The Regularity Lemma [16] is a powerful tool in Graph Theory and its applications. It basically says that every graph can be well approximated by the union of a constant number of random-looking bipartite graphs called regular pairs (see the definitions below). These bipartite graphs share many local properties with random bipartite graphs, e.g. most degrees are about the same, most pairs of ve...

The possibility of finite-time, dispersive blow up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the possible explanations for oceanic and optical rogue waves. In dimension one, the possibility of dispersive blow up for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways....

tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we investigate the lower bounds for the blow-up time of the non-negative solutions of porous medium equation with Neumann boundary conditions. We find that the blow-up time are bounded below b...

Abstract We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension d ≥ 3 find two critical curves intersecting at one point which separate the global existence blow up of weak solutions to problem. More precisely, above these (i.e. subcritical case), problem admits solution obtained by limits strong an approximated system. Based on second moment solutions, initial ...

Abstract We describe the small-time heat kernel asymptotics of real powers $\operatorname {\Delta }^r$ , $r \in (0,1)$ a non-negative self-adjoint generalized Laplacian }$ acting on sections Hermitian vector bundle $\mathcal {E}$ over closed oriented manifold M . First, we treat separately asymptotic diagonal $M \times M$ and in compact set away from it. Logarithmic terms appear only if n is od...

The equation ut = ∆u + u with homegeneous Dirichlet boundary conditions has solutions with blow-up if p > 1. An adaptive time-step procedure is given to reproduce the asymptotic behvior of the solutions in the numerical approximations. We prove that the numerical method reproduces the blow-up cases, the blow-up rate and the blow-up time. We also localize the numerical blow-up set.