Finite time blowup of solutions to the nonlinear Schrödinger equation without gauge invariance

Abstract

A lifespan estimate and a condition of the initial data for finite time blowup for the nonlinear Schrödinger equation are presented from a view point of ordinary differential equation (ODE) mechanism.

Sharp upper bound for lifespan of solutions to some critical semilinear parabolic, dispersive and hyperbolic equations via a test function method

“…The critical case p = 1 + 2 N of Proposition 2.3 has not been proved so far. The assertion of 2.3 can be regarded as a refinement of the results of Ikeda-Wakasugi [16] and Fujiwara-Ozawa [8]. It is worth noticing that the technique by Lai-Zhou [26] seems to be difficult to apply to (2.5) because they use the positivity of heat kernel for heat equations.…”

A note for the global nonexistence of semirelativistic equations with nongauge invariant power type nonlinearity

“…Proposition follows from Lemma by using an ordinary differential equation approach introduced by the author and Ozawa . Indeed, it is shown that for some R > 0, is a super solution of an ordinary differential equation taking the form of f ′ = f p , coming from without (−Δ) 1/2 u .…”

“…In general, ϕ ≥ 0 does not imply (−Δ) s/2 ϕ ≥ 0, and therefore, (13) does not imply (9). D'Abbicco and Reissig 12 studied global nonexistence for structural damped wave equation possessing fractional derivative by generalizing (13). For the study of structural damped wave equation, (13) works well because we have nonnegative solutions(D'Abbicco and Reissig 12, lemma 1 ), which we cannot expect for (1).…”

“…D'Abbicco and Reissig 12 studied global nonexistence for structural damped wave equation possessing fractional derivative by generalizing (13). For the study of structural damped wave equation, (13) works well because we have nonnegative solutions(D'Abbicco and Reissig 12, lemma 1 ), which we cannot expect for (1). Lemma 1 implies the following statement, which is our main statement:…”

“…: Proposition 4 follows from Lemma 1 by using an ordinary differential equation approach introduced by the author and Ozawa. 13 Indeed, it is shown that for some R > 0,…”