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 𝑅 𝑁 . Some conditions which guarantee the solvability of the problem are given.

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.
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 2 International Journal of Partial Differential Equations 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 of the main results.

Basic Assumptions and Main Theorem
In this section, we introduce some assumptions and give the main results in this paper.

Preliminary Lemmas
In this section, we introduce some lemmas.First, we introduce some notions.

Proof of Theorem 6
In this section, we will give the proof of Theorem 6.In order to do this, we divide the proof into three parts.
Lemma 14.Let all the assumptions in Theorem 6 hold.Then, for  ≥ 2, ∃  ∈   such that Proof.To prove the lemma, we first observe from (35) that , and set , where  ≥  0 .
Suppose that the assertion is false.Then, without loss of generality, we assume lim Putting   in place of V in (78), we have Consequently, dividing both sides of (88) by ‖  ‖ 2 H and passing to the limit as  → ∞, from Lemma 12, the fact that  ∈ ( H)  , (80), and (84), we obtain that 1 ≤ 0. This is a contradiction.Hence (80) does not hold and (79) is true.
Proof of Theorem 6.Since H is a separable Hilbert space, we see from (79) and Lemmas 9 and 13 that there exist a subsequence (for the sake of simplicity, we take to be a full sequence) and a function  * ∈ H with the following properties [11]: Next, for given V ∈ H, we replace V  with   (V) (defined as (32)) in (95).From (29) and Lemma 8, ‖  (V) − V‖ H → 0 as  → ∞.From (95) and Lemma 12, it is then an easy matter to obtain that ⟨   * , V⟩ and the proof of Theorem 6 is established.