Nonexistence of global solutions for fractional temporal Schrodinger equations and systems

Abstract

We, first, consider the nonlinear Schrodinger equation $$ i^\alpha {}_0^C D_t^\alpha u+\Delta u= \lambda |u|^p+\mu a(x)\cdot\nabla |u|^q, \quad t>0,\; x\in \mathbb{R}^N, $$ where 0q>1$, $u(t,x)$ is a complex-valued function, and $a: \mathbb{R}^N\to \mathbb{R}^N$ is a given vector function. We provide sufficient conditions for the nonexistence of global weak solution under suitable initial data. Next, we extend our study to the system of nonlinear coupled equations $$\displaylines{ i^\alpha {}_0^C D_t^\alpha u+\Delta u = \lambda |v|^p+\mu a(x)\cdot\nabla |v|^q, \quad t>0,\;x\in \mathbb{R}^N,\cr i^\beta {}_0^C D_t^\beta v+\Delta v = \lambda |u|^\kappa+\mu b(x)\cdot\nabla |u|^\sigma, \quad t>0,\; x\in \mathbb{R}^N, }$$ where $0q>1$, $\kappa>\sigma>1$, and $a,b: \mathbb{R}^N\to \mathbb{R}^N$ are two given vector functions. Our approach is based on the test function method.

Keywords

• Fractional temporal Schrodinger equation
• nonexistence
• global weak solution