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.
Highlights
We mainly study the following fourth-order parabolic equations with variable exponents: vt + div(|∇4v| p( x)−2 ∇4v) − |4v|q( x)−2 4v = g( x, v), ( x, t) ∈ Ω T, (1)
There have been some results related to the existence, uniqueness and properties of solutions to the fourth-order degenerate parabolic equations
For the problems in variable exponent spaces, the papers [6,7,8] have studied the existence of some fourth-order parabolic equations with a variable exponent, and [9] has given the Fujita type conditions for fast diffusion equation
Summary
There have been some results related to the existence, uniqueness and properties of solutions to the fourth-order degenerate parabolic equations (see [1,2]). For the constant exponent case of (1), the paper [5] has given the existence and uniqueness of weak solutions. For the problems in variable exponent spaces, the papers [6,7,8] have studied the existence of some fourth-order parabolic equations with a variable exponent, and [9] has given the Fujita type conditions for fast diffusion equation. We apply the idea of the entropy method to deal with the corresponding problems with variable exponents. We apply the Leray-Schauder’s fixed point theorem to prove the existence of weak solutions of the corresponding elliptic problem of (1)–(3) in order to deal with the nonlinear source. We will show the effect of the variable exponents and the second-order nonlinear diffusion to the degenerate parabolic Equation (1)
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