7 research outputs found
Wave equation with concentrated nonlinearities
In this paper we address the problem of wave dynamics in presence of
concentrated nonlinearities. Given a vector field on an open subset of
\CO^n and a discrete set Y\subset\RE^3 with elements, we define a
nonlinear operator on L^2(\RE^3) which coincides with the free
Laplacian when restricted to regular functions vanishing at , and which
reduces to the usual Laplacian with point interactions placed at when
is linear and is represented by an Hermitean matrix. We then consider the
nonlinear wave equation and study the
corresponding Cauchy problem, giving an existence and uniqueness result in the
case is Lipschitz. The solution of such a problem is explicitly expressed
in terms of the solutions of two Cauchy problem: one relative to a free wave
equation and the other relative to an inhomogeneous ordinary differential
equation with delay and principal part . Main properties of
the solution are given and, when is a singleton, the mechanism and details
of blow-up are studied.Comment: Revised version. To appear in Journal of Physics A: Mathematical and
General, special issue on Singular Interactions in Quantum Mechanics:
Solvable Model
BLOW UP SOLUTIONS FOR THE SCHR\"ODINGER EQUATION WITH A CONCENTRATED NONLINEARITY IN DIMENSION THREE
Blow up solutions for the Schroedinger equation with a concentrated nonlinearity in dimension three.
We present some results on the blow-up phenomenon for the Schrödinger equation in dimension three with a nonlinear term
supported in a fixed point. We find sufficient conditions for the blow-up exploiting the moment of inertia of the solution and
the uncertainty principle. In the critical case, we discuss the additional symmetries of the equation and construct a family of
explicit blow-up solutions