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

摘要

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 L~2-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.