Abstract Solutions to nonlinear Schrödinger equations may blow up in finite time. We study the influence of the introduction of a potential on this phenomenon. For a linear potential (Stark effect), the blow-up time remains unchanged, but the location of the collapse is altered. The main part of our study concerns isotropic quadratic potentials. We show that the usual (confining) harmonic poten...

We construct blow-up solutions of the energy critical wave map equation on R → N with polynomial blow-up rate (t for blow-up at t = 0) in the case when N is a surface of revolution. Here we extend the blow-up range found by Carstea (ν > 1 2 ) based on the work by Krieger, Schlag and Tataru to ν > 0. This work relies on and generalizes the recent result of Krieger and the author where the target...

In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems.

We use Karamata regular variation theory to study the exact asymptotic behavior of large solutions near the boundary to a class of quasilinear elliptic equations with convection terms ⎧⎨ ⎩ Δpu±|∇u|q(p−1) = b(x) f (u), x ∈Ω,

Blow-up behaviour for the fourth-order quasilinear porous medium equation with source, (0.1) ut = −(|u|u)xxxx + |u|u in R × R+, n > 0, p > 1, is studied. Countable and finite families of similarity blow-up patterns of the form uS(x, t) = (T − t)− 1 p−1 f(y), where y = x/(T − t) , β = p−(n+1) 4(p−1) , which blow-up as t → T− < ∞, are described. These solutions explain key features of regional (f...

The tight transverse emittance budget for the LHC beam imposes severe constraints on the blow-up produced by injection errors. The modifications to the beam distributions and their possible parameterisation in terms of blow-up factors are discussed for betatron and dispersion mismatch as well as for injection offsets. The cases of initial uniform or gaussian distributions are considered in deta...