Lifespan of strong solutions to the periodic nonlinear Schrödinger equation without gauge invariance

Estimates of Lifespan and Blow-up Rates for the Wave Equation with a Time-dependent Damping and a Power-type Nonlinearity

“…For detail, see Proposition 2.3. Next, to prove Proposition 1.3, we follow [9,10] and consider the localized average I φ (t) = R n u(t, x)φ(x)dx, and derive the ordinary differential inequality (1.15) from the equation (1.1). Finally, for Corollary 1.4, we choose a special family of cut-off functions and apply a scaling argument to reduce its proof to Proposition 1.3.…”

“…To overcome these difficulties, we employ the method developed by the first author and Ozawa [9,10] in which the lifespan of the solution for a nonlinear Schrödinger equation is studied. They considered a localized average of the solution R n u(t, x)φ(x)dx with a cut-off function φ(x), and derive an ordinary differential inequality for it.…”

Sharp local well-posedness of the two-dimensional periodic nonlinear Schrödinger equation with a quadratic nonlinearity $|u|^2$

“…Indeed, the second author [18] proved a finite-time blowup result for (1.1). See also [9]. This argument can be easily extended to L 2 (T 2 ), yielding the following proposition.…”

Quasiperiodicity and blowup in integrable subsystems of nonconservative nonlinear Schrödinger equations

“…For comparison to another nonlinear Schrödinger equation lacking gauge invariance one may consider the equation iu t = u + |u| p . By studying the dynamics of the zeroth Fourier mode it has been shown that a robust set of initial data will blowup in finite time [Oh12,II15,FO17]. For the nonlinearity |u| 2 recent work has shown that initial data u 0 ∈ L 2 (T 1 ) will have a global solution if and only if u 0 (x) = iµ 0 for µ 0 ≥ 0 [FG20].…”