The generalized anisotropic dynamical Wentzell heat equation with nonstandard growth conditions

Abstract: The aim of this paper is to establish the solvability and global regularity theory for a new class of generalized anisotropic heat-type boundary value problems with (pure) dynamical anisotropic Wentzell boundary conditions. We first prove that the Wentzell operator with the above boundary conditions generates a nonlinear order-preserving submarkovian C_0-semigroup {T_{\sigma}(t)} over \mathbb{X!}^{\,r(\cdot)}(\overline{\Omega}):=L^{r(\cdot)}(\Omega)\times L^{r(\cdot)}(\Gamma) for all measurable functions r(\cdot) on \overline{\Omega} with 1\leq r^-\leq r^+<\infty. Consequently, the corresponding anisotropic dynamical Wentzell problem is well-posed over \mathbb{X!}^{\,r(\cdot)}(\overline{\Omega}). Furthermore, we show that the nonlinear C_0-semigroup {T_{\sigma}(t)} enjoys a H\"older-type ultracontractivity property in the sense that there exist constants C_1,\,C_2,\,\kappa>0, and \gamma\in(0,1), such that ||T_{\sigma}(t)\mathbf{u}0-T{\sigma}(t)\mathbf{v}0||{{\infty}}\,\leq\, C_1\,e^{C_2t}t^{-\kappa}||\mathbf{u}_0-\mathbf{v}_0||^{\gamma}{_{r(\cdot),s(\cdot)}} for every \mathbf{u}_0,\,\mathbf{v}_0\in\mathbb{X!}^{\,r(\cdot),s(\cdot)}(\overline{\Omega}) and for all t>0.