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In this paper, we consider the long time behavior for the solution of a classof variable coefficient wave equation with nonlinear damping and logarithmicsource. The existence and uniqueness of local weak solution can be obtained byusing the Galerkin method and contraction mapping principle. However, the longtime behavior of the solution is usually complicated and it depends on thebalance mechanism between the damping and source terms. When the dampingexponent $(p+1)$ (see assumption (H3)) is greater than the source term exponent$(q-1)$ (see equation (1.1)), namely, $p+2>q$, we obtain the global existenceand accurate decay rates of the energy for the weak solutions with any initialdata. Moreover, whether the weak solution exists globally or blows up in finitetime, it is closely related to the initial data. In the framework of modifiedpotential 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, weprove that the weak solution exists globally and has similar decay rates as theprevious results. For $p+2<q$ and the initial data belonging to the unstableset, we prove that the weak solution blows up in finite time for a littlespecial damping $g(u_{t})=|u_{t}|^{p}u_{t}$.

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