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

Abstract

AbstractGiven a bounded domain G ⊂ Rm, m ≧ 2, we study weak formulations of parabolic problems (Si) L1 and L2 are uniformly elliptic differential operators with time and space dependent coefficients; γ1 = 1, γ2 a real function. Problem (S3) is the “time‐reversed” version of (S2): L3 = −L2, γ3 = γ2. Equations (S1), (S2) are supplemented with an initial condition, a decay condition for ∣x∣ → ∞ and conditions for smoothness or “matching” on the interface ∂G × ℝ+. Since (S3) 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 (S1) 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.