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

. For a class of semilinear parabolic equations with discontinuous coeﬃcients, the strong solvability of the Dirichlet problem is studied in this paper. The problem 􏽐 ni,j � 1 a ij ( t,x ) u x i x j − u t + g ( t, x,u ) � f ( t, x ) ,u | Γ ( Q T ) � 0 , in Q T � Ω × ( 0 , T ) is the subject of our study, where Ω is bounded C 2 or a convex subdomain of E n + 1 , Γ ( Q T ) � z Q T \ t � T { } . The function g ( x,u ) is assumed to be a Caratheodory function satisfying the growth condition | g ( t, x,u )| ≤ b 0 | u | q , for b 0 > 0 , q ∈ ( 0 , ( n + 1 ) / ( n − 1 )) ,n ≥ 2, and leading coeﬃcients satisfy Cordes condition b 0 > 0 ,q ∈ ( 0 , ( n + 1 ) / ( n − 1 )) ,n ≥ 2.


Introduction
Let E n be an n-dimensional Euclidean space of points x � (x 1 , x 2 , . . . , x n ) and Ω be a bounded domain in E n with boundary zΩ of the class C 2 or simply a convex domain. Set Q T � Ω × (0, T) and Γ(Q T ) � zQ T \ t � T { }. Consider in Q T the Dirichlet problem: It is assumed that the coefficients a ij (t, x), i, j � 1, 2, . . . , n, of the operator are bounded measurable functions satisfying the uniform parabolicity for c ∈ (0, 1), ∀(t, x) ∈ Q T , ∀ξ ∈ E n , and the Cordes-type condition n i,j�1 a 2 ij (t, x) n i�1 a ii (t, x) Here, μ � (ess inf n i�1 a ii (t, x))/(ess sup n i�1 a ii (t, x)), and the number δ ∈ (0, (1/(n + 1))). e nonlinear term, function g(t, x, u): Q T ⟶ E 1 , satisfies the Caratheodory condition, that is, g is a measurable function with respect to variables (t, x) ∈ Ω, and for almost all (t, x) ∈ Q T continuously depend on the variable u ∈ E 1 . Also, the growth condition is satisfied.
Here, u i , u t , and u ij denote the weak derivatives u x i , u t , and u x i x j , respectively, i, j � 1, . . . , n. e 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.
For p ∈ [1, ∞], we denote by ‖v‖ L p (Q T ) or simply ‖v‖ p the norm of a Banach space L p [0, T; L p (Ω)] defined as is called the strong solution (almost everywhere) of problems (1) and (2) if it satisfies equation (1), a.e., in Q T .
In this study, we will make essential use of the existence results given in eorem 1.1 of [1] (see, also [2]) for Cordestype parabolic equations satisfying (5). In [1], the estimate was proved for all u ∈ _ W 2,1 p (Q T ), and when T ≤ T 0 with T 0 � T 0 (n, L, Ω) to be sufficiently small and positive constant C depends on n, Ω, L.
If the trace of matrix ‖a ij (t, x)‖ is constant, condition (5) is exactly Cordes condition (see, e.g., [7,[11][12][13]): For the strong solvability problem in _ W 2 p (Ω) 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 whenever p > n. e aim pursued in this paper is to prove the strong solvability of Dirichlet problems (1) and (2) in the space _ W 2,1 2 (Q T ) for T to be sufficiently small, the ‖f(t, x)‖ L 2 (Q T ) norm to be sufficiently small, and the coefficients to satisfy (5).

Main Result
In order to carry out the proof of main eorem 1, we need the following assertion from [1]. (2), (4), and (5) be fulfilled for u(t, x) and coefficients of the operator L; the domain Ω is of C 2 class or simply convex. en, there exists sufficiently small T 0 depending on L, n, Ω such that, for T ≤ T 0 , estimate (8) holds with the constant C depending on L, n, Ω. e following assertion is the main result of this paper. Theorem 1. Let n > 4, 0 < q < (n + 1/n − 1), and conditions (4)-(6) be fulfilled, and zΩ ∈ C 2 . Let T 0 be a number in Lemma 1 and T ≤ T 0 . en, problems (1) and (2) 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 L 2q (Q T ) as a basic Banach space. In this space, we where the number K will be chosen later. Show that V 2 is compact in L 2q (Q T ). By using the condition 2q < 2(n + 1)/(n − 1) and Sobolev-Kondrachov's compact embedding theorem, the space W 1 2 (Q T ) is imbedded into L 2q (Q T ) compactly. On the contrary, W 2,1 2 (Q T )W 1 2 (Q T ) is continuous. erefore, Show V 2 is convex. For any u 1 , u 2 ∈ V 2 and t ∈ [0, 1], it holds u � tu 1 + (1 − t)u 2 ∈ V 2 : For For fixed u(t, x) ∈ V 2 and f ∈ L 2 (Q T ), problems (12) and (13) are uniquely solvable in the space _ W 2,1 2 (Q T ); 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]): where F � f(t, x) − g(t, x, ) ∈ L 2 (Q T ).
We have 2 Journal of Mathematics By using the chain of imbeddings, Insert an operator A: u ⟶ v acting on L 2q (Q T ), where v is a solution of problems (12) and (13): Show that operator A is completely continuous in en, We have x, u), and show that For that, from u m ⟶ u 0 in L 2q (Q T ) follows the convergnce in measure in Q T .
is and the Caratheodory condition imply that the convergence in measure (g m − g 0 ) 2 ⟶ 0. To prove (19), it remains to show the equicontinuity of g 2 m , which follows from equicontinuity of |u m | 2q . e convergence u m ⟶ u 0 in L 2q (Q T ) implies equicontinuity of |u m | 2q .
Applying Vitali's theorem, we get To show v m ⟶ v 0 in L 2q (Q T ), we use the estimate from Lemma 1 for sufficiently small T 0 with T ≤ T 0 : e complete continuity of operator A in L 2q (Q T ) has been shown. Now, we have to show u ∈ V 2 implies v � Au ∈ V 2 . For this, applying Lemma 1, it follows that Using Holder's inequality and the imbedding chain it follows that Using Lemma 1, this is exceeded: Using estimate (26) in (23), we get Let K be such that For such number K to exist, condition (10) is sufficient. To prove it, set the notation Inequality (28) takes the form e function f(K) � aK q − K, K ≥ 0, takes its minimal in K 0 � (1/qa) 1/(q− 1) . Indeed, df/dK � aqK q− 1 − 1; then, for K q− 1 0 (30) is solvable with respect to K. To finish the proof, it remains to set sufficiently small T 0 so that condition (10) is satisfied. It is possible since mes n+1 Q T � Tmes n Ω, the power on mes n+1 Q T , is positive, i.e., (1/2) − (q(n − 1)/2(n + 1)) > 0.
is completes the proof of eorem 1.

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.

Journal of Mathematics 3
Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.