On Solutions of a Parabolic Equation with Nonstandard Growth Condition

Huashui Zhan

doi:10.1155/2020/9397620

Abstract

A parabolic equation with nonstandard growth condition is considered. A kind of weak solution and a kind of strong solution are introduced, respectively; the existence of solutions is proved by a parabolically regularized method. The stability of weak solutions is based on a natural partial boundary value condition. Two novelty elements of the paper are both the dependence of diffusion coefficient bx,t on the time variable t, and the partial boundary value condition based on a submanifold of ∂Ω×0,T. How to overcome the difficulties arising from the nonstandard growth conditions is another technological novelty of this paper.

Highlights

IntroductionWe are concerned with the initial-boundary value problem of a parabolic equation with nonstandard growth condition
We are concerned with the initial-boundary value problem of a parabolic equation with nonstandard growth condition ut = div bðx, tÞjujrðxÞ−1∇u + div !a urðxÞ, x, t + f ðu, x, tÞ, ðx, tÞ ∈ QT = Ω × ð0, TÞ ð1Þ
It has at the same time tested as the testing ground from the development of new methods of analytical investigation, since it offers a variety of surprising phenomena that strongly deviate from the heat equation standard, for example, free boundary, limited regularity, mass loss, and extinction or quenching

Summary

Introduction

We are concerned with the initial-boundary value problem of a parabolic equation with nonstandard growth condition. Uðx, tÞ is said to be a weak solution of the initialboundary value problem of equation (1), provided that initial value (6) is true in the sense ð lim t→0 ðuðx, tÞ − u0ðxÞÞOðxÞdx = 0, OðxÞ ∈ C∞ 0 ðΩÞ, ð15Þ and the partial boundary value condition (10) is true in the sense of the trace. 0 ≤ bðx, tÞ ∈ CðQT Þ satisfies (4), f ðs, x, tÞ is a continuous function on R × QT and when s < 0 f ðs, x, tÞ > 0, ðx, tÞ ∈ Ω × ð0, TÞ, ð20Þ the initial boundary value problem of equation (1) has a nonnegative weak solution. CðMÞ, ð21Þ for any jsj ≤ M + 1, M = ku0ðxÞkL∞ðΩÞ, f ðs, x, tÞ is a continuous function satisfies (20), the initial boundary value problem of equation (1) has a nonnegative strong solution. Just as one reviewer has suggested, one can study Theorem 3 and Theorem 4 when rðxÞ ≥ r− > 0 and f ðs, x, tÞ is just a Carathodory function

The Proof of Theorems 3-4

The Stability Based on the Partial Boundary Value Condition

The Explanation of the Partial Boundary Value Condition

Conclusion