L²‐convergence results for linear dissipative wave equations in unbounded domains

Article type: Research Article

Authors: Ikehata, Ryo

Affiliations: Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi‐Hiroshima 739‐8524, Japan E‐mail: ikehatar@hiroshima‐u.ac.jp

Note: [] Partially supported by Grant‐in‐Aid for Scientific Research (C)(2)14540208 of Japan Society for the Promotion of Science.

Abstract: Hyperbolic linear Cauchy problem εu″+Au+u′=0, u(0)=u0, u′(0)=u1, with “nonnegative” selfadjoint operator A in a real Hilbert space H is first considered. It is shown that the solution uε tends to some solution v as ε↓0 for the parabolic equation v′+Av=0 in a certain sense. Some applications are given. Finally, we present hyperbolic‐hyperbolic convergence results such as the solution for the damped wave equations goes to some solution for the free wave equations as the effect of the damping vanishes in a concrete context.

Keywords: dissipative wave equation, unbounded domain, singular limit, [TeX:] $L^{2}$‐convergence result

Journal: Asymptotic Analysis, vol. 36, no. 1, pp. 63-74, 2003