Cauchy problem for quasi-linear wave equations with viscous damping
In this paper, the author studied the global existence and asymptotic behavior of weak solutions to the following Cauchy problem for quasi-linear wave equations with viscous damping: \[ x_{tt}-\Delta u_t+\lambda u_t- \sum^N_{i=1} {\partial\over\partial x_i} \sigma_i(u_{x_t})+ f(u_t)= g(u)\quad\text{on }\mathbb{R}^N\times (0,\infty), \] \[ u(x,0)= u_0(x),\quad u_t(x, 0)= u_1(x),\quad x\in\mathbb{R}^N, \] where \(\lambda> 0\) is a real number, \(\sigma_i\) \((i=1,2,3,\dots, N)\), \(f\) and \(g\) are given nonlinear functions with some conditions. It proved that when \(p\geq\max\{m,\alpha\}\), where \(m+1\), \(\alpha+ 1\) are respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem gives a global weak solution, which decays to zero accroding to the rate of polynomial as \(t\to\infty\). As the special case, \(N= 1\), the solutions are regularized, and so generalized and classical solution both prove to be unique. If we compare with the previous results, the present paper shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear term, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.
scientific article