Cauchy problem for a Schr\"odinger-type equation related to the Riemann zeta function

Abstract

We study the Cauchy problem in the space $H^1(\Sigma)$ for a nonlinear damped Schr\"odinger equation of the form \begin{equation}\tag{NLS-$\zeta$}\label{nls} i u_t + \Delta u + i \lambda u \, \zeta(|u|+1) = 0, \quad u(0,x) = u_0, \end{equation} where $\zeta$ 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 $H^1(\Sigma)$, 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.