We consider the 3d cubic focusing nonlinear Schrödinger equation (NLS) i∂tu+∆u+ |u|u = 0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H axial(R) of initial data, that blow-up in finite time with singular set a circle in xy plane. Our construction is modeled on Raphaël’s construction...

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...

We study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half-line with a nonlinear boundary condition,    ut = uxx − λ(u + 1) log(u + 1) (x, t) ∈ R+ × (0, T ), −ux(0, t) = (u + 1) log(u + 1)(0, t) t ∈ (0, T ), u(x, 0) = u0(x) x ∈ R+, with p, q, λ > 0. We describe in terms of p, q and λ when the solution is global in time and when it blows up in f...

This is the first part of our comprehensive study on the structure of doubly periodic solutions for the Chern-Simons-Higgs equation with a small coupling constant. We first classify the bubbling type of the blow-up point according to the limit equations. Assuming that all the blow-up points are away from the vortex points, we prove the noncoexistence of different bubbling types in a sequence of...

Radial blow-up, including inhomogeneous versions, of boundary faces of a manifold (always with corners) is an important tool for resolving singularities, degeneracies and competing notions of homogeneity. These constructions are shown to be particular cases of generalized boundary blow-up in which a new manifold and blow-down map are constructed from, and conversely determine, combinatorial dat...

We rigorously construct radial H solutions to the 3d cubic focusing NLS equation i∂tψ + ∆ψ + 2|ψ|ψ = 0 that blow-up along a contracting sphere. With blow-up time set to t = 0, the solutions concentrate on a sphere at radius ∼ t but focus towards this sphere at the faster rate ∼ t. Such dynamics were originally proposed heuristically by Degtyarev-Zakharov-Rudakov [2] in 1975 and independently la...

This paper deals with the blow-up properties of the solution to the degenerate nonlinear reaction diffusion equation with nonlocal source xut − (xux)x = ∫ a 0 u pdx in (0, a) × (0, T ) subject to the homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally or blows up in fini...