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

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.