The generalized anisotropic dynamical Wentzell heat equation with nonstandard growth conditions

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_σ(t)} over $\mathbb{X}\text{\hspace{0.17em}⁣}$^{,r(\cdot)}(\overlineΩ):=L^{r(\cdot)}(Ω)\times L^{r(\cdot)}(Γ) for all measurable functions r(\cdot) on \overlineΩ with 1\leq r^-\leq r^+<\infty. Consequently, the corresponding anisotropic dynamical Wentzell problem is well-posed over $\mathbb{X}\text{\hspace{0.17em}⁣}$^{,r(\cdot)}(\overlineΩ). Furthermore, we show that the nonlinear C_0-semigroup {T_σ(t)} enjoys a Hölder-type ultracontractivity property in the sense that there exist constants C_1,,C_2,,κ>0, and γ\in(0,1), such that ||T_σ(t)$\mathbf{u}$0-T_σ(t)$\mathbf{v}$0||{{\infty}},\leq, C_1,e^{C_2t}t^{-κ}||$\mathbf{u}$0-$\mathbf{v}$0||^γ{{r(\cdot),s(\cdot)}} for every $\mathbf{u}$_0,,$\mathbf{v}$_0\in$\mathbb{X}\text{\hspace{0.17em}⁣}$^{,r(\cdot),s(\cdot)}(\overlineΩ) and for all t>0.