The well-posedness of an anisotropic parabolic equation based on the partial boundary value condition
Huashui Zhan · Boundary Value Problems · 2017
Consider the anisotropic parabolic equation with the variable exponent $$ {u_{t}}=\sum_{i=1}^{N} \bigl(a_{i}(x)|u_{x_{i}}|^{p_{i}(x)-2}u_{x_{i}} \bigr)_{x _{i}}, $$ with $a_{i}(x)$ , $p_{i}(x)\in C^{1}(\overline{\Omega})$ , $p_{i}(x)>1$ , $a_{i}(x)\geq0$ . If some of $\{a_{i}(x)\}$ are degenerate on the boundary, a partial boundary value condition is imposed, the stability of weak solutions can be proved based on the partial boundary value condition.