In this paper we prove the existence of Kähler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold which already carries a Kähler constant scalar curvature metric. Necessary conditions of the number and locations of the blow up points are given. 1991 Math. Subject Classification: 58E11, 32C17.

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,...

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 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...

This article is concerned with the existence, uniqueness and numerical approximation of boundary blow up solutions for elliptic PDE’s as ∆u = f(u) where f satisfies the so-called Keller-Osserman condition. We characterize existence of such solutions for non-monotone f . As an example, we construct an infinite family of boundary blow up solutions for the equation ∆u = u(1 + cos u) on a ball. We ...

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...

We prove the finite time blow-up for solutions of the 3D incompressible Euler equations, which happens along the fluid particle trajectories starting from a set of points. This set is specified by the relation between the deformation tensor and the Hessian of pressure both coupled with the vorticity directions, associated with the initial data. As a corollary of this result we prove the finite ...

We study the Cauchy problem for a system of two coupled nonlinear focusing Schrödinger equations arising in nonlinear optics. We discuss when the solutions are global in time or blow-up in finite time. Some results, in dependence of the data of the problem, are proved; in particular we give a bound, depending on the coupling parameter, for the blow-up threshold. 2000 Mathematics Subject Classif...

We study the simultaneous blow-up rates of a system of two heat equations coupled through the boundary in a nonlinear way. We complete the previous known results by covering the whole range of possible parameters.