On a degenerate parabolic equation from double phase convection

Author:

Zhan Huashui

Abstract

AbstractThe initial-boundary value problem of a degenerate parabolic equation arising from double phase convection is considered. Let $a(x)$ a ( x ) and $b(x)$ b ( x ) be the diffusion coefficients corresponding to the double phase respectively. In general, it is assumed that $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x Ω and the boundary value condition should be imposed. In this paper, the condition $a(x)+b(x)>0$ a ( x ) + b ( x ) > 0 , $x\in \overline{\Omega }$ x Ω 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 $u_{t}\in L^{2}(Q_{T})$ u t L 2 ( Q T ) is shown. The stability of weak solutions is studied according to the different integrable conditions of $a(x)$ a ( x ) and $b(x)$ 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 $a(x)b(x)|_{x\in \partial \Omega }=0$ a ( x ) b ( x ) | x Ω = 0 is found for the first time.

Publisher

Springer Science and Business Media LLC

Subject

Applied Mathematics,Algebra and Number Theory,Analysis

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