Global $L_p$-$L_q$ estimates for solutions to the third initial-boundary value problem for the heat equation in a bounded domain

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 $W_{p,q}^{2,1}$ 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.

arXiv comment34 pages