Cauchy problem for a Schrödinger-type equation related to the Riemann zeta function
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910044559245312 |
|---|---|
| author | Mohamed, Bensaid |
| author_facet | Mohamed, Bensaid |
| contents | We study the Cauchy problem in the space $H^1(Σ)$ for a nonlinear damped Schrödinger equation of the form \begin{equation}\tag{NLS-$ζ$}\label{nls}
i u_t + Δu + i λu \, ζ(|u|+1) = 0, \quad u(0,x) = u_0, \end{equation} where $ζ$ denotes the Riemann zeta function. We first establish the uniqueness of solutions in the sense of distributions. Then, by considering a regularized problem, we prove the existence of a global solution in $H^1(Σ)$, using uniform estimates and compactness arguments. Finally, we show that the limiting solution indeed satisfies the original equation in the weak sense. In the addition we proof that, the one-dimensional case, we show that it becomes zero in finite time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20128 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cauchy problem for a Schrödinger-type equation related to the Riemann zeta function Mohamed, Bensaid Analysis of PDEs Mathematical Physics We study the Cauchy problem in the space $H^1(Σ)$ for a nonlinear damped Schrödinger equation of the form \begin{equation}\tag{NLS-$ζ$}\label{nls} i u_t + Δu + i λu \, ζ(|u|+1) = 0, \quad u(0,x) = u_0, \end{equation} where $ζ$ denotes the Riemann zeta function. We first establish the uniqueness of solutions in the sense of distributions. Then, by considering a regularized problem, we prove the existence of a global solution in $H^1(Σ)$, using uniform estimates and compactness arguments. Finally, we show that the limiting solution indeed satisfies the original equation in the weak sense. In the addition we proof that, the one-dimensional case, we show that it becomes zero in finite time. |
| title | Cauchy problem for a Schrödinger-type equation related to the Riemann zeta function |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2511.20128 |