Blowup and global existence for some complex Ginzburg-Landau equations
speaker: Thierry Cazenave (Paris VI)abstract:
In this talk I will discuss some recent work in collaboration with S.~Correia, F.~Dickstein and F.~B.~Weissler on the family of complex Ginzburg-Landau equation \(u_t = e^{i\theta }\Delta u + e^{i\gamma }
u
^\alpha u \) set on the whole space, where \(-\frac {\pi } {2}\le \theta ,\gamma \le \frac {\pi } {2}\). In the particular case \(\gamma =\theta \), we prove finite-time blowup when the initial value satisfies some energy condition, and study the behavior of the blow-up time when \(\theta \to \pm \frac {\pi } {2}\), i.e., in the NLS limit. In another special case, namely for \(\theta =\pm \frac {\pi } {2}\) and \(-\frac {\pi } {2} < \gamma < \frac {\pi } {2}\), we prove that \(\alpha =\frac {N} {2}\) is a critical exponent of Fujita type. I will mention a number of related open questions.
Finally, I will briefly discuss the existence of standing waves for the more general
equation \(u_t = e^{i\theta }\Delta u + e^{i\gamma }
u
^\alpha u + ku \), set either on the whole space, or on a bounded domain with Dirichlet boundary conditions.
timetable:
Thu 6 Nov, 10:30 - 11:20, Aula Dini
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