Well-posedness and decay estimates for 1D nonlinear Schr\"odinger equations with Cauchy data in $L^p$

Ryosuke Hyakuna

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arXiv · 1811.09752

Well-posedness and decay estimates for 1D nonlinear Schr\"odinger equations with Cauchy data in $L^p$

Ryosuke Hyakuna

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Abstract

In this paper, we establish a standard $L^p$-theory of solutions to one dimensional nonlinear Schr\"odinger equations with the power like nonlinearity. More precisely, we extend the following three well-known results in the $L^2$ space into $L^p$ setting: 1. Large data local well-posedness for subcritical nonlinearities, 2. Small data global well-posedness for critical nonlinearities, 3. Large data global well-posedness if the subcritical nonlinearity is given by $|u|^{\alpha-1}u$.

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Ryosuke Hyakuna. 2018-11-24. Well-posedness and decay estimates for 1D nonlinear Schr\"odinger equations with Cauchy data in $L^p$. https://arxiv.org/abs/1811.09752

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