Existence of Solutions to a Nonlinear Parabolic Equation of Fourth-Order in Variable Exponent Spaces

Abstract

This paper is devoted to studying the existence and uniqueness of weak solutions for an initial boundary problem of a nonlinear fourth-order parabolic equation with variable exponent v t + div ( | ∇ ▵ v | p ( x ) − 2 ∇ ▵ v ) − | ▵ v | q ( x ) − 2 ▵ v = g ( x , v ) . By applying Leray-Schauder’s fixed point theorem, the existence of weak solutions of the elliptic problem is given. Furthermore, the semi-discrete method yields the existence of weak solutions of the corresponding parabolic problem by constructing two approximate solutions.

Keywords

• fourth-order parabolic equation
• semi-discretization
• variable exponent