The blow-up dynamics for the divergence Schrödinger equations with inhomogeneous nonlinearity
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| Format: | Preprint |
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2024
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| _version_ | 1866915024495181824 |
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| author | Zheng, Bowen Ozawa, Tohru |
| author_facet | Zheng, Bowen Ozawa, Tohru |
| contents | This paper is dedicated to the blow-up solution for the divergence Schrödinger equations with inhomogeneous nonlinearity (dINLS for short) \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)=-|x|^c|u|^pu,\quad\quad u(x,0)=u_0(x),\] where $2-n<b<2$, $c>b-2$, and $np-2c<(2-b)(p+2)$. First, for radial blow-up solutions in $W_b^{1,2}$, we prove an upper bound on the blow-up rate for the intercritical dNLS. Moreover, an $L^2$-norm concentration in the mass-critical case is also obtained by giving a compact lemma. Next, we turn to the non-radial case. By establishing two types of Gagliardo-Nirenberg inequalities, we show the existence of finite time blow-up solutions in $\dot{H}^{s_c}\cap \dot{W}^{1,2}_b$, where $\dot{H}^{s_c}=(-Δ)^{-\frac{s_c}{2}}L^2$, and $\dot{W}_b^{1,2}=|x|^{-\frac{b}{2}}(-Δ)^{-\frac{1}{2}}L^2$. As an application, we obtain a lower bound for this blow-up rate, generalizing the work of Merle and Raphaël [Amer. J. Math. 130(4) (2008), pp. 945-978] for the classical NLS equations to the dINLS setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11333 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The blow-up dynamics for the divergence Schrödinger equations with inhomogeneous nonlinearity Zheng, Bowen Ozawa, Tohru Analysis of PDEs 35Q55, 35B44 This paper is dedicated to the blow-up solution for the divergence Schrödinger equations with inhomogeneous nonlinearity (dINLS for short) \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)=-|x|^c|u|^pu,\quad\quad u(x,0)=u_0(x),\] where $2-n<b<2$, $c>b-2$, and $np-2c<(2-b)(p+2)$. First, for radial blow-up solutions in $W_b^{1,2}$, we prove an upper bound on the blow-up rate for the intercritical dNLS. Moreover, an $L^2$-norm concentration in the mass-critical case is also obtained by giving a compact lemma. Next, we turn to the non-radial case. By establishing two types of Gagliardo-Nirenberg inequalities, we show the existence of finite time blow-up solutions in $\dot{H}^{s_c}\cap \dot{W}^{1,2}_b$, where $\dot{H}^{s_c}=(-Δ)^{-\frac{s_c}{2}}L^2$, and $\dot{W}_b^{1,2}=|x|^{-\frac{b}{2}}(-Δ)^{-\frac{1}{2}}L^2$. As an application, we obtain a lower bound for this blow-up rate, generalizing the work of Merle and Raphaël [Amer. J. Math. 130(4) (2008), pp. 945-978] for the classical NLS equations to the dINLS setting. |
| title | The blow-up dynamics for the divergence Schrödinger equations with inhomogeneous nonlinearity |
| topic | Analysis of PDEs 35Q55, 35B44 |
| url | https://arxiv.org/abs/2411.11333 |