Nonlinear optics and supercritical wave equation. (English) Zbl 1034.35137

Lebeau, Gilles

Bull. Soc. R. Sci. Liège 70, No. 4-6, 267-306 (2001).

The Cauchy problem for the nonlinear wave equation in \(\mathbb{R}^3\) \[ (\partial^2_t-\Delta_x) u+u^p=0,\quad u=u(t,x),\quad t\in\mathbb{R}, \quad x\in\mathbb{R}^3 \]
\[ u|_{t=0}= u_0(x)\in H^1\cap L^{p+1};\;\partial_t u|_{t=0} =u_1(x)\in L^2 \] is studied, for which the existence and uniqueness of strong solutions is well known for the sub-critical case \(p\leq 3\). The present work extends the study to the supercritical case and shows that the local behavior of solutions then exhibits new phenomena.

Reviewer: Alan Jeffrey (Newcastle upon Tyne)

MSC:

35Q60	PDEs in connection with optics and electromagnetic theory
78A60	Lasers, masers, optical bistability, nonlinear optics

Keywords:

existence; uniqueness; strong solutions; local behavior