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...

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...

Many interesting geometric structures can be defined by specifying a smooth subbundle V of the complexified tangent bundle of the underlying smooth manifold, subject to an integrability condition. Examples include foliations, complex structures, and CR structures. In general such structures are called involutive, or formally integrable, and their study has been the starting point of far-reachin...

infantile hemiplegia refers to brain injuries that occur before or at birth and lead to hemiplegia/ total paralysis of one side of the body, including the face, arm and leg. the main purpose of this article is to provide valuable information to pediatric dentists about the review and treatment alternatives for patients with infantile hemiplegia. this article reports the case of a 12-year-old gi...

Many interesting geometric structures can be defined by specifying a smooth subbundle V of the complexified tangent bundle of the underlying smooth manifold, subject to an integrability condition. Examples include foliations, complex structures, and CR structures. In general such structures are called involutive, or formally integrable, and their study has been the starting point of far-reachin...