The asymptotic behavior of the wave and heat-conduction equations in two-dimensional exterior spaces

The time-asymptotic behavior of the initial-value and boundary-value problems of wave and heat-conduction equations with time-independent right-hand terms (f) is investigated analytically for the two-dimensional space outside a closed curve, considering both Neumann boundary conditions and the corresponding Dirichlet problems and applying a spectral-theory approach. The solutions for the Neumann case are found to increase logarithmically as t goes to infinity as long as the integral of f dx is not equal to zero; in the Dirichlet case, the solutions converge to the solution of the corresponding static problem. The results for the wave equation are shown to be identical with those obtained by Muravei (1978) by a different method.