On the evolutionary p-Laplacian equation with a partial boundary value condition

Abstract

Abstract Consider the equation ut=div(dα|∇u|p−2∇u)+∂bi(u,x,t)∂xi,(x,t)∈Ω×(0,T), $${u_{t}} = \operatorname{div} \bigl(d^{\alpha} \vert \nabla u \vert ^{p - 2}\nabla u\bigr) + \frac{\partial b_{i}(u,x,t)}{\partial{x_{i}}},\quad (x,t) \in\Omega \times(0,T), $$ where Ω is a bounded domain, d(x) $d(x)$ is the distance function from the boundary ∂Ω. Since the nonlinearity, the boundary value condition cannot be portrayed by the Fichera function. If αp−1 $\alpha>p-1$, the stability of the weak solutions may be proved independent of the boundary value condition.

Keywords

• Nonlinearity
• Stability
• p-Laplacian equation
• Partial boundary value condition