On Multiple Degenerate Parabolic Equation with Variable Exponent

A multiple degenerate parabolic equation related to the \(p(x,t)\)-Laplacian is considered. Since it is with multiple degeneracy, how to obtain the \(L^{\infty }\)-estimate becomes difficult, and the usual Dirichlet boundary value condition may be invalid or overdetermined. By adding some restrictions on the growth order, using the maximum value principle, the corresponding \(L^{\infty }\)-estimate of the weak solution is obtained first time. Since the solution is so weak that its trace on the boundary cannot be defined in the conventional manner. By employing the weak characteristic function method introduced in (Zhan and Feng in J. Differ. Equ. 268:389–413, 2020)), the classical trace in \(W_{0}^{1,p(\cdot )}(\Omega )\) is generalized to the function space \(W_{loc}^{1, p(\cdot )}(\Omega )\bigcap L^{\infty }(\Omega )\). Through this framework, the partial boundary value condition is imposed on a submanifold of \(\partial \Omega \times (0,T)\), thereby establishing the stability of weak solutions.