The generalized anisotropic dynamical Wentzell heat equation with nonstandard growth conditions

TL;DR

This paper establishes the existence, uniqueness, and regularity of solutions for a new class of anisotropic heat equations with nonstandard growth and dynamical Wentzell boundary conditions, using nonlinear semigroup theory.

Contribution

It introduces a novel framework for analyzing anisotropic heat equations with complex boundary conditions and proves the generation of a nonlinear semigroup with ultracontractivity properties.

Findings

Proves the Wentzell operator generates a nonlinear order-preserving submarkovian C_0-semigroup.

Shows the problem is well-posed in variable exponent Lebesgue spaces.

Establishes ultracontractivity estimates for the semigroup.

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.