In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem ut = ∆u+ λf(u) ( ∫ Ω f(u)dx )p , x ∈ Ω, t > 0, with homogeneous Dirichlet boundary condition, where λ > 0, p > 0, f is nonincreasing. It is found that: (a) For 0 < p ≤ 1, u(x, t) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0; (b) For 1 < p < 2, u(x, t) is...

A PERSON tempted to kill himself may need one, two or all three of the following kinds of help: medical treatment, non-medical counselling, non-professional befriending. In many cases, e.g. endogenous depression, the life cannot be saved without the first of these; yet the most important of these is the last. If two kinds of help are needed, befriending is one of them. If only one kind of help ...

Let f be a positive smooth function on a close Riemann surface (M,g). The f − energy of a map u from M to a Riemannian manifold (N, h) is defined as E f (u) = M f |∇u| 2 dV g. In this paper, we will study the blow-up properties of Palais-Smale sequences for E f. We will show that, if a Palais-Smale sequence is not compact, then it must blows up at some critical points of f. As a sequence, if an...

Murine peritoneal macrophages exuding early after stimulation with the activators of the alternative complement pathway are able to destroy some of the tumor cells. This destruction is inhibited by anti-Mac-1 and anti-C3. The tumor cell killing needs longer incubation time (> 15 hr), and anti-Mac-1 affects the reaction even if it were put into the reaction mixture at later time of incubation. T...

In this paper, I consider nonlinear parabolic problems under nonlinear boundary conditions. I establish respectively the conditions on nonlinearities to guarantee that ( , ) u x t exists globally or blows up at some finite time. If blow-up occurs, an upper bound for the blow-up time is derived, under somewhat more restrictive conditions, lower bounds for the blow-up time are also derived.

We study solutions of some supercritical parabolic equations which blow up in finite time but continue to exist globally in the weak sense. We show that the minimal continuation becomes regular immediately after the blow-up time and if it blows up again, it can only do so finitely many times.