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...

* 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 establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (i...

We use techniques from reaction-diffusion theory to study the blow-up and existence of solutions of the parabolic Monge–Ampère equation with power source, with the following basic 2D model (0.1) u t = −|D 2 u| + |u| p−1 u in R 2 × R + , where in two-dimensions |D 2 u| = u xx u yy − (u xy) 2 and p > 1 is a fixed exponent. For a class of " dominated concave " and compactly supported radial initia...

BACKGROUND Molecular tools are now widely used to address crucial management and conservation questions. To date, dart biopsying has been the most commonly used method for collecting genetic data from cetaceans; however, this method has some drawbacks. Dart biopsying is considered inappropriate for young animals and has recently come under scrutiny from ethical boards, conservationists, and the...

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...