Local well-posedness for the inhomogeneous nonlinear Schrödinger equation. (English) Zbl 1479.35759
Summary: We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation \(i\partial_t u +\Delta u = \mu |x|^{-b}|u|^\alpha u,\; u(0)\in H^s(\mathbb{R}^N),\; N\geq 1,\; \mu\in\mathbb{C},\; b>0\) and \(\alpha>0.\) Only partial results are known for the local existence in the subcritical case \(\alpha<(4-2b)/(N-2s)\) and much more less in the critical case \(\alpha = (4-2b)/(N-2s)\). In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for \(b = 0\) and \(b>0\).
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
Keywords:
inhomogeneous nonlinear Schrödinger equation; local well-posedness; continuous dependence; unconditional uniqueness; blow-up rate; life span; global existenceReferences:
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