Global existence and nonexistence of solutions for a system of nonlinear damped wave equations

We consider the Cauchy problem for the system of semilinear damped wave equations with small initial data:{∂t2u1−Δu1+∂tu1=|uk|p1,t>0,x∈Rn,∂t2u2−Δu2+∂tu2=|u1|p2,t>0,x∈Rn,⋮∂t2uk−Δuk+∂tuk=|uk−1|pk,t>0,x∈Rn,uj(0,x)=aj(x),∂tuj(0,x)=bj(x),x∈Rn(1⩽j⩽k). We show that a critical exponent which classifies the global existence and the finite time blow up of solutions indeed coincides with the one to a corresponding semilinear heat systems with small data. The proof of the global existence is based on the Lp–LqLp–Lq estimates of fundamental solutions for linear damped wave equations [K. Nishihara, Lp–LqLp–Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244 (2003) 631–649; K. Marcati, P. Nishihara, The Lp–LqLp–Lq estimates of solutions to one-dimensional damped wave equations and their application to compressible flow through porous media, J. Differential Equations 191 (2003) 445–469; T. Hosono, T. Ogawa, Large time behavior and Lp–LqLp–Lq estimate of 2-dimensional nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118; T. Narazaki, LpLp–LqLq estimates for damped wave equations and their applications to semilinear problem, J. Math. Soc. Japan 56 (2004) 585–626]. And the blow-up is shown by the Fujita–Kaplan–Zhang method [Q. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris 333 (2001) 109–114; F. Sun, M. Wang, Existence and nonexistence of global solutions for a nonlinear hyperbolic system with damping, Nonlinear Anal. 66 (12) (2007) 2889–2910; T. Ogawa, H. Takeda, Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal. 70 (10) (2009) 3696–3701].