Zur Asymptotik der wellengleichung und der wärmeleitungsgleichung in zweidimensionalen außenräumen
Peter Christian Werner · Mathematical Methods in the Applied Sciences · 1985
Abstract We study initial and boundary value problems for the wave equation and the heat equation with a time‐independent right‐hand term f in two space dimensions in the exterior of a closed curve C. In the case of Neumann's boundary condition ∂u/∂n = 0 on C, the solutions increase with a logarithmic rate as t → ∞ if ∫ fdx ≠ 0. In contrast to this, the solutions of the corresponding Dirichlet problems converge to the solution of the related static problem as t → ∞. In the case of the wave equation, these results have already been obtained by L: A. Muravei under the additional assumption that the curvature of C is positive, by using high frequency estimates for the reduced wave equation Δ U + ϰ2 U = 0. The analysis presented here is based on different methods, which can be applied to arbitrary smooth curves.