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Blowup for nonlinear wave equations describing boson stars. (English) Zbl 1135.35011
The purpose of the present paper is to address the problem of proving blowup of solutions for the following nonlinear wave equation $$i\partial_t u= \sqrt{-\Delta_x+ m^2}u-\biggl({1\over|x|}* |u|^2\biggr)u\quad\text{on }\bbfR^3,\tag1$$ where $u(t,x)$ is a complex-valued wave field, the operator $\sqrt{-\Delta_x+ m^2}$ is defined via its symbol $\sqrt{k^2+ m^2}$ and the symbol $*$ stands for convolution on $\bbfR^3$. The authors establish the following result: any spherically symmetric initial data $u_0(x)\in C^\infty_0(\bbfR^3)$ with negative energy gives rise to a solution $u(t,x)$ of (1) that blows up within a finite time, more precisely $\lim_{t\nearrow T}\Vert u(t,\cdot)\Vert_{H^{1/2}}= \infty$ for some $0< T< \infty$. Moreover, the authors consider more general Hartree-type nonlinearities. As an application, they exhibit instability of ground solitary waves at rest if $m= 0$.

35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35S10Initial value problems for pseudodifferential operators
35L70Nonlinear second-order hyperbolic equations
85A05Galactic and stellar dynamics
35Q53KdV-like (Korteweg-de Vries) equations
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