The Global Existence and Attractor for m(x)-Laplacian Equation with Nonlinear Boundary Conditions

Abstract

In this paper, we consider a doubly m(x)-Laplacian equation
∂α(v)/∂t −div(|∇v|^m(x)−2∇v)+F(v) = G,       in Ω ×(0,+∞),
with nonlinear boundary conditions and initial data given. Firstly, we use the regularization method to determine the existence and uniqueness of weak solutions in the Sobolev space with variable exponents. Secondly, in the frame of the dynamical systems approach, a standard limiting process and a method to generate a series of approximation solutions are used to study the long behavior of solutions for the above problem (1.1). We formulate our problem as a dynamical system, and then, by using H¨older continuity solutions and assuming appropriate hypotheses, we prove also the existence of a global attractor in L²(Ω).

Keywords: m(x)-Laplacian; nonlinear boundary conditions; existence; uniqueness; variable exponents; global attractors.

2010 Mathematics Subject Classification. 35K55; 46E35; 35D30