Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity

Abstract:

We consider the following semirelativistic nonlinear Schrödinger equation (SNLS): \begin{equation} \left \{ \begin {array}{ll} i\partial _t u \pm (m^2-\Delta )^{1/2} u = \lambda |u|^{p}, & (t,x)\in [0,T)\times \mathbb {R}^d, \\ u(0,x)=u_0(x), & x \in \mathbb {R}^d, \end{array} \right . \notag \end{equation} where $m\geq 0$, $\lambda \in \mathbb {C} \setminus \{ 0\}$, $d\in \mathbb {N}$, $T>0$, and $\partial _t=\partial /\partial t$. Here $(m^2-\Delta )^{1/2}:=\mathcal {F}^{-1} (m^2+|\xi |^2 )^{1/2} \mathcal {F}$, where $\mathcal {F}$ denotes the Fourier transform. Fujiwara and Ozawa proved the nonexistence of global weak solutions to SNLS for some initial data in the case of $d=1$, $m=0$, and $1<p\leq 2$ by a test function method. In this paper, we extend their result to a more general setting: for example, $m\geq 0$, $d\in \mathbb {N}$, or $p>1$. Moreover, we obtain the upper estimates of weak solutions to SNLS. The key to the proof is to choose an appropriate test function.