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_σ(t)\} over \mathbb{X\!}^{\,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\!}^{\,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\!}^{\,r(\cdot),s(\cdot)}(\overlineΩ) and for all t>0.

36 pages