On a degenerate parabolic equation from double phase convection

Mingguang Li

doi:10.1186/s13662-021-03659-4

Abstract

The initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let a(x) and b(x) be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that a(x)+b(x)>0, xin overline{Omega } and the boundary value condition should be imposed. In this paper, the condition a(x)+b(x)>0, xin overline{Omega } is weakened, and sometimes the boundary value condition is not necessary. The existence of a weak solution u is proved by parabolically regularized method, and u_{t}in L^{2}(Q_{T}) is shown. The stability of weak solutions is studied according to the different integrable conditions of a(x) and b(x). To ensure the well-posedness of weak solutions, the classical trace is generalized, and that the homogeneous boundary value condition can be replaced by a(x)b(x)|_{xin partial Omega }=0 is found for the first time.

Highlights

In the last section of this paper, by giving a generalization of the classical trace of u ∈ BV (QT ), we will use a reasonable partial boundary value condition instead of condition (2.16) (or (2.19)) to study the stability of weak solutions
Consider the parabolic equation with a nonlinear convective term ut = div a(x)|∇u|p–2∇u + b(x)|∇u|q–2∇u + N i=1∂fi(u, x, t), ∂ xi (x, t) ∈ QT, (1.1)which arises from the double phase problems, as well as from the flows of incompressible turbulent fluids etc. [3]
We have found that, to study the uniqueness of weak solution of equation (1.10), condition (1.11) can take place of the boundary value condition (1.8) [26, 29,30,31,32]

Summary

Introduction

In the last section of this paper, by giving a generalization of the classical trace of u ∈ BV (QT ), we will use a reasonable partial boundary value condition instead of condition (2.16) (or (2.19)) to study the stability of weak solutions. By the following proposition, u can be defined as the trace on the boundary ∂ , u is a solution of equation (1.1) with the initial-boundary value conditions (1.7)–(1.8).

Objectives

Results

Conclusion