n−ax,t,∇T_k(u)∇T_ku_n− ∇T_k(u)^θdx dt=0. (2.36)

Then there exists a subsequence also denoted by (u_n) such that

∇u_n−→ ∇u a.e. inQ. (2.37)

Step 3. From (2.16), we deduce that

Q

hx,u_nⁿGx,t,u_n,∇u_ndx dt≤C, (2.38)

which gives for everyβ >0,

|h(x,Tβ(un))|>k

Gx,t,Tβ un

,∇Tβ

undx dt≤ C

kⁿ, (2.39)

L

wherek >1. Lettingn→+∞forkfixed, we deduce by using Fatou’s lemma

|h(x,Tβ(u))_|>k

Gx,t,T_β(u),∇T_β(u)dx dt=0, (2.40)

and so, by (2.4)

hx,Tβ(u)≤1 a.e. inQ. (2.41)

So

q−(x)≤T_βu(x)≤q+(x) a.e. inQ. (2.42) Letting nowβ→+∞, we deduce also that

q−(x)≤u(x)≤q+(x) a.e. inQ. (2.43) Step 4. Strong convergence of the truncations.

We will prove that

nlim→∞

Q

ax,t,∇T_ku_n−ax,t,∇T_k(u)∇T_ku_n− ∇T_k(u)dx dt=0. (2.44)

Fixk >0 and letϕ(s)=exp(δs²),δ >0. Letl > kand define the functionRl(s)=_s

0ρl(t)dt.

Let us considerω^m_μ =T_k(H_m(u)_μ), wherev_μ is the mollification with respect to timev.

Lettingv^m_μ^,n=ρl(Hm(un))ϕ(Tk(Hm(un))−ω^m_μ) as test function in the problem (Rn), we get

_T

0

∂H_mu_n

∂t ,ρ_lH_mu_nϕT_kH_mu_n−ω^m_μdt +

Qax,t,∇u_n∇u_nh²u_nρ_lH_mu_nϕT_kH_mu_n−ω^m_μdx dt +

Qax,t,∇u_n∇T_kH_mu_n− ∇ω^m_μ

×h_mu_nρ_lH_mu_nϕT_kH_mu_n−ω^m_μdx dt +

Qax,t,∇un

∇unh_mun ρl

Hm un

ϕTk Hm

un

−ω_μ^mdx dt +

Q

hx,unⁿ⁻¹hx,unGx,t,un,∇un

×hm un

ρl Hm

un ϕTk

Hm un

−ω_μ^mdx dt

=

Qf v^m_μ^,nh_mu_ndx dt.

(2.45)

We deal now with the estimate of each term of the last equalities.