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 ∈ R{double-struck}ⁿ, u_j (0, x) =a_j(x), ∂_t u_j (0, x) =b_j (x), x ∈ R{double-struck}ⁿ, where u(t,x)=(u₁(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].