The local well-posed results of Kirchhoff parabolic equation with nonlocal condition

Abstract

This article is devoted to the Kirchhoff type parabolic problem with a nonlocal condition. Our problem has many applications in physical and biological phenomena. Applying Banach fixed point theory, we first show that the local existence of the mild solution under some prior assumptions on the input data f. Next, we investigate the convergence of the mild solution as ϵ goes to zero, and we have proven the solution of our problem to the solution of backward in time problem when T is enough small.