The diffusion phenomenon for damped wave equations with space-time dependent coefficients

Abstract

We introduce a method to study the long-time behavior of solutions to damped wave equations, where the coefficients of the equations are space-time dependent. We show that solutions exhibit the diffusion phenomenon, connecting their asymptotic behaviors with the asymptotic behaviors of solutions to corresponding parabolic equations. Sharp decay estimates for solutions to damped wave equations are given, and decay estimates for derivatives of solutions are also discussed.

Optimal energy decay rates for some wave equations with double damping terms

“…recently so many new results are intermittently announced. In particular, we can cite Karch [21], D'Abbicco-Ebert [4,5,6], D'Abbicco-Reissig [8], D'Abbicco-Ebert-Picon [7], Charaõ-da Luz-Ikehata [1], Ikehata-Natsume [14], Ikehata-Takeda [19], and all these papers have contributed to derive several decay estimates and asymptotic profiles of solutions to problem (4)-(2) with θ ∈ (0, 1] (for θ = 0, one can cite Matsumura [23], Nishihara [25], Racke [27], Sobajima-Wakasugi [29], Taylor [30], Todorova-Yordanov [31], and the references therein). In this connection, to the best of authors' knowledge, the paper due to Lu-Reissig [22] first presented a Cauchy problem of (4) with a more generalized time dependent structural damping b(t)(−∆) θ u t to study parabolic aspects of the equation from the viewpoint of energy estimates.…”

“…The inequality (30) also holds true for ξ = 0. By (27) and (30) with t = 0, one has arrived at the significant estimate. The lemma below will be used in the proofs of Theorems 1.1, 1.2 and 1.3.…”