Nonglobal Existence of Solutions for a Generalized Ginzburg–Landau Equation Coupled with a Poisson Equation

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“…Indeed, there exists ε > 0 arbitrarily small such that u(ε) ∈ H 2 (R n ) (by the above cited result of Snoussi [8]). Hence, by theorem 2.2 of Weissler [11] we have: u ∈ C 1 ((ε, T max ); H 1 (R n )).…”

“…Thanks to the results of Snoussi [8], we know that for any u 0 ∈ H 1 (R n ) with n ≤ 2, there exists a unique maximal solution u of (P) such that…”

On a coupled system of a Ginzburg–Landau equation with a quasilinear conservation law

“…is the infinitesimal generator of an analytic semigroup of contractions {T θ (t)} t≥0 in L 2 (R), with domain H 2 (R) and verifying the estimate (16), the result follows by considering the Duhamel formulas of the auxiliary system…”

Finite-time Blowup for some Nonlinear Complex Ginzburg–Landau Equations

“…Therefore, the righthand side of (4.34) becomes negative for t large, which implies that T max < ∞. [38] in the case γ ≤ 0 (with extra conditions on the parameters), in [6] in the case γ = 0, and in [5] in the case γ > 0. The upper bound is established in [6] in the case γ = 0.…”

“…On the other hand, this strategy does not give any information on how the blowup occurs, nor on the mechanism that leads to blowup. Concerning the family (1.1), this is the type of approach used in [15,19,2] for the nonlinear heat equation; in [45,12,17,41,30,31] for the nonlinear Schrödinger equation; in [38,6,5] for the intermediate case of (1.1).…”