Finite time blow-up of non-radial solutions for some inhomogeneous Schrödinger equations
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866918043302494208 |
|---|---|
| author | Bai, Ruobing Saanouni, Tarek |
| author_facet | Bai, Ruobing Saanouni, Tarek |
| contents | This work studies the inhomogeneous Schrödinger equation $$ i\partial_t u-\mathcal{K}_{s,λ}u +F(x,u)=0 , \quad u(t,x):\mathbb{R}\times\mathbb{R}^N\to\mathbb{C}. $$ Here, $s\in\{1,2\}$, $N>2s$ and $λ>-\frac{(N-2)^2}{4}$. The linear Schrödinger operator reads $\mathcal{K}_{s,λ}:= (-Δ)^s +(2-s)\fracλ{|x|^2}$ and the focusing source term is local or non-local $$F(x,u)\in\{|x|^{-2τ}|u|^{2(q-1)}u,|x|^{-τ}|u|^{p-2}(J_α*|\cdot|^{-τ}|u|^p)u\}.$$ The Riesz potential is $J_α(x)=C_{N,α}|x|^{-(N-α)}$, for certain $0<α<N$. The singular decaying term $|x|^{-2τ}$, for some $τ>0$ gives a inhomogeneous non-linearity. One considers the inter-critical regime, namely $1+\frac{2(1-τ)}N<q<1+\frac{2(1-τ)}{N-2s}$ and $1+\frac{2-2τ+α}{N}<p<1+\frac{2-2τ+α}{N-2s}$. The purpose is to prove the finite time blow-up of solutions with datum in the energy space, non necessarily radial or with finite variance. The assumption on the data is expressed in terms of non-conserved quantities. This is weaker than the ground state threshold standard condition. The blow-up under the ground threshold or with negative energy are consequences. The proof is based on Morawetz estimates and a non-global ordinary differential inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15210 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finite time blow-up of non-radial solutions for some inhomogeneous Schrödinger equations Bai, Ruobing Saanouni, Tarek Analysis of PDEs This work studies the inhomogeneous Schrödinger equation $$ i\partial_t u-\mathcal{K}_{s,λ}u +F(x,u)=0 , \quad u(t,x):\mathbb{R}\times\mathbb{R}^N\to\mathbb{C}. $$ Here, $s\in\{1,2\}$, $N>2s$ and $λ>-\frac{(N-2)^2}{4}$. The linear Schrödinger operator reads $\mathcal{K}_{s,λ}:= (-Δ)^s +(2-s)\fracλ{|x|^2}$ and the focusing source term is local or non-local $$F(x,u)\in\{|x|^{-2τ}|u|^{2(q-1)}u,|x|^{-τ}|u|^{p-2}(J_α*|\cdot|^{-τ}|u|^p)u\}.$$ The Riesz potential is $J_α(x)=C_{N,α}|x|^{-(N-α)}$, for certain $0<α<N$. The singular decaying term $|x|^{-2τ}$, for some $τ>0$ gives a inhomogeneous non-linearity. One considers the inter-critical regime, namely $1+\frac{2(1-τ)}N<q<1+\frac{2(1-τ)}{N-2s}$ and $1+\frac{2-2τ+α}{N}<p<1+\frac{2-2τ+α}{N-2s}$. The purpose is to prove the finite time blow-up of solutions with datum in the energy space, non necessarily radial or with finite variance. The assumption on the data is expressed in terms of non-conserved quantities. This is weaker than the ground state threshold standard condition. The blow-up under the ground threshold or with negative energy are consequences. The proof is based on Morawetz estimates and a non-global ordinary differential inequality. |
| title | Finite time blow-up of non-radial solutions for some inhomogeneous Schrödinger equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2306.15210 |