# Bounded Solutions to Nonlinear Parabolic Equations

Authors: Jaouad Igbida
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

We deal with existence results for nonlinear parabolic equations with general quadratic gradient terms and with absorption term which depend on the solution. We note that no boundedness is assumed on the data of the problem. We prove an existence result of distributional solution via test-function method. A priori estimates and compactness arguments are our main ingredient; the method of sub-supersolution does not apply her.


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## Body

## 1. Introduction

This work is devoted to deal with existence results concerning nonlinear parabolic equations with both first-order terms having quadratic growth with respect to the gradients and superlinear absorption terms which depend on the solution. Let us consider the following Cauchy-Dirichlet problem:(1.1)ut+Λu+A(x,u)=β(u)|∇u|2+f(x,t)inΩ×]0,T[,u=0on⁡∂Ω×]0,T[,u(x,0)=u0(x)inΩ,
where Λu=-div(a(x,t,u,∇u)) is a pseudomonotone, coercive, uniformly elliptic operator acting from L2(0,T;H01(Ω)) to its dual. Ω is a bounded open set in ℝN, N≥1 and T>0. The operator A(x,u) grows like |u|r. The unknown real function u depends on x∈Ω and t∈]0,T[. Let a:Q×ℝ×ℝN→ℝN be a Carathéodory function (i.e., a(…,σ,ζ) is measurable in Q for every (x,t) in Q).(A1)
There exists a constantβ>0 such that
(1.2)|a(x,t,σ,ζ)|≤β[k(x,t)+|σ|+|ζ|],
for almost every (x,t) in Q and every (σ,ζ) in ℝ×ℝN, where k(⋯) is a nonnegative function in L2(Q).(A2)
There exists a constantα such that
(1.3)a(x,t,σ,ζ)≥α|ζ|2,
for almost every (x,t) in Q and every (σ,ζ) in ℝ×ℝN.(A3)(1.4)[a(x,t,σ,ζ)-a(x,t,σ,η)]⋅(ζ-η)>0,for almost every(x,t) in Q, for every σ in ℝ, for every ζ,η in ℝN,ζ≠η.Let us define the differential operatorA(x,u) as follows:
(1.5)A(x,u)=a(x)u|u|r-1.The functionf satisfies
(1.6)f(x,t)∈Lr(0,T;Lq(Ω)),withr>1,Nq+2r<2.
The initial data satisfy the following hypothesis:
(1.7)ψ(u0)∈L1(Ω),
where
(1.8)ψ(s)=∫0seγ(r)dr,withγ(s)=∫0sβ(σ)dσ.The source termf satisfies the same condition considered by Aronson and Serrin in [1] to show the existence of a solution for the classical problem (∂u/∂t)-Δu=f(x,t).  The condition on the source term f is optimal. Indeed, if f∈L∞(0,T;Lq(Ω)) or if f(x,t)=f(x)∈Lq(Ω), with q>(N/2), the condition (1.6) is still satisfied.Let us note that we studied the elliptic problem associated to(1.1) in [2]. This kind of problems has been extensively studied in the last years by many authors (see, e.g., [3–13] and the references therein). In these works, the hypothesis on the function β implies grosso modo that β is bounded, with some restrictions as in [4, 10, 14]. A special condition is assumed in [10], where β is supposed to tend to +∞ for s tending to ∞.There is many results for particular situations of our problem (see, e.g., [4, 7, 12, 13]). All this work have studied the question of existence of distributional solutions for this problem in the case where a≡0, u0∈L∞(Ω), and f(x,t)∈Lr(0,T;Lq(Ω)) with (q(r-1)/r)>(N/2) The existence has also been studied in the case where u0∈L∞(Ω) and f(x,t)=f(x)∈Lm(Ω×]0,T[), m>1+(N/2), which are special cases of our conditions.In the case wherea≡0, ψ(u0)∈L2(Ω) and for more general condition on β, existence results of a solution for parabolic convection diffusion problems
(1.9)∂u∂t-Δu=β(u)|∇u|2+f(x,t)
have been given in [15], [8].We can reduce this problem with the change ofu with the Col-Hopf change of variable
(1.10)v=ψ(u)=∫0uexp⁡(∫0rβ(s)ds)dr.
to the following
(1.11)∂v∂t-Δv=f(x,t)(1+|v||log⁡|v||α),In this stage, we have an existence result of distributional solutions via test function method. That gives the a priori estimates for the approximate problem associated with (1.11) which also provide a priori estimates for the approximate problem associated with (1.9) and, therefore, an existence result of distributional solution for problem (1.9).One cannot perform such a change of variable, when trying to extend the previous results to our more general situation, where one has a general first-order term which grows quadratically with respect to the gradient and with superlinear reaction terms which grow like|u|r. Therefore, we shall use some convenient test functions to prove the a priori estimates and use compactness arguments to prove an existence result of distributional solutions of (P). We point out that for this class of problems, the regularity assumed on the data f and u0, can not expect bounded solutions.We also point out that we are interested in solutions having finite norms inL2(0,T;H01(Ω)). The techniques used in this paper are mainly based on a linear operator and on the concept of distributional solutions. These approachs allow to have, in the case of both subcritical growth and a reaction terms with u, existence results. The first ingredient of our proof consists in obtaining certain a priori bounds on the solutions of approximate problems and some suitable L1-norm of diffusion terms. A convenient use of Young's inequality will give a uniform estimate of the L2(0,T;H01(Ω))-norms and, therefore, the weak convergence up to a subsequence. We will prove that there exists u such that, up to a subsequence the solution un of the approximate problems converges to u, all everywhere convergence of gradient of un to gradient of u, up to a subsequence, which is important in the study of the limiting process. Next, we will prove the convergence of the superlinear reaction term and the quadratic gradient term in L1(Ω×]0,T[). Another interesting approach is in some sense the combination of the previous, in studies of the behavior of sequences of approximating solutions. Likewise, we will see that the solutions of the approximates problems converge to the solution of the model problem in C([0,T];L1(Ω)), which gives meaning to the initial condition.


## 2. Basic Results

LetΩ be a bounded domain in ℝN, N≥1. We denote by QT for T>0 the set Ω×]0,T[ and by Γ the set ∂Ω×]0,T[.We consider the following nonlinear problem that we denote by(P)(2.1)ut+Λu+A(x,u)=B(u,∇u)+f(x,t)inΩ×]0,T[,u=0on⁡∂Ω×]0,T[,u(x,0)=u0(x)inΩ,
where the unknown function u=u(x,t) is a real function depending on x∈Ω and t∈]0,T[. Λ,A and B are differential operators such that
(2.2)Λu:=-div⁡(a(x,t,u,∇u)),B(u,∇u)=β(u)|∇u|2.Let us consider the following assumptions.(H1)
The real functionβ is such that
(2.3)βiscontinuousnonincreasing,withβ(0)=0.(H2)
The real functionsu0 and f are satisfying (1.7) and (1.6), respectivelyThe operatorA(x,u) is such that
(2.4)A(x,u)=a(x)u|u|r-1,
where(H3)
r>1 and a(⋯) is a real function such that
(2.5)a∈L∞(Ω),a(x)≥a0>0a.e.x∈Ω.By a weak solution of the problem (1.1), we mean a function u∈L2(0,T;H01(Ω)) such that
(2.6)a(x)u|u|∈Lr(QT),β(u)|∇u|2∈L1(QT)
and satisfying
(2.7)∫QTutϕ+∫QTa(x,t,u,∇u)∇ϕ+∫QTa(x)u|u|r-1ϕ=∫QTβ(u)|∇u|2ϕ+∫QTfϕ,
for any test function ϕ in Cc1(QT) (the C1 functions with compact support).In the sequel we denote byθ a truncation function satisfying θ∈C∞(ℝ), 0≤θ≤1, θ(η)=1 for |η|≤ρ, θ(η)=0 for |η|⩾2ρ and |∇θ|≤(2/ρ), where ρ is a positive real. By c we denote different constants in ℝ which may vary from line to line. The main result in this paper is the following.Theorem 2.1.
If the hypotheses(A1)–(A3) and (H1)–(H3) are satisfied, then the problem (1.1) admits at least one solution u, such that
(2.8)u∈L2(0,T;H01(Ω))∩C0([0,T];L2(Ω)),a(x)u|u|∈Lr(QT),β(u)|∇u|2∈L1(QT).To prove the main result, we approximate our problem by a sequence of regular problems and show a priori estimates of solutions. Next, we shall prove the convergence of approximating solutions to some function that solves our problem.Let us recall the classical inequality of Poincaré and Sobolev (see [16, Chapters 7.7 and 7.8]).Lemma 2.2.
LetΩ be an open subset of ℝN with finite Lebesgue measure. Then, for every p such that 1≤p<∞, one has the following inequality:
(2.9)‖u‖Lp(Ω)≤(|Ω|ωN)1/N‖∇u‖Lp(Ω),for everyu∈W01,P(Ω),
where ωN is the measure of the unit ball in ℝN. Furthermore, there exists a constant c=c(N,p) such that, for allu∈W01,P(Ω),
(2.10)‖u‖Lp*(Ω)≤c‖∇u‖Lp(Ω),for anyp<N,‖u‖L∞(Ω)≤|Ω|(1/(N-1))/P‖∇u‖Lp(Ω),forp>N,
where p*=Np/(N-p).Remark 2.3.
By an approximation argument, the same inequality holds true if we replace|Ω| by |{x∈Ω:u(x)≠0}|.Let us recall next the Gagliardo-Nirenberg's inequality for evolution spaces.Lemma 2.4 (see, e.g., [17]).
LetΩ be a bounded open set of ℝN and T a real positive number. Let v(x,t) be a function such that v∈L∞(0,T;L2(Ω))∩L2(0,T;H01(Ω)). Then
(2.11)v∈Lρ(0,T;Lσ(Ω)),where,2≤σ≤2NN-2,2≤ρ≤∞,Nσ+2ρ=N2,
and the following estimate holds
(2.12)∫0T‖v(t)‖Lσ(Ω)ρdt≤c(N)‖v‖L∞(0,T;L2(Ω))ρ-2∫0T‖∇v(t)‖L2(Ω,RN)2dt.We are interested in studding a sequence of regular problems approximating the model problem. We prove the existence of bounded solutions for the approximating problems, and this bound does not depend onn. We shall prove some a priori estimates on the solutions of this sequence of problems which serves in the limiting process.


## 3. Approximating Problems

We regularize the problem (1.1) by considering the following sequence of problems:
(3.1)∂un∂t-Λun+An(x,un)=Bn(un,∇un)+fn(x,t)inQn,un(x,t)=0on⁡Γn,un(x,0)=u0n(x)inΩn,
where
(3.2)Bn(un,∇un)=βn(un)|∇un|2,βn(η)={nifβ(η)>n,β(η)if-n≤β(η)≤n,-nifβ(η)<-n.
Let us consider
(3.3)An(x,un)=a(x)un|un|r-1.Next, we consider the truncated function(3.4)fn=inf⁡(|f|,n)sign⁡(f).We denote byΩn a strictly increasing sequence of bounded sets Ω∩Bn invading Ω. Next we denote
(3.5)Qn=Ωn×]0,T[,Γn=∂Ωn×]0,T[.From standard results (see, e.g., [14]), the following problem:
(3.6)∂un∂t-Λun+Υn(x,un,∇un)=fn(x,t)inQn,un(x,t)=0on⁡Γn,un(x,0)=u0n(x)inΩn,
where
(3.7)Υn(x,un,∇un)=An(x,un)-Bn(un,∇un),
admits at least one solution un satisfying
(3.8)un∈L2(0,T;H01(Ωn))∩C(0,T;L2(Ωn)),unt∈L2(0,T;H-1(Ωn)),|un|r∈L1(Qn).Then one has the following estimates:(3.9)∫0T∫Ω|∇un|2+∫0T∫Ωa(x)un|un|r-1≤c(ρ,λ,q,T).Indeed, let us define the following function:(3.10)ϕ(s)=∫0sH(σ)dσ,fors>0,ϕ(s)=-ϕ(-s),fors<0,
wehere
(3.11)H(s)=1(1+s)q+1,0<q<1.We introduce the functionψ defined by
(3.12)ψ(s)=∫0sϕ(σ)dσ.
Let us consider the following sequences:
(3.13)I1,n=∫Ωnψ(un(x,T))θλ+1,I2,n=α∫Qn|∇un|2ϕ′(un)θλ+1,I3,n=β(λ+1)∫Qn|∇un||∇θϕ(un)|θλ,I4,n=∫Qna(x)un|un|r-1ξ.Takingξ=ϕθλ+1 with λ>0, we obtain
(3.14)I1,n+I2,n+I3,n+I4,n≤∫B2ρf(x,t)ξ+∫B2ρψ(u0(x))θλ+1.Applying Young's inequality, one has the following inequality:(3.15)|∇u|θλ≤ϵ2|∇u|2θλ+1(1+|u|)q(λ+1)+c(λ)2ϵθλ-1(1+|u|)q+1.We now chooseq such that q<r, which is possible, since r>1 and q>0. We use again Young's inequality twice and we obtain
(3.16)|∇u|θλ≤ϵϕ′(u)|∇u|2θλ+12(λ+1)+ϵa0λ+1|u|r-1uϕ(u)θλ+1+c(ρ,λ,q,a0)≤12|∇u|2θλ+1(1+|u|)q+a(x)2|u|r-1uϕ(u)θλ+1+c(ρ,λ,q,a0).
Then
(3.17)1βI3,n≤12∫Qn|∇un|2ϕ′(un)θλ+1+∫Qna(x)un|un|r-1ϕ(u)θλ+1+c(ρ,λ,q).From (3.14), we obtain(3.18)∫Ωnψ(un(x,T))θλ+1+12∫Qn|∇un|2ϕ′(un)θλ+1+12∫Qna(x)un|un|r-1ϕ(u)θλ+1≤c(ρ,λ,q,T).In consequence,(3.19)∫0T∫Ω|∇un|2+∫0T∫Ω|un|r≤c(ρ,λ,q,T).Next, we substituteT by t for any t, 0≤t≤T in (3.18), which is possible. We obtain
(3.20)∫Bn|un(x,t)|r≤c,
where c is a constant that does not depend on n. then, the approximate problem admits at least one solution which is bounded independently on n in L∞(0,T;Lr(Ω)).Let us now prove that the sequenceβn(un)|∇un|2 is bounded in L1(QT). We denote
(3.21)ψ(r)=∫0re|γ(σ)|dσ,
where
(3.22)γ(r)=∫0rβ(σ)dσ.We define, for anyk>0 fixed, the functions hk(r) defined as follows:(3.23)hk(r)=χ[|r|>k](s)∫krβ(σ)e(|γ(σ)|-γ(k))dσ,ϕk(r)=∫0rhk(σ)dσ.
Let us consider the following sequences:
(3.24)T1,n=∫Ω∩[|un(T)|>k]ϕk(un(T,x))dx,T2,n=α∫[|un|>k]|∇un|2βn(un)e(|γ(un)|-γ(k)),T3,n=∫Ω∩[|u0n|>k]ϕk(u0n(x))dx,T4,n=∫[|un|>k]|∇un|2βn(un)(e(|γ(un)|-γ(k))-1).We can choosehk(un) as test function, and we obtain
(3.25)T1,n+T2,n-T3,n≤∫[|un|>k]f(x,t)sign⁡(un)(e(|γ(un)|-γ(k))-1)+T4,n.
Then, we deduce
(3.26)T2,n≤∫[|un|>k]f(x,t)sign⁡(un)(e(|γ(un)|-γ(k))-1)+∫Ω∩[|u0n|>k]ϕk(u0n(x))dx≤∫[|un|>k]f(x,t)e|γ(un)|+∫Ω∩[|u0n|>k]ϕk(u0n(x))dx.
In consequence,(3.27)α∫[|un|>k]|∇un|2βn(un)≤c‖fχ[|un|>k]‖r,q‖e|γ(un)|‖r′,q′+∫Ω∩[|u0n|>k]ψ(u0n).Finally, we obtain(3.28)‖e|γ(un)|‖r′,q′≤c‖1+|ψ(un)|2‖r′,q′≤c(1+‖ψ(un)‖2r′,2q′2).
Then, the sequence βn(un)|∇un|2 is bounded in L1(QT).We require the all everywhere convergence of gradientun to the gradient of u. Let us consider
(3.29)ψ(s)={inf⁡(s,ϵ)ifs≥0,-inf⁡(s,ϵ)ifs≤0.Substitutingu in the approximating problem successively with un and um, we consider the following function:
(3.30)ϕ=ψ(un-um)θ.
After substraction, for n, m≥4ρ, we obtain the following inequality:
(3.31)∫QT∂(un-um)∂tϕ+α∫QT|∇un-∇um|2ϕ+∫QTa(x)(un|un|r-1-um|um|r-1)ϕ≤∫QT(βn(un)|∇un|2-βm(um)|∇um|2)ϕ+∫QT(fn-fm)ϕ.
Let us now consider for n,m sufficiently large the following sequence:
(3.32)ϝn=fn-a(x)un|un|r-1+βn(un)|∇un|2.
Then we get
(3.33)∫0T∫[|un-um|≤ϵ]∩Bρ|∇un-∇um|2≤cϵ∫0T∫B2ρ(|ϝn|+|ϝm|)dx.
Since (3.20) holds, then a(x)un|un|r-1 is in L1(QT). So that,ϝn is bounded in L1(QT), then one has
(3.34)∫0T∫[|un-um|≤ϵ]∩Bρ|∇un-∇um|2≤cϵ.Using Holder's inequality, we obtain(3.35)∫0T∫Bρ|∇un-∇um|≤(∫0T∫[|un-um|≤ϵ]∩Bρ|∇un-∇um|2)1/2c+∫0T∫[|un-um|≥ϵ]∩Bρ|∇un-∇um|.Using (3.34), one has
(3.36)∫0T∫[|un-um|≤ϵ]∩Bρ|∇un-∇um|2⟶0asn,m⟶+∞.On the other hand, since the measure of[|un-um|≥ϵ]∩Bρ converges to 0 as n, then, as m converges to +∞, then (3.37)∫0T∫[|un-um|≥ϵ]∩Bρ|∇un-∇um|→0.Therefore(3.38)∫QT|∇un-∇um|⟶0asn,m⟶+∞.


## 4. Limiting Process

We denote byun the solution of the approximate problems (Pn) on Ωn with initial condition U0n. To prove the main result, we deal with the limiting process of the approximating problems. First of all, we will prove that there exists u such that, up to a subsequence, (un) converges to u, for almost every (x,t)∈QT. First, we will prove the all everywhere convergence of the gradients of un to the gradient of u, up to a subsequence, in QT. Next, we will prove the convergence of the superlinear reaction term and the quadratic gradient term in L1(QT). Finally, we will see that (un) converges to u in C([0,T];L1(Ω)), which gives meaning to the initial condition.From (3.38) and up to a subsequence (un), we have
(4.1)∇un⟶∇ua.e.inQT.By consequence, since∇un is bounded in L1(QT), Vitali's theorem implies that
(4.2)∇un⟶∇uinL1(QT).Since one has(4.3)un∈L2(0,T;H01(Bn))∩C([0,T];L1(Bn)),By a diagonal process, we may select a subsequence, also denoted by{un}, such that
(4.4)un⟶uweaklyinL2(0,T;H01(Ω)),
and also
(4.5)un⟶ua.e.inΩ.From the construction ofu0n and fn, we have
(4.6)u0n⟶u0inL1(Ω),fn⟶finLr(0,T;Lq(Ω)).
Since unt∈L2(0,T;H0-1(Ωn))+L1(Qn), Then using compactness arguments (see [18]), we have
(4.7)un⟶ustronglyinL1(QT).From (4.5) and the fact that a(x)un is bounded in Lr(QT), the equi-integrability of a(x)un|un|r-1 is derived and then from Vitali's theorem, we have
(4.8)a(x)un|un|r-1⟶a(x)u|u|r-1inL1(QT).
By consequence,
(4.9)An(x,un)⟶A(x,u)inL1(QT).Since(4.10)un⟶ua.e.inQT,∇un⟶∇ua.e.inQT,
then
(4.11)βn(un)|∇un|2⟶β(u)|∇u|2a.e.inQT,Bn(un,∇un)⟶B(u,∇u)a.e.inQT.
Therefore, from (4.1) and Vitali's theorem, we conclude that
(4.12)βn(un)|∇un|2⟶β(u)|∇u|2inL1(QT),Bn(un,∇un)⟶B(u,∇u)inL1(QT).Finally, the sequence(un)n belongs to C0(0,T;H-1(Ω)) and un(x,0)=u0n(x); this implies that the initial condition u(x,0)=u0(x) is satisfied.



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