Asymptotic expansion of L 2 -norms of solutions to the heat and dissipative wave equations

Abstract

Let v(t, x) and u(t, x) be solutions of the heat equation v_t−Δv=0 and dissipative wave equation u_tt+u_t−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), u_t(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.