In the first part of the paper, we give an explicit algorithm to compute the (genus zero) Gromov-Witten invariants of blow-ups of an arbitrary convex projective variety in some points if one knows the Gromov-Witten invariants of the original variety. In the second part, we specialize to blow-ups of Pr and show that many invariants of these blow-ups can be interpreted as numbers of rational curv...

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...

In this paper we study the asymptotic behaviour of a semidiscrete numerical approximation for the heat equation, ut = ∆u, in a bounded smooth domain, with a nonlinear flux boundary condition at the boundary, ∂u ∂η = up. We focus in the behaviour of blowing up solutions. First we prove that every numerical solution blows up in finite time if and only if p > 1 and that the numerical blow-up time ...

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...

We consider blow-out bifurcations of synchronous chaotic attractors on invariant subspaces in coupled chaotic systems with symmetries. Through a blow-out bifurcation, the synchronous chaotic attractor becomes unstable with respect to perturbations transverse to the invariant subspace, and then a new asynchronous chaotic attractor may appear. However, the system symmetry may be preserved or viol...

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...

We obtain some conditions under which the positive solution for semidiscretizations of the semilinear equation ut uxx − a x, t f u , 0 < x < 1, t ∈ 0, T , with boundary conditions ux 0, t 0, ux 1, t b t g u 1, t , blows up in a finite time and estimate its semidiscrete blow-up time. We also establish the convergence of the semidiscrete blow-up time and obtain some results about numerical blow-u...

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...

After a brief discussion of known global well-posedness results for semilinear systems, we introduce a class of quasilinear systems and obtain spatially local estimates which allow us to prove that if one component of the system blows up in finite time at a point x∗ in space then at least one other component must also blow up at the same point. For a broad class of systems modelling one-step re...