For n+1 ≥ 3, we construct complete solutions to Ricci flow on R which encounter global singularities at a finite time T . The singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t)−2λ for λ ≥ 1. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converg...

We investigate blow-up of the focusing nonlinear Schrödinger equation, in the critical and supercritical cases. Numerical simulations are performed to examine the dependence of the time at which blow-up occurs on properties of the data or the equation. Three cases are considered: dependence on the scale of the nonlinearity when the initial data are fixed; dependence upon the strength of a quadr...

We study the global and blow-up solutions for a strong degenerate reaction–diffusion system modeling the interactions of two biological species. The local existence and uniqueness of a classical solution are established. We further give the critical exponent for reaction and absorption terms for the existence of global and blow-up solutions. We show that the solution may blow up if the intraspe...

where D⊂RN (N≥ 2) is a bounded domain with smooth boundary ∂D. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for ‘blow-up time’, and an upper estimate of ‘blow-up rate’ are specified unde...

In [] Rammaha and Sakuntasathien studied the global well posedness of the solution of problem (.). Agre and Rammaha [] studied the global existence and the blow up of the solution of problem (.) for k = l = θ = = , and also Alves et al. [] investigated the existence, uniform decay rates and blow up of the solution to systems. After that, the blow up result was improved by Houari []. Al...

Assume that Ω is a bounded domain in Rn with n ≥ 2. We study positive solutions to the problem, ∆u = up in Ω, u(x)→∞ as x→ ∂Ω, where p > 1. Such solutions are called boundary blow-up solutions of ∆u = up. We show that a boundary blow-up solution exists in any bounded domain if 1 < p < n n−2 . In particular, when n = 2, there exists a boundary blow-up solution to ∆u = up for all p ∈ (1,∞). We al...

According to Jill Walker, associate professor of humanistic informatics at the University of Bergen, a blog (short for "web log") "is a frequently updated website consisting of dated entries arranged in reverse chronological order so the most recent post appears first (see temporal ordering). Typically, weblogs are published by individuals and their style is personal and informal. Weblogs first...

Free will and human agency are considered foundational for ascriptions of criminal responsibility in Anglo-American jurisprudence. As United States Supreme Court Justice Oliver Wendell Holmes famously observed, "even a dog distinguishes major support from the law firm of Leonard, Street and Deinard, and then presented at a conference on "Neuroethics and Empirical Moral Psychology" at the Depart...