Hyperbolic and parabolic equations with several opposite-sign source at critical initial energy level in heat and vibrating systems

Abstract

We study semilinear hyperbolic equations and parabolic equations with several opposite sign source terms under the Dirichlet boundary condition in heat and vibrating systems. For the above problems, we prove the global nonexistence of solution and give the sharp conditions at critical energy level. Furthermore, we prove that the existence of global solution of parabolic equation vanishes to zero as time goes to infinity, which is different from the behavior of solution of the hyperbolic equation.