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L 2 -estimates of solutions for damped wave equations with space–time dependent damping term

2009; Elsevier BV; Volume: 248; Issue: 2 Linguagem: Inglês

10.1016/j.jde.2009.09.022

1090-2732

Jiayun Lin, Kenji Nishihara, Jian Zhai,

Nonlinear Partial Differential Equations

In this paper, we consider the damped wave equation with space–time dependent potential b(t,x) and absorbing semilinear term |u|ρ−1u. Here, b(t,x)=b0(1+|x|2)−α2(1+t)−β with b0>0, α,β⩾0 and α+β∈[0,1). Based on the local existence theorem, we obtain the global existence and the L2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the damped wave equation with space dependent potential and absorbed semilinear term, preprint] in the case of β=0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent damped wave equations, preprint] in the case of α=0.