The role of potential, Morawetz estimate and spacetime bound for quasilinear Schr\"{o}dinger equations

Abstract

In this paper, we deal with the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t = \Delta u + 2uh'(|u|^2)\Delta h(|u|^2) + V(x)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0) = u_0(x), \quad x \in \mathbb{R}^N. \end{array}\right. \end{equation*} Here \(h(s)\) and \(V(x)\) are some real functions. We take the potential \(V(x)\in L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)\) as criterion of the blowup and global existence of the solution to (1.1). In some cases, we can classify it in the following sense: If \(V(x)\in S(I)\), then the solution of (1.1) is always global existence for any \(u_0\) satisfying \(0<E(u_0)<+\infty\); If \(V(x)\in S(II)\), then the solution of (1.1) may blow up for some initial data \(u_0\). Here \[ S(I)=\cup_{q>q_c}[L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)],\quad S(II)=\left\{\cup_{q<q_c}[L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)]\right\}\setminus S(I).\] Under certain assumptions, we also establish Morawetz estimates and spacetime bounds for the global solution, for example, \begin{align*} &\int_0^{+\infty} \int_{\mathbb{R}^N }\frac{[|\nabla h(|u|^2)|^2 + |V(x)||u|^2]}{(|x|+t)^{\lambda}}dxdt\leq C,\\ & \|u\|_{L^{\bar{q}}_t (\mathbb{R}) L^{\bar{r}}_x(\mathbb{R}^N)} = \left(\int_0^{+\infty} \left(\int_{\mathbb{R}^N}|u|^{\bar{r}} dx\right)^{\frac{\bar{q}}{\bar{r}}} dt\right)^{\frac{1}{\bar{q}}} \leq C. \end{align*}