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We discuss the unique solvability of the third initial-boundary value problemfor the heat equation in a bounded domain. This problem has uniquely atime-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 aboveproperties by $L_{p}$ estimates for steady solutions to the heat equation, thetheory of analytic semigroups on Banach spaces and the operator-valued Fouriermultiplier theorem on UMD spaces.

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