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 in is the subject of our study, where is bounded or a convex subdomain of . The function is assumed to be a Caratheodory function satisfying the growth condition , for , and leading coefficients satisfy Cordes condition .

1. Introduction

Let be an -dimensional Euclidean space of points and be a bounded domain in with boundary of the class or simply a convex domain. Set and Consider in the Dirichlet problem:

It is assumed that the coefficients , of the operatorare bounded measurable functions satisfying the uniform parabolicityfor and the Cordes-type condition

Here, , and the number The nonlinear term, function , satisfies the Caratheodory condition, that is, is a measurable function with respect to variables , and for almost all continuously depend on the variable Also, the growth conditionis satisfied.

The space , , is a closure of function class with respect to norm

Here, , and denote the weak derivatives , and , respectively, . The conjugate number is denoted by , i.e., , . By the same letter , we denote different positive constants, and the value of is not essential for purposes of this study.

For , we denote by or simply the norm of a Banach space defined as

A function is called the strong solution (almost everywhere) of problems (1) and (2) if it satisfies equation (1), a.e., in .

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 estimatewas proved for all , and when with to be sufficiently small and positive constant depends on .

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 is solvable in for no (see [38]) 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 matrix is constant, condition (5) is exactly Cordes condition (see, e.g., [7, 1113]):

For the strong solvability problem in for any for parabolic equations with discontinuous coefficients, we refer [8, 14, 15], where the leading coefficients are taken from the class. We refer [16] on exact growth conditions for strong solvability of nonlinear elliptic equations in whenever .

The aim pursued in this paper is to prove the strong solvability of Dirichlet problems (1) and (2) in the space for to be sufficiently small, the norm to be sufficiently small, and the coefficients to satisfy (5).

2. Main Result

In order to carry out the proof of main Theorem 1, we need the following assertion from [1].

Lemma 1. Let be a function in and conditions (2), (4), and (5) be fulfilled for and coefficients of the operator ; the domain is of class or simply convex. Then, there exists sufficiently small depending on such that, for , estimate (8) holds with the constant depending on

The following assertion is the main result of this paper.

Theorem 1. Let , and conditions (4)–(6) be fulfilled, and . Let be a number in Lemma 1 and . Then, problems (1) and (2) have at least one strong solution in the space for any satisfying

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]).
Set as a basic Banach space. In this space, we define the set , where the number will be chosen later. Show that is compact in . By using the condition and Sobolev–Kondrachov’s compact embedding theorem, the space is imbedded into compactly. On the contrary, is continuous. Therefore, is compact.
Show is convex. For any and , it holds :For , denote the solution of the Dirichlet problem:For fixed and , problems (12) and (13) are uniquely solvable in the space ; because of the assumptions on domain and , we get the Dirichlet problem for equation (1) (for its solvability, we refer [1, 2, 9, 10]):where .
We haveBy using the chain of imbeddings, and , the norm is finite.
Insert an operator acting on , where is a solution of problems (12) and (13):Show that operator is completely continuous in . Let be a convergence sequence in with . Show that its image is convergent in with , where .
Then,We haveSet , , and show thatFor that, from in follows the convergnce in measure in . This and the Caratheodory condition imply that the convergence in measure To prove (19), it remains to show the equicontinuity of which follows from equicontinuity of The convergence in implies equicontinuity of
Applying Vitali’s theorem, we getTo show in , we use the estimate from Lemma 1 for sufficiently small with :By virtue of , it follows thatThe complete continuity of operator in has been shown.
Now, we have to show implies . For this, applying Lemma 1, it follows thatUsing Holder’s inequality and the imbedding chainit follows thatUsing Lemma 1, this is exceeded:Using estimate (26) in (23), we getLet be such thatFor such number to exist, condition (10) is sufficient. To prove it, set the notationInequality (28) takes the formThe function , , takes its minimal in . Indeed, ; then, for . Therefore, for , inequality (30) is solvable with respect to . To finish the proof, it remains to set sufficiently small so that condition (10) is satisfied. It is possible since , the power on , is positive, i.e.,
This completes the proof of Theorem 1.

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.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank Professor Farman Mamedov for assistance in preparing this paper and his valuable suggestions.