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A singular perturbation problem for an envelope equation in plasma physics. (English) Zbl 0884.35097
Summary: We investigate the so-called Langmuir wave envelope approximation, which consists in taking the limit $$\omega\rightarrow\infty$$ in the nonlinear plasma wave equation $1/\omega^2\partial^2_tE_\omega-2i\partial_tE_\omega-\Delta E_\omega=f(|E_\omega|^2)E_\omega,$ stated for nonlinearities satisfying $$|f(\rho)|\leq K \rho^\sigma$$. For any finite value of $$\omega>1$$, the solution $$E_\omega$$ with the initial data $$E_\omega(x,t=0) \in H^2(\mathbb{R}^n),\partial_tE_\omega(x,t=0)\in H^1(\mathbb{R}^n)$$ is shown to exist locally in time and to be unique. Under some specific conditions including $$\omega$$ below a threshold value, we construct solutions $$E_\omega$$ that blow up in a finite time with a divergent $$L^2$$ norm; nevertheless, in the so-called subcritical case ($$\sigma n <2$$), the solution defined for fixed initial data is global provided that $$\omega$$ should be large enough. We demonstrate the strong convergence of $$E_\omega$$ towards the nonlinear Schrödinger solution $$E$$ reached as $$\omega\rightarrow\infty$$, as long as $$E$$ exists. In this same limit, we finally discuss the behavior of the time derivative of $$E_\omega$$ and compare the blow-up times associated with $$E_\omega$$ and with its time-enveloped counterpart $$E$$.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35Q55 NLS equations (nonlinear Schrödinger equations) 76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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