On a Fractional in Time Nonlinear Schrödinger Equation with Dispersion Parameter and Absorption Coefficient

This paper is concerned with the nonexistence of global solutions to fractional in time nonlinear Schrödinger equations of the form $i^{\alpha }{\partial }_t^{\alpha }\omega (t,z)+a_1\left(t\right)\Delta \omega (t,z)+i^{\alpha }{a}_2\left(t\right)\omega (t,z)=\xi {\left|\omega (t,z)\right|}^p,(t,z)\in (0,\infty )\times {\mathbb{R}}^N$ , where $N\ge 1$ , $\xi \in \mathbb{C}\backslash \left\{0\right\}$ and $p>1$ , under suitable initial data. To establish our nonexistence theorem, we adopt the Pohozaev nonlinear capacity method, and consider the combined effects of absorption and dispersion terms. Further, we discuss in details some special cases of coefficient functions $a_1,a_2\in L_{loc}^1([0,\infty ),\mathbb{R})$ , and provide two illustrative examples. View Full-Text