BLOW-UP PHENOMENA FOR NONLINEAR PSEUDO-PARABOLIC EQUATIONS WITH GRADIENT TERM

摘要

This paper is concerned with the pseudo-parabolic problem {u_t - λ△u_t =k(t)div(g(|▽u|~2)▽u) + f(t,u, |▽u|) in Ω × (0,t*), u=0 on ∂Ω ×(0,t*), u(x, 0) =u_0(x) in Ω, where Ω is a bounded domain in R~n, n ≥ 2, with smooth boundary ∂Ω, 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.