Existence of Solutions for a Class of Quasilinear Parabolic Equations with Superlinear Nonlinearities

Zhong-Xiang Wang,Gao Jia,Xiao-Juan Zhang

doi:10.1155/2014/436369

Abstract

Working in a weighted Sobolev space, this paper is devoted to the study of the boundary value problem for the quasilinear parabolic equations with superlinear growth conditions in a domain of RN. Some conditions which guarantee the solvability of the problem are given.

Highlights

We deal with the existence of solutions for the quasilinear parabolic problem: ρDtu + Mu = λ1uρ + f (x, u) ρ − G, (1)
(x, t) ∈ Ω, u ∈ H (Ω, Γ), where Ω = Ω×T, Ω is an open set in RN (N ≥ 1), T = (−π, π), H(Ω, Γ) is a weighted Sobolev space, λ1 is the first eigenvalue of L, and Mis a singular quasilinear operator defined by
There are a number of results concerning solvability of different boundary problems for quasilinear equations in which the nonlinearities satisfy sublinear or linear conditions in the weighted Sobolev space, for example, [1,2,3,4,5,6]

Summary

Introduction

There are a number of results concerning solvability of different boundary problems for quasilinear equations (elliptic and parabolic) in which the nonlinearities satisfy sublinear or linear conditions in the weighted Sobolev space, for example, [1,2,3,4,5,6]. In [1], Shapiro established a new weighted compact Sobolev embedding theorem and proved a series of existence problems for weighted quasilinear elliptic equations and parabolic equations. In [2], working in Sobolev space Hp1,ρ(Ω, Γ) only for the first eigenvalue, Rumbos and Shapiro on the basis of [3] by using the generalized Landesman-Lazer conditions discussed the existence of the solutions for weighted quasilinear elliptic equations.

Basic Assumptions and Main Theorem

Preliminary Lemmas

Proof of Theorem 6