Blow-up and decay for a class of variable coefficient wave equation with nonlinear damping and logarithmic source

Abstract

In this paper, we consider the long time behavior for the solution of a class of variable coefficient wave equation with nonlinear damping and logarithmic source. The existence and uniqueness of local weak solution can be obtained by using the Galerkin method and contraction mapping principle. However, the long time behavior of the solution is usually complicated and it depends on the balance mechanism between the damping and source terms. When the damping exponent $(p+1)$ (see assumption (H3)) is greater than the source term exponent $(q-1)$ (see equation (1.1)), namely, $p+2>q$, we obtain the global existence and accurate decay rates of the energy for the weak solutions with any initial data. Moreover, whether the weak solution exists globally or blows up in finite time, it is closely related to the initial data. In the framework of modified potential well theory, we construct the stable and unstable sets (see (2.8)) for the initial data. For the initial data belonging to the stable set, we prove that the weak solution exists globally and has similar decay rates as the previous results. For $p+2<q$ and the initial data belonging to the unstable set, we prove that the weak solution blows up in finite time for a little special damping $g(u_{t})=|u_{t}|^{p}u_{t}$.