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

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
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