We study finite-time blow-up for pseudo-relativistic Hartreeand Hartree-Fock equations, which are model equations for the dynamical evolution of white dwarfs. In particular, we prove that radially symmetric initial configurations with negative energy lead to finite-time blow-up of solutions. Furthermore, we derive a mass concentration estimate for radial blow-up solutions. Both results are math...

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 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...

In this paper we study the blowup problem of nonlinear heat equations. Our result show that for a certain family of initial conditions the solution will blowup in finite time, the blowup parameters satisfy some dynamics which are asymptotic stable, moreover we provide the remainder estimates. Compare to the previous works our approach is analogous to one used in bifurcation theory and our techn...

We study the relation between the symplectomorphism group SympM of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp f M and Diff f M of its one point blow up f M . There are three main arguments. The first shows that for any oriented M the natural map from π1(M) to π0(Diff M) is often injective. The second argument applies when M is simply connec...

We give a solution of the blow-up problem for equation u = e, with data close to constants, in any number of space dimensions: there exists a blow-up surface, near which the solution has logarithmic behavior; its smoothness is estimated in terms of the smoothness of the data. More precisely, we prove that for any solution of u = e with Cauchy data on t = 1 close to (ln 2; 2) in H(R) H (R), s is...

We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasigeostrophic equation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall w...

A new method to detect finite-time blow-up in systems of ordinary differential equations is presented. This simple algorithmic procedure is based on the analysis of singularities in complex time and amounts to checking the real-valuedness of the leading order term in the asymptotic series describing the behavior of the general solution around movable singularities. Illustrative examples and an ...