We study radially symmetric finite time blow-up dynamics for the aggregation equation with degenerate diffusion ut = ∆u m − ∇ · (u ∗ ∇(K ∗ u)) in R, where the kernel K(x) is of power-law form |x|−γ . Depending on m, d, γ and the initial data, the solution exhibits three kinds of blow-up behavior: self-similar with no mass concentrated at the core, imploding shock solution and near-self-similar ...

Abstract. In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than (T − t), the expected one. Moreover,...

We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution u(x, t), the graph x 7→ T (x) of its blow-up points and S ⊂ R the set of all characteristic points and show that S has an empty interior. Finally, given x0 ∈ S, we show that in self...

We study the behaviour of nonnegative solutions of the reaction-diffusion equation    ut = (u)xx + a(x)up in R× (0, T ), u(x, 0) = u0(x) in R. The model contains a porous medium diffusion term with exponent m > 1, and a localized reaction a(x)up where p > 0 and a(x) ≥ 0 is a compactly supported function. We investigate the existence and behaviour of the solutions of this problem in dependenc...

This work connects the idea of a “blow-up” of a quiver with that of injectivity, showing that for a class of monic maps Φ, a quiver is Φ-injective if and only if all blow-ups of it are as well. This relationship is then used to characterize all quivers that are injective with respect to the natural embedding of Pn into Cn.

We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in the case when the dissipation degree is sufficiently small.

We analyze the asymptotic behavior of blowing up solutions for the SU(3) Toda system in a bounded domain. We prove that there is no boundary blow-up point, and that the blow-up set can be localized by the Green function.