Abstract
The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach’s fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019].
Highlights
In this paper, we consider the following Kirchhoff-type problem for parabolic equation systems: ⎧ ⎪⎪⎨ ∂u ∂t = L( ∇u L2, ∇v L2 ) u,⎪⎪⎩u∂∂vt(x=, tL) =(∇u v(x, L2, t) =
Using the Banach fixed point theorem, we show that our problem has a unique mild solution
2 Some preliminaries and the mild solution of problem (1.1)-(1.2) we introduce some properties of the eigenvalues of the operator – ; see, for example, [6]
Summary
1 Introduction In this paper, we consider the following Kirchhoff-type problem for parabolic equation systems: The main tool in the paper is the Fourier series technique in Hs spaces, combined with Banach’s fixed point theorem. The first contribution result is the proof of the existence and uniqueness of a solution of our backward problem. 2, we introduce some preliminaries and the mild solution of problem (1.1)–(1.2).
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