# American Institute of Mathematical Sciences

August  2017, 22(6): 2291-2300. doi: 10.3934/dcdsb.2017096

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

 Università di Cagliari, Dipartimento di Matematica e Informatica, V.le Merello 92,09123 Cagliari, Italy

Received  April 2016 Revised  December 2016 Published  March 2017

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.
Citation: Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096
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