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.
Categories: math.AP