Local existence, global existence, and scattering for the nonlinear Schrödinger equation

Abstract

In this paper, we construct for every $\alpha >0$ and $\lambda \in \mathbb{C}$ a space of initial values for which there exists a local solution of the nonlinear Schrödinger equation \begin{equation*} \begin{cases} iu_t + Δu + λ|u|^αu= 0 \\ u(0,x) = u_0 \end{cases} \end{equation*} on ${\mathbb{R}}^N$. Moreover, we construct for every $\alpha >\frac{2}{N}$ a class of (arbitrarily large) initial values for which there exists a global solution that scatters as $t\to \infty$.

References in corpus (1)

• A Fujita-type blowup result and low energy scattering for a nonlinear Schrö\-din\-ger equation

Cited by in corpus (5)

• Modified scattering for the critical nonlinear Schrödinger equation
• Asymptotic behavior for a dissipative nonlinear Schrödinger equation
• On a class of solutions to the generalized KdV type equation
• Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity
• Local smooth solutions of the nonlinear Klein-gordon equation