Oscillations of concentration effects in semilinear dispersive wave equations. (English) Zbl 0868.35075

Gérard, Patrick

J. Funct. Anal. 141, No. 1, 60-98 (1996).

Summary: Let \((u_n)\) be a sequence of smooth solutions to a dispersive nonlinear wave equation, \[ \partial^2_t u_n-\Delta u_n+f(u_n)= 0 \] in \(\mathbb{R}^{1+3}\), with uniformly compactly supported Cauchy data converging weakly to 0 in \(H^1(\mathbb{R}^3)\times L^2(\mathbb{R}^3)\). Let \((v_n)\) be the sequence of solutions to the linear wave equation with the same Cauchy data. We show that \(u_n-v_n\) goes strongly to 0 in the energy space \(C([0,T],H^1)\cap C^1([0,T],L^2)\) if \(f\) is a subcritical nonlinearity. In the critical case \(f(u)=u^5\), we show that this property is equivalent to \(v_n\to 0\) in \(L^\infty([0,T],L^6)\). Then we give sharp sufficient conditions on microlocal measures associated to the data. The proof relies on a microlocal version of P.-L. Lions’ concentration-compacticity.

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

dispersive nonlinear wave equation; microlocal measures; concentration-compacticity