We consider the FPU model with nonlinearity starting with terms of order n ≥ 3. We compute the resonant normal form in the region where only one low frequency modes is excited and deduce rigorous results on the correspondence between the dynamics of the normal form and that of the complete system. As n varies, we give a criterion in order to deduce whether the FPU phenomenon (formation of a met...

We study the asymptotic behaviour of classes of global and blow-up solutions of a semilinear parabolic equation of Cahn-Hilliard type ut = −∆(∆u + |u|u) in R ×R+, p > 1, with bounded integrable initial data. We show that in some {p, N}-parameter ranges it admits a countable set of blow-up similarity patterns. The most interesting set of blow-up solutions is constructed at the first critical exp...

Let H = (V (H), E(H)) be a simple connected graph of order n with the vertex set V (H) and the edge set E(H). We consider a blow-up graph G[H ]. We are interested in the following problem. We have to decide whether there exists a blow-up graph G[H ], with edge densities satisfying special conditions (homogeneous or inhomogeneous), such that the graph H does not appear in a blow-up graph as a tr...

Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the H3/2+ǫ Sobolev norm. It is shown that their model can be reduced to the dyadic inviscid Burgers equation where nonlinear interactions are restricted to dyadic wavenumbers. The inviscid Burgers equation exhibits finite time blow-up in Hα, for α ≥ 1/2, but its dyadic restrict...

Let H = (V, E) be a 3-uniform linear hypergraph with one hypercycle C3. We consider a blow-up hypergraph B[H]. We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph B[H] of the hypergraph H, with hyperedge densities satisfying special conditions, such that the hypergraph H appears in a blow-up hypergraph as a transversal. We present an efficient...

* Correspondence: [email protected] School of Automation, Southeast University, Nanjing 210096, China Full list of author information is available at the end of the article Abstract In this article, we investigate the Dirichlet problem for a porous medium equation with a more complicated source term. In some cases, we prove that the solutions have global blow-up and the rate of blow-up is unifo...

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.

In order to study wall crossing formula of Donaldson type invariants on the blown-up plane, Nakajima-Yoshioka constructed a sequence blow-up/blow-down diagrams connecting moduli space torsion free framed sheaves projective and that its blow-up. this paper, we prove Nakajima-Yoshioka's diagram realizes minimal model program. Furthermore, obtain fully-faithful embedding between derived categories...

This paper deals with a class of heat emission processes in a medium with a nonegative source, a nonlinear decreasing thermal conductivity and a linear radiation (Robin) boundary condition. For such heat emission problems, using a differential inequality technique, we establish conditions on the data sufficient to guarantee that the blow-up of the solutions does occur or does not occur. In addi...

We consider the classical problem of the blowing-up of solutions of the nonlinear heat equation. We show that there exist infinitely many profiles around the blow-up point, and for each integer k, we construct a set of codimension 2k in the space of initial data giving rise to solutions that blow-up according to the given profile.