Evolutionary p(x)-Laplacian Equation with a Convection Term

Abstract

The paper studies an evolutionary p(x)-Laplacian equation with a convection term $$u_{t}=\operatorname{div}\left(\rho^{\alpha}|\nabla u|^{p(x)-2} \nabla u\right)+\sum_{i=1}^{N} \frac{\partial b_{i}(u)}{\partial x_{i}},$$ where ρ(x)= dist(x, ∂Ω), essinf p(x) = p− > 2. To assure the well-posedness of the solutions, the paper shows only a part of the boundary, ∑p ⊂ ∂Ω, on which we can impose the boundary value. ∑p is determined by the convection term, in particular, when \(1<\alpha<\frac{p^{-}-2}{2}\), ∑p = {x ∈ ∂Ω: bi′(0)ni(x) < 0}. So, there is an essential difference between the equation and the usual evolutionary p-Laplacian equation. At last, the existence and the stability of weak solutions are proved under the additional conditions \(\alpha<\frac{p^{-}-2}{2}\) and ∑p = ∂Ω.