Asymptotic expansion of L²-norms of solutions to the heat and dissipative wave equations

Article type: Research Article

Authors: Volkmer, Hans

Affiliations: Department of Mathematical Sciences, University of Wisconsin–Milwaukee, PO Box 413, Milwaukee, WI 53201, USA. Tel.: +1 414 229 5950; Fax: +1 414 229 4907; E-mail: [email protected]

Abstract: Let v(t, x) and u(t, x) be solutions of the heat equation vt−Δv=0 and dissipative wave equation utt+ut−Δu=0, respectively. The paper finds the asymptotic expansions of the squared L2-norms of v, u and u−v as well as of their derivatives as t→∞. Suitable conditions on the initial values u(0, x), ut(0, x) and v(0, x) lead to cancellation of the leading terms of the asymptotic expansion of u−v explaining the diffusion phenomenon for linear hyperbolic waves.

Keywords: heat equation, dissipative wave equation, diffusion phenomenon, asymptotic expansion of L^2-norm

DOI: 10.3233/ASY-2010-0980

Journal: Asymptotic Analysis, vol. 67, no. 1-2, pp. 85-100, 2010