[{"type":"report","title":"On the global well-posedness of the quadratic NLS on $L^2(\\mathbb{R})+H^1(\\mathbb{R})$","issued":{"date-parts":[["2019"]]},"DOI":"10.5445\/IR\/1000093752","author":[{"family":"Chaichenets","given":"Leonid"},{"family":"Hundertmark","given":"Dirk"},{"family":"Kunstmann","given":"Peer"},{"family":"Pattakos","given":"Nikolaos"}],"publisher":"KIT","publisher-place":"Karlsruhe","ISSN":"2365-662X","collection-title":"CRC 1173","abstract":"We study the one dimensional nonlinear Schro\u0308dinger equation with power nonlinearity $|u|^{\\alpha-1}$ for $\\alpha \\in [2, 5]$ and initial data $u_0 \u2208 L^2(\\mathbb{R})+H^1(\\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity $(\\alpha = 2)$ we obtain unconditional global well-posedness in the space $C(\\mathbb{R}, L^2(\\mathbb{R})+H^1(\\mathbb{T}))$ via Gronwall\u2019s inequality.","keyword":"nonlinear Schro\u0308dinger equation, local well-posedness, global well-posedness, Gronwall\u2019s inequality, Strichartz estimates","number-of-pages":23,"kit-publication-id":"1000093752"}]