Decay property of solutions for damped wave equations with space–time dependent damping term

Abstract

We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t, x) and absorbing semilinear termUsing the weighted energy method, we can obtain the L 2 decay rate of the solution, which is almost optimal in the case ρ > ρ c (N, α, β) := 1 + 2/(N − α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L 2 -estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρ c (N, α, β) is a critical exponent. Note that when α = β = 0, ρ c (N, α, β) coincides to the Fujita exponent ρ F (N) := 1 + 2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.

A Modified Test Function Method for Damped Wave Equations

“…On the other hand, if we have a space-dependent damping b(x)u t the critical exponent p C is modified by the decaying behavior of b(x), see [15]. For the existence result with damping term depending on time and space variables, we address to [18,29]. If we consider global existence of classical solutions for semilinear waves with time-dependent propagation speed a(t), no damping and f = 1, the range of admissible exponents p for large data depend on a(t), see [7,8,12].…”

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

“…They used a weighted energy method with a special weight. Similar weighted energy methods were implemented in [11,12,13,18,19,26]. Nishihara and Zhai [19] and Nishihara [18] considered (3) with a defocusing nonlinear term −|u| p−1 u and a slowly changing damping coefficient that was either space-or time-dependent .…”

“…Nishihara and Zhai [19] and Nishihara [18] considered (3) with a defocusing nonlinear term −|u| p−1 u and a slowly changing damping coefficient that was either space-or time-dependent . This was followed by Lin, Nishihara and Zhai [12,13], and Khader [11] who studied (3) with a slowly changing, radially symmetric damping coefficient and a defocusing nonlinear term −|u| p−1 u.…”

Small data global existence for the semilinear wave equation with space–time dependent damping

“…(2.13) Following Nishihara [19], related to the size of 1 + |x| 2 and the size of (1 + t) 2 , we divide the space R n into two different zones Ω(t; K, t 0 ) and Ω c (t; K, t 0 ), where Ω = Ω(t; K, t 0 ) := {x ∈ R n ; (t 0 + t) 2 ≥ K + |x| 2 },…”