Global existence and blow up of solutions for the Cauchy problem of some nonlinear wave equations

In this paper, we study the global existence and blow up for the Cauchy problem for some hyperbolic system u(ktt )+ delta u(kt) - phi Delta u(k) + f(k) (u(1), u(2)) = lambda vertical bar u(k)vertical bar(beta-1) u(k). k = 1, 2. Under certain conditions we prove the global existence of solutions by adapting the method of modified potential well in a functional setting of generalized Sobolev spaces, and we prove that the solution decays exponentially by introducing an appropriate Lyapunov function. By the concave method, we discuss the blow-up behavior of weak solution with certain conditions and give some estimates for the lifespan of solutions.

Automated summary

In this paper, we study the global existence and blow up phenomenon of certain hyperbolic systems. We prove the global existence of solutions and exponential decay of solutions by using the method of modified potential well and introducing an appropriate Lyapunov function. Moreover, we discuss the blow-up behavior of weak solutions and provide estimates for the lifespan of solutions using the concave method.