# On Degenerate Parabolic Equations

Authors: Mohammed Kbiri Alaoui
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

The paper deals with the existence of solutions of some generalized Stefan-type equation in the framework of Orlicz spaces.


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## Body

## 1. Introduction

In this paper, we deal with the following boundary value problems(P)∂u∂t+A(θ(u))=f,inQ,u=0,on⁡∂Q=∂Ω×(0,T),u(x,0)=u0(x),inΩ,
where
(1.1)A(u)=-div⁡(a(⋅,t,∇u)),Q=Ω×[0,T],T>0, and Ω is a bounded domain of RN, with the segment property, f is a smooth function, u0∈L2(Ω),θ is a positive real function increasing but not necessarily strictly increasing, θ(0)=0, and θ(u0)∈L2(Ω).a:Ω×RN→RN is a Carathéodory function (i.e., measurable with respect to x in Ω for every (t,ξ) in R×R×RN, and continuous with respect to ξ in RN for almost every x in Ω) such that for all ξ,ξ*∈RN,ξ≠ξ*,
(1.2)a(x,t,ξ)ξ≥αB(|ξ|),(1.3)[a(x,t,ξ)-a(x,t,ξ*)][ξ-ξ*]>0,(1.4)|a(x,t,ξ)|≤c(x,t)+k1B¯-1B(k2|ξ|).
There exist an N-function M such that
(1.5)B(θ(t))≪M(t),
where c(x,t) belongs to EB¯(Q),c≥0 and ki(i=1,2) to R+, and α to R*+.Some examples of such operator are in particular the case where(1.6)a(⋅,∇θ(u))=B(|∇θ(u)|)|∇θ(u)|2∇θ(u),
where B is an N-function.Many physical models in hydrology, infiltration through porous media, heat transport, metallurgy, and so forth lead to the nonlinear equations (systems) of the form(E)∂tu=∇ψ(∇β(u)),
where β,ψ are monotone, ψ(s)s is even and convex for s≥so>0,  |ψ(s)|→∞,|β(s)|→∞ for |s|→∞ (for the details see [1]). Jäger and Kačur treated the porous medium systems where β is strictly monotone in [2] and Stefan-type problems where β is only monotone. For the last model, there exists a large number of references. Among them, let us mention the earlier works [3–5] for a variational approach and [6] for semigroup.In [7], a different approach was introduced to study the porous and Stefan problems.The enthalpy formulation and the variational technique are used. Nonstandard semidiscretization in time is used, and Newton-like iterations are applied to solve the corresponding elliptic problems.Due to the possible jumps ofθ, problem P enters the class of Stefan problems. In the present paper, we are interested in the parabolic problem with regular data. It is similar in many respects to the so-called porous media equation. However, the equation we consider has a more general structure than that in the references above.Two main difficulties appear in the study of existence of solutions of problemP. The first one comes from the diffusion terms in P since they do not depend on u but on θ(u), and, moreover, at the same time, P poses big problems, since in general we have not information on u but on θ(u). For the last reason, the authors in [8] define a new notion of weak solution to overcome this problem.In the above cited references, the authors have shown the existence of a weak solution when the functiona(x,t,ξ) was assumed to satisfy a polynomial growth condition with respect to ∇u. When trying to relax this restriction on the function a(·,ξ), we are led to replace the space Lp(0,T;W1,p(Ω)) by an inhomogeneous Sobolev space W1,xLB built from an Orlicz space LB instead of Lp, where the N-function B which defines LB is related to the actual growth of the Carathéodory’s function.Our goal in this paper is, on the one hand, to give a generalization ofE in the case of one equation in the framework of Leray-Lions operator in Orlicz-Sobolev spaces. on the second hand, we prove the existence of solutions in the BV(Q) space.


## 2. Preliminaries

LetM:R+→R+ be an N-function, that is, M is continuous, convex, with M(t)>0 for t>0, M(t)/t→0 as t→0, and M(t)/t→∞ as t→∞. The N-function M¯ conjugate to M is defined by M¯(t)=sup{st-M(s):s>0}.LetP and Q be two N-functions. P≪Q means that P grows essentially less rapidly than Q, that is, for each ε>0,
(2.1)P(t)Q(εt)⟶0,ast⟶∞.
Let Ω be an open subset of RN. The Orlicz class ℒM(Ω) (resp., the Orlicz space LM(Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that ∫ΩM(u(x))dx<+∞ (resp., ∫ΩM(u(x)/λ)dx<+∞ for some λ>0).Note thatLM(Ω) is a Banach space under the norm ∥u∥M,Ω=inf{λ>0:∫ΩM(u(x)/λ)dx≤1} and ℒM(Ω) is a convex subset of LM(Ω). The closure in LM(Ω) of the set of bounded measurable functions with compact support in Ω¯ is denoted by EM(Ω). In general, EM(Ω)≠LM(Ω) and the dual of EM(Ω) can be identified with LM¯(Ω) by means of the pairing ∫Ωu(x)v(x)dx, and the dual norm on LM¯(Ω) is equivalent to ∥·∥M¯,Ω.We say thatun converges to u for the modular convergence in LM(Ω) if, for some λ>0, ∫ΩM((un-u)/λ)dx→0. This implies convergence for σ(LM,LM¯).The inhomogeneous Orlicz-Sobolev spaces are defined as follows:W1,xLM(Q)={u∈LM(Q):Dxαu∈LM(Q)forall|α|≤1}. These spaces are considered as subspaces of the product space ΠLM(Q) which have as many copies as there are α-order derivatives, |α|≤1. We define the space W01,xLM(Q)=𝒟(Q)¯σ(ΠLM,ΠLM¯). (For more details, see [9].)Fork>0, we define the truncation at height k,Tk:R→R by
(2.2)Tk(s)={s,if|s|≤k,ks|s|,if|s|>k.


## 3. Main Result

Before giving our main result, we give the following lemma which will be used.Lemma 3.1 (see [10]).
Under the hypothesis (1.2)–(1.4), θ(s)=s, the problem P admits at least one solution u in the following sense:
(3.1)u∈W01,xLB(Q)∩L2(Q),〈∂u∂t,v〉+∫Qa(⋅,∇u)∇v=∫Qfvdxdt,
for all v∈W01,xLB(Q)∩L2(Q) and for v=u.Theorem 3.2.
Under the hypothesis (1.2)–(1.5), the problem (P0) admits at least one solution u in the following sense:
(3.2)u∈BVloc⁡(Q),θ(u)∈W01,xLB(Q)∩L2(Q),〈∂u∂t,v〉+∫Qa(⋅,∇θ(u))∇v=∫Qfvdxdt,
for all v∈W01,xLB(Q).Proof.
Step 1 (approximation and a priori estimate).
Consider the approximate problem:(3.3)∂un∂t-div⁡(a(⋅,∇θ(un)))-1nΔM(un)=f,inQ,un(x,0)=u0n(x),inΩ,
where -ΔM(u)=-div((M(|∇un|)/|∇un|2)∇un) is the M-Laplacian operator and (u0n) is a smooth sequence converging strongly to u0 in L2(Q).
The approximate problem has a regular solutionun and in particular un∈W01,xLM(Q) (by Lemma 3.1).
LetΘ(s)=∫0sθ(t)dt.
Letv=θ(un)χ(0,τ) as test function, one has
(3.4)∫ΩΘ(un(τ))dx+α∫QτB(|∇θ(un)|)+≤∫Qτfθ(un)+∫ΩΘ(un(0))dx,
then, (θ(un))n bounded in W01,xLB(Q).
There exist a measurable functionv and a subsequence, also denoted (un), such that,
(3.5)θ(un)⇀v,a.einQandweaklyinW01,xLB(Q).
Let us consider theC2 function defined by(3.6)ηk(s)={s|s|≤k2,ksign⁡(s)|s|≥k.
Multiplying the approximating equation byηk′(un), we get(3.7)∂ηk(un)∂t-div⁡(a(⋅,∇θ(un))ηk′(un))+a(⋅,∇θ(un))ηk′′(un),-1ndiv⁡(M(|∇un|)|∇un|2∇unηk′(un))+1nM(|∇un|)|∇un|2∇unηk′′(un)=fηk′(un)
in the distributions sense. We deduce, then, ηk(un) is bounded in W01,xLM(Q) and ∂ηk(un)/∂t in W-1,xLM¯(Q)+L2(Q). Then, ηk(un) is compact in L1(Q).
Following the same way as in [11], we obtainθ(un)⇀θ(u), weakly in W01,xLB(Q) for σ(ΠLB,ΠEB¯), strongly in L1(Q) and a.e in Q.Step 2 (passage to the limit).
Let setb(·,∇u)=(M(|∇u|)/|∇u|2)∇u.
Letv∈W01,xLB(Q), one has
(3.8)〈∂un∂t,θ(un)-v〉+∫Qa(⋅,∇θ(un))∇(θ(un)-v)+∫Q1nb(⋅,∇un)∇(θ(un)-v)dx=∫Qf(θ(un)-v)dxdt.
By using the following decomposition:
(3.9)a(⋅,∇θ(un))∇(θ(un)-v)=a(⋅,∇θ(un))-a(⋅,θ(∇v))∇(θ(un)-v)+a(⋅,θ(∇v))∇(θ(un)-v),b(⋅,∇un)∇(θ(un)-v)=b(⋅,∇un)-b(⋅,∇v)∇(θ(un)-v)+b(⋅,∇v)∇(θ(un)-v),∇(θ(un)-v)=(θ′(un)∇un-∇v)=θ′(un)∇un-θ′(un)∇v+θ′(un)∇v-∇v,
and by the monotonicity of the operator defined by a and b, we obtain
(3.10)〈∂un∂t,θ(un)-v〉+∫Qa(⋅,θ(∇v))(∇θ(un)-∇v)+∫Q1nb(⋅,∇v)(∇θ(un)-∇v)+∫Q1n(b(⋅,∇un)-b(⋅,∇v))(θ′(un)-1)∇v≤∫Qf(θ(un)-v),
by passage to the limit with a standard argument as in [10, 11], and using the above convergence of θ(un), we have
(3.11)〈∂u∂t,θ(u)-v〉+∫Qa(⋅,∇θ(v))(∇θ(u)-∇v)dx≤∫Qf(θ(u)-v).
Taking now v=θ(u)-tψ, with ψ∈W01,xLB(Q) and t∈(-1,1), we deduce that u is solution of the problem (1.2).Step 3 (u∈BVloc(Q)).
LetK be a compact in Q, and let φ∈D(Q) with K⊂supp(φ) such that
(3.12)φ=1,on⁡K,|∇φ|≤1.
Using φ as test function in (3.7), we get
(3.13)∫Q∂ηk(un)∂tφdxdt+∫Qa(⋅,∇θ(un))∇φ⋅ηk′(un)dxdt+∫Qa(⋅,∇θ(un))φ⋅ηk′′(un)dxdt+1n∫QM(|∇un|)|∇un|2∇un∇φ⋅ηk′(un)dxdt+1n∫QM(|∇un|)|∇un|2∇unφ⋅ηk′′(un)dxdt=I1+I2+I3+I4+I5=∫Qfηk′(un)φdxdt.
The terms I2,I3,I4,I5 are bounded, so
(3.14)∫K|∂ηk(un)∂t|dxdt≤C.
Letting k tend to infinity, we have
(3.15)∫K|∂un∂t(x,t)|dxdt≤C.
We deal now with the following estimation which ends the proof.
For all compactK⊂Q,(3.16)∫K|Dun(x,t)|dxdt≤C.
Indeed, we differentiate the approximate problem with respect to xi, we multiply the obtained equation by ηk′(∂xiun), and one has the following equality in the distributions sense
(3.17)∂(∂xiun)∂tηk′(∂xiun)-div⁡(∂xia(⋅,∇θ(un)))ηk′(∂xiun)-1n∂xiΔM(un)ηk′(∂xiun)=∂xifηk′(∂xiun),
which is equivalent to
(3.18)∂ηk(∂xiun)∂t-div⁡(∂xia(⋅,∇θ(un)))ηk′(∂xiun)-1ndiv⁡(∂xib(⋅,∇un))ηk′(∂xiun)=∂xifηk′(∂xiun).
We recall thatηk,ηk′, and ηk′′ are bounded on R,(θ(un)) is bounded in W01,xLB(Q), and (a(·,∇θ(un))) is bounded in LB¯(Q).
Using now the test functionφ (defined below), we obtain, as for (3.14),
(3.19)∫K|∂ηk∂t(∂un∂xi)|dxdt≤C.
With the same way as above, we conclude the result, u∈BVloc(Q).Remark 3.3.
As in Theorem3.2, one can prove the same result in the case where we replace the initial condition in the problem P by θ(u(x,0))=θ(u0(x)) and θ(u0(x))∈L2(Q).



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*Source: 506857-2011-10-18.xml*