# First Boundary Value Problem for Cordes-Type Semilinear Parabolic Equation with Discontinuous Coefficients

Authors: Aziz Harman; Ezgi Harman
Journal: Journal of Mathematics
 (2020)

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## Abstract

For a class of semilinear parabolic equations with discontinuous coefficients, the strong solvability of the Dirichlet problem is studied in this paper. The problem∑i,j=1naijt,xuxixj−ut+gt,x,u=ft,x,uΓQT=0, in QT=Ω×0,T is the subject of our study, where Ω is bounded C2 or a convex subdomain of En+1,ΓQT=∂QT\∖t=T. The function gx,u is assumed to be a Caratheodory function satisfying the growth condition gt,x,u≤b0uq, for b0>0,q∈0,n+1/n−1,n≥2, and leading coefficients satisfy Cordes condition b0>0,q∈0,n+1/n−1,n≥2.


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## Body

## 1. Introduction

LetEn be an n-dimensional Euclidean space of points x=x1,x2,…,xn and Ω be a bounded domain in En with boundary ∂Ω of the class C2 or simply a convex domain. Set QT=Ω×0,T and ΓQT=∂QT\t=T. Consider in QT the Dirichlet problem:(1)∑i,j=1naijt,xuxixj−ut+gt,x,u=ft,x,t,x∈QT,(2)uΓQT=0.It is assumed that the coefficientsaijt,x,i,j=1,2,…,n, of the operator(3)L=∑i,j=1naijt,x∂2∂xi∂xj−∂∂t,are bounded measurable functions satisfying the uniform parabolicity(4)γξ2≤∑i,j=1naijt,xξiξj≤γ−1ξ2,for γ∈0,1,∀t,x∈QT,∀ξ∈En, and the Cordes-type condition(5)∑i,j=1naij2t,x∑i=1naiit,x2≤1n−μ2−δ.Here,μ=essinf∑i=1naiit,x/esssup∑i=1naiit,x, and the number δ∈0,1/n+1. The nonlinear term, function gt,x,u:QT⟶E1, satisfies the Caratheodory condition, that is, g is a measurable function with respect to variables t,x∈Ω, and for almost all t,x∈QT continuously depend on the variable u∈E1. Also, the growth condition(6)gt,x,u≤b0uq,b0>0,is satisfied.The spaceW˙p2,1QT, p>1, is a closure of function class u∈C∞Q¯T∩CQ¯T,uΓQT=0 with respect to norm(7)uW˙p2,1QT=uLpQT+∑i=1n∂xiuLpQT+∂tuLpQT+∑i,j=1n∂xi∂xjuLpQT.Here,ui,ut, and uij denote the weak derivatives uxi,ut, and uxixj, respectively, i,j=1,…,n. The conjugate number is denoted by p′, i.e., 1<p<∞, 1/p′+1/p==1. By the same letter C, we denote different positive constants, and the value of C is not essential for purposes of this study.Forp∈1,∞, we denote by vLpQT or simply vp the norm of a Banach space Lp0,T;LpΩ defined as gp=∫0Tgt,⋅LpΩpdt1/p.A functionut,x∈W˙p2,1QT is called the strong solution (almost everywhere) of problems (1) and (2) if it satisfies equation (1), a.e., in QT.In this study, we will make essential use of the existence results given in Theorem 1.1 of [1] (see, also [2]) for Cordes-type parabolic equations satisfying (5). In [1], the estimate(8)uW˙22,1QT≤CLuL2QT,was proved for all u∈W˙p2,1QT, and when T≤T0 with T0=T0n,L,Ω to be sufficiently small and positive constant C depends on n,Ω,L.In the stationary case, i.e., the solution does not depend on the time variable (the elliptic equation), from examples ([3], p. 48), it is followed that the equation Lu=f is solvable in W˙p2,1QT for no p>1 (see [3–8]) if the coefficients are discontinuous. In the absense of g (t, x, u), the strong solvability of the Dirichlet problem for quasi-linear parabolic equations under more restrictive then (5) conditions see, e.g. [9, 10].If the trace of matrixaijt,x is constant, condition (5) is exactly Cordes condition (see, e.g., [7, 11–13]):(9)∑i,j=1naij2t,x∑i=1naiit,x2≤1n−1−δ.For the strong solvability problem inW˙p2Ω for any p>1 for parabolic equations with discontinuous coefficients, we refer [8, 14, 15], where the leading coefficients are taken from the VMO class. We refer [16] on exact growth conditions for strong solvability of nonlinear elliptic equations Δu=gx,u,ux in W˙p2Ω whenever p>n.The aim pursued in this paper is to prove the strong solvability of Dirichlet problems (1) and (2) in the space W˙22,1QT for T to be sufficiently small, the ft,xL2QT norm to be sufficiently small, and the coefficients to satisfy (5).


## 2. Main Result

In order to carry out the proof of main Theorem1, we need the following assertion from [1].Lemma 1.
Letut,x be a W˙22,1QT function in QT=Ω×0,T and conditions (2), (4), and (5) be fulfilled for ut,x and coefficients of the operator L; the domain Ω is of C2 class or simply convex. Then, there exists sufficiently small T0 depending on ℒ,n,Ω such that, for T≤T0, estimate (8) holds with the constant C depending on ℒ,n,Ω.The following assertion is the main result of this paper.Theorem 1.
Letn>4,0<q<n+1/n−1, and conditions (4)–(6) be fulfilled, and ∂Ω∈C2. Let T0 be a number in Lemma 1 and T≤T0. Then, problems (1) and (2) have at least one strong solution in the space W˙22,1QT for any ft,x∈L2QT satisfying(10)fL2QT≤Cb0−1/q−1mesn+1QTqn−1/n+1−11/2q−1.Proof.
In order to get the solvability of problem (1) and (2), we apply the Schauder fixed point theorem on completely continuous mappings of a compact subset in the Banach space (see, e.g. [4], p. 257, or [17]).
SetL2qQT as a basic Banach space. In this space, we define the set V2=u∈W˙22,1QTuW2,12QT≤K, where the number K will be chosen later. Show that V2is compact in L2qQT. By using the condition 2q<2n+1/n−1 and Sobolev–Kondrachov’s compact embedding theorem, the space W21QT is imbedded into L2qQT compactly. On the contrary, W22,1QTW21QT is continuous. Therefore, V2L2qQT is compact.
ShowV2 is convex. For any u1,u2∈V2 and t∈0,1, it holds u=tu1+1−tu2∈V2:(11)uW22,1QT≤tu1W22,1QT+1−tu2W22,1QT≤K.
Forut,x∈V2, denote vt,x∈W˙22,1QT the solution of the Dirichlet problem:(12)Lv+gt,x,u=ft,x,t,x∈QT,(13)vΓQT=0.
For fixedut,x∈V2 and f∈L2QT, problems (12) and (13) are uniquely solvable in the space W˙22,1QT; because of the assumptions on domain and q, we get the Dirichlet problem for equation (1) (for its solvability, we refer [1, 2, 9, 10]):(14)Lv=Ft,x,t,x∈QT,uΓQT=0,where F=ft,x−gt,x,∈L2QT.
We have(15)FL2QT≤fL2QT+gL2QT≤fL2QT+b0uqL2QT.
By using the chain of imbeddings,W22,1QTW21QTL2qQT and u∈W˙22,1QT, the norm uqL2QT is finite.
Insert an operatorA:u⟶v acting on L2qQT, where v is a solution of problems (12) and (13):(16)Au=v.
Show that operatorA is completely continuous in L2qQT. Let um be a convergence sequence in L2qQT with um⟶u0. Show that its image is convergent in L2qQT with vm⟶v0, where v0=Au0,vm=Aum.
Then,(17)Lvm=−gt,x,um+f,⋮Lv0=−gt,x,u0+f.
We have(18)Lvm−v0=−gt,x,um−gt,x,u0.
Setgm=gt,x,um, g=gt,x,u, and show that(19)gm−gL2QT⟶0form⟶∞.
For that, fromum⟶u0 in L2qQT follows the convergnce in measure in QT. This and the Caratheodory condition imply that the convergence in measure gm−g02⟶0. To prove (19), it remains to show the equicontinuity of gm2, which follows from equicontinuity of um2q. The convergence um⟶u0 in L2qQT implies equicontinuity of um2q.
Applying Vitali’s theorem, we get(20)gm−gL2QT⟶0asm⟶∞.
To showvm⟶v0 in L2qQT, we use the estimate from Lemma 1 for sufficiently small T0 with T≤T0:(21)vm−v0W22,1QT≤CLvm−v0L2QT=Cgm−gL2QT⟶0.
By virtue ofW˙22,1QT↪L2qQT, it follows that(22)vn−v0Lr,2Ω⟶0asn⟶∞.
The complete continuity of operatorA in L2qQT has been shown.
Now, we have to showu∈V2 implies v=Au∈V2. For this, applying Lemma 1, it follows that(23)vW22,1QT≤CFL2QT≤Cδ,γ,ngL2QT+fL2QT.
Using Holder’s inequality and the imbedding chain(24)W22,1QT↪W21QT↪L2qQT,it follows that(25)gL2QT≤∫QTb02u2qdxdt1/2=b0uL2qQTq≤Cb0u2n+1/n−1qmesn+1QT1/2−qn−1/2n+1≤Cb0mesn+1QT1/2−qn−1/2n+1uW21QTq≤C2b0mesn+1QT1/2−qn−1/2n+1uW22,1QTq.
Using Lemma1, this is exceeded:(26)C1b0mesn+1QT1/2−qn−1/2n+1ℒuL2QTq.
Using estimate (26) in (23), we get(27)vW22,1QTC1b0mesn+1QT1/2−qn−1/2n+1ℒuL2QTq+fL2Ω≤C3Kqb0mesn+1QT1/2−qn−1/2n+1fL2Ω.
LetK be such that(28)C2.5Kqb0mesn+1QT1/2−qn−1/2n+1+fL2Ω≤K.
For such numberK to exist, condition (10) is sufficient. To prove it, set the notation(29)a=b0mesn+1QT1/2−qn−1/2n+1,b=fL2Ω.
Inequality (28) takes the form(30)aKq+b≤K,aKq−K+b≤0,K>0.
The functionfK=aKq−K, K≥0, takes its minimal in K0=1/qa1/q−1. Indeed, df/dK=aqKq−1−1; then, for K0q−1=1/qa,df/dKK0=0;d2f/dK2K0>0. Therefore, for b≤fK0, inequality (30) is solvable with respect to K. To finish the proof, it remains to set sufficiently small T0 so that condition (10) is satisfied. It is possible since mesn+1QT=TmesnΩ, the power on mesn+1QT, is positive, i.e., 1/2−qn−1/2n+1>0.
This completes the proof of Theorem1.


## 3. Conclusion

In this paper, the strong solvability problem for a class of second-order semilinear parabolic equations is studied. For the strong solvability of the first boundary value problem for a class of parabolic equations having a nonlinear term, a sufficient condition is found for the power growth condition. In the proof, the Schauder fixed point theorem in the Banach space is used. Also, some a priori estimates are shown in order to realize the legitimate.



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*Source: 1019038-2020-06-19.xml*