On a degenerate parabolic equation from double phase convection

Abstract

Abstract The 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 ∈ Ω ‾ $x\in \overline{\Omega }$ and the boundary value condition should be imposed. In this paper, the condition a ( x ) + b ( x ) > 0 $a(x)+b(x)>0$ , x ∈ Ω ‾ $x\in \overline{\Omega }$ 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 ∈ L 2 ( Q T ) $u_{t}\in L^{2}(Q_{T})$ is shown. The stability of weak solutions is studied according to the different integrable conditions of a ( x ) $a(x)$ 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 ∈ ∂ Ω = 0 $a(x)b(x)|_{x\in \partial \Omega }=0$ is found for the first time.

Keywords

• Degenerate parabolic equation
• Double phase convection
• Boundary value condition
• Trace