Asymptotic behavior of solutions of certain parabolic problems with space and time dependent coefficients

Given a bounded domain G ⊂ R m , m ≧ 2, we study weak formulations of parabolic problems\documentclass{article}\pagestyle{empty}\begin{document}$$ \mathop u\limits^. = L_i u\;{\rm in }G \times R_ +, $$\end{document} (S i )\documentclass{article}\pagestyle{empty}\begin{document}$$ \Delta \left({\gamma _i u} \right) = 0\;{\rm in}\left({{\rm R}^{\rm m} \backslash G} \right) \times R_ +,\;i = 1,2,3. $$\end{document}L 1 and L 2 are uniformly elliptic differential operators with time and space dependent coefficients; γ 1 = 1, γ 2 a real function. Problem (S 3 ) is the “time‐reversed” version of (S 2 ): L 3 = − L 2 , γ 3 = γ 2 . Equations (S 1 ), (S 2 ) are supplemented with an initial condition, a decay condition for ∣x∣ → ∞ and conditions for smoothness or “matching” on the interface ∂ G × ℝ + . Since (S 3 ) is a backward equation the prescription of initial values is not appropriate and has to be replaced by a normalization condition for the initial values. We prove existence of unique weak solutions of all three problems and show that the solutions of (S 1 ) decay exponentially, while the solutions of the other two problems stay bounded for all times. In an appendix we characterize exterior harmonic functions of given decay rates at infinity.