Existence and Nonexistence of Global Solutions for Higher-Order Nonlinear Viscoelastic Equations

Abstract

The initial-boundary value problem for some nonlinear higher-order viscoelastic equation with damping and source terms in a bounded domain is studied. The existence of global solutions for this problem is established by constructing a stable set in $H_0^l(\Omega )$ and the decay of energy based on a difference inequality due to M. Nakao is obtained. Meanwhile, under suitable conditions on relaxation function $g(\cdot )$ and the positive initial energy, it is proved that the solution blows up in the finite time and the lifespan estimates of solutions are also given.