Nonlinear dissipative wave equations with space–time dependent potential

Abstract

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space–time dependent friction a ( t , x ) u t and nonlinear absorption | u | p − 1 u (Ikawa (2000) [17]). We consider 1 < p < ( n + 2 ) / ( n − 2 ) and separable a ( t , x ) = λ ( x ) η ( t ) with λ ( x ) ∼ ( 1 + | x | ) − α and η ( t ) ∼ ( 1 + t ) − β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L 2 and L p + 1 norms of solutions. We also observe the following behavior: if α ∈ [ 0 , 1 ) , β ∈ ( − 1 , 1 ) and 0 < α + β < 1 , there are three different regions for the decay of solutions depending on p ; if α ∈ ( − ∞ , 0 ) and β ∈ ( − 1 , 1 ) , there are only two different regions for the decay of the solutions depending on p .