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 term | u | ρ − 1 u . Here, b ( t , x ) = b 0 ( 1 + | x | 2 ) − α 2 ( 1 + t ) − β with b 0 > 0 , α , β ⩾ 0 and α + β ∈ [ 0 , 1 ) . Using 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.