Limit of the blow-up solution for the inhomogeneous nonlinear Schrödinger equation. (English) Zbl 1540.35378
Summary: We study the \(H^1\) blow-up profile for the inhomogeneous nonlinear Schrödinger equation
\[
i \partial_t u = - \Delta u-|x|^k |u|^{2\sigma} u,\quad (t,x) \in \mathbb{R} \times \mathbb{R}^N,
\]
where \(k \in (-1,2N-2)\) and \(N \geq 3\). We develop a new version of Gagliardo-Nirenberg inequality for \(\sigma \in [\frac{2+k}{N}, \frac{2+k}{N-2}]\) and \(k \in (-1,2N-2)\), and show that for the \(L^2\)-critical exponent \(\sigma = \frac{2+k}{N}, u(t)\) has no \(L^2\)-limit as \(t \to T^\ast\) when \(\| u(t) \|_{H^1}\) blows up at \(T^\ast \). Moreover, we investigate \(L^2\) concentration at the origin in the radial case. Additionally, if \(\frac{2+k}{N} < \sigma < \min \{2,\frac{2+k}{N-2}, \frac{2(1+k)+N}{2N}\}\), we show that there exists a unique \(u^\ast \in L^2 (\mathbb{R}^N)\) such that \(\Gamma (-t) u(t) \to \Gamma (-T^\ast) u^\ast\) in \(L^r (\mathbb{R}^N) (r\in [2,2^\ast))\) as \(t \to T^\ast\). Our results extend the work for \(k=0\) by F. Merle in earlier time.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
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