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

We consider the Cauchy problem of the semilinear damped wave system:{∂t2u−Δu+∂tu=F(u),t>0,x∈Rn,uj(0,x)=aj(x),∂tuj(0,x)=bj(x),x∈Rn, where u(t,x)=(u1(t,x),…,um(t,x))u(t,x)=(u1(t,x),…,um(t,x)) with m⩾2m⩾2 and j=1,…,mj=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 LpLp–LqLq 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].