Global $L_{p}$-$L_{q}$ estimates for solutions to the third initial-boundary value problem for the heat equation in a bounded domain

TL;DR

This paper establishes global $L_{p}$-$L_{q}$ estimates and unique solvability for the third initial-boundary value problem of the heat equation in bounded domains, using advanced functional analysis techniques.

Contribution

It provides the first comprehensive $L_{p}$-$L_{q}$ estimates for solutions to this problem in bounded domains, including exponential weights and operator theory methods.

Findings

Unique solvability in anisotropic Sobolev spaces

Exponential weighted $L_{p}$-$L_{q}$ estimates for solutions

Application of $L_{p}$ estimates, semigroup theory, and Fourier multipliers

Abstract

We discuss the unique solvability of the third initial-boundary value problem for the heat equation in a bounded domain. This problem has uniquely a time-global solution in the anisotropic Sobolev space $W^{2,1}_{p,q}$ for any $1<p<\infty$, $1<q<\infty$. Moreover, exponentially weighted $L_{p}$-$L_{q}$ estimates for time-global solutions can be established. We prove the above properties by $L_{p}$ estimates for steady solutions to the heat equation, the theory of analytic semigroups on Banach spaces and the operator-valued Fourier multiplier theorem on UMD spaces.