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

This paper is concerned with the pseudo-parabolic problem { u(t) - lambda Delta u(t) =k(t)div(g(vertical bar del u vertical bar(2))del u) + f (t, u, vertical bar del u vertical bar) in Omega x (0,t*), u =0 on partial derivative Omega x (0, t*), u(x, 0) =u(0)(x) in Omega, where Omega is a bounded domain in R-n, n >= 2, with smooth boundary partial derivative 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