Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term

Abstract

This paper is concerned with the pseudo-parabolic problem$\left\{ \begin{array}{l}\begin{split}u_t- \lambda \triangle u_t=& k(t) \text{div}(g(| \nabla u|^2) \nabla u) +f(t,u,| \nabla u| ) \quad {\rm in} \ \Omega \times (0, t^*), \\[6pt] u=&0 \ \qquad {\rm on} \ \partial \Omega \times (0,t^*),\\[6pt] u ({ x},0) =& u_0 ({ x}) \quad {\rm in} \ \Omega,\\[6pt]\end{split}\end{array} \right.$where $\Omega$ is a bounded domain in $\mathbb{R}^n, \ n\geq 2$, with smooth boundary $ \partial \Omega$, $ k$ is a positive constant or in general positive derivable function of $t$. The solution $u(x,t)$ may or may not blow up in finite time. Under suitable conditions on data, a lower bound for $t^*$ is derived, where $[0,t^*)$ is the time interval of existence of $u(x,t).$ We indicate how some of our results can be extended to a class of nonlinear pseudo-parabolic systems.