Large time behavior of solutions for a system of nonlinear damped wave equations

Abstract

We consider the Cauchy problem of the semilinear damped wave system: { ∂ t 2 u − Δ u + ∂ t u = F ( u ) , t > 0 , x ∈ R n , u j ( 0 , x ) = a j ( x ) , ∂ t u j ( 0 , x ) = b j ( x ) , x ∈ R n , where u ( t , x ) = ( u 1 ( t , x ) , … , u m ( t , x ) ) with m ⩾ 2 and j = 1 , … , m . We show the asymptotic behavior of solutions under the sharp condition on the nonlinear exponents, which is a natural extension of the results for the single nonlinear damped wave equations Nishihara (2003) [21], Hayashi et al. (2004) [9], Hosono and Ogawa (2004) [10]. The proof is based on the L p – L q type decomposition of the fundamental solutions of the linear damped wave equations into the dissipative part and hyperbolic part Hosono and Ogawa (2004) [10], Nishihara (2003) [21].