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

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 . Some conditions which guarantee the solvability of the problem are given. 1. Introduction In this paper, we deal with the existence of solutions for the quasilinear parabolic problem: where , is an open set in , , is a weighted Sobolev space, is the first eigenvalue of , and is a singular quasilinear operator defined by where . The nonlinear part satisfies the superlinear growth condition where , is a nonnegative constant, , and . 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, [1–6]. 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 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 where In [4], Jia and Zhao obtained the existence of a nontrivial solution for a class of singular quasilinear elliptic equations in weighted Sobolev spaces. However, past research results regarding this kind of parabolic equations on superlinearity in the weighted Sobolev space like (1) are very limited. Two notable exceptions are found in [7, 8], where they discuss the periodic solutions for quasilinear parabolic equations when the nonlinearity may grow superlinearly. Our goal here is to extend these results to the case of quasilinear parabolic operators. In fact, (1) is one of the most useful sets which describe the motion of viscous fluid substances. They are widely used in the design of aircrafts and cars, the study of blood flow, the design of power stations, and so forth. Furthermore, coupled with Maxwell’s equations, the Navier-Stokes equations can also be used to model and study magnetohydrodynamics. The main tools applied in our approaches consist of Galerkin method, Brouwer’s theorem, and a new weighted compact Sobolev-type embedding theorem due to Shapiro. This paper is organized as follows. In Section 2, we introduce some necessary assumptions and basic results. In Section 3, five fundamental lemmas are established. The subsequent Section 4 contains proofs