Existence of Weak Solutions for a Class of Parabolic Problems in Weighted Sobolev Space

Abstract

In this paper, we established the existence and uniqueness of weak solutions for some parabolic problems with nonlinear perturbation term in weighted Sobolev space. First, we investigated the compact imbedding in weighted Sobolev space, which can be imbedded compactly into $H^1_0(\Omega)$ and $L^2(\Omega)$ spaces. By exploiting Sobolev interpolation inequalities and extending Galerkin's method to a new class of nonlinear problems, we  proofed the energy estimates of the equations and furthermore obtained the unique weak solution .