Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term
The author studies the global existence as well as the exponential decay (as \(t\) tends to \(\infty \)) of weak solutions to the problem \[ u_{tt}+ \Delta^2u + \lambda u_t = \sum_{i=1}^N\frac{\partial}{\partial x_i}\sigma_i u_{x_i} \quad \text{in }\Omega \times (0, \infty), \] \[ u=0, \;\;\frac{\partial u}{\partial n} = 0 \quad \text{on }\partial \Omega \times [0,\infty), \] \[ u(x,0)=u_0(x), \;\;u_t(x,0)=u_1(x), \quad x\in \Omega, \] where \(\lambda \geq 0\) is a constant. Under different assumptions the solutions blow up in finite time.
scientific article; zbMATH DE number 1884235