An Optimal Estimate for the Time Singular Limit of an Abstract Wave Equation

Abstract

R. Ikehata recently proved some integral estimate for the difference between the solution of an abstract heat equation and the solution of an abstract wave equation which results from the heat equation by a time singular perturbation. The estimate is obtained if the initial values are chosen appropriately. We prove a pointwise estimate which improves the above result for large times into several directions, and we also establish the optimality of this estimate for the wave equation in an exterior domain. Our proofs rely on the spectral theorem for unbounded self-adjoint operators.