I am currently investigating an equation that is : $$\partial ^2_t u = \Delta _{3/2} u,$$ where $u : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}$ and $\Delta _{3/2}$ stands for the $3/2$-laplacian i.e $\Delta _{3/2} u := \operatorname{div} (|\nabla u| ^{1/2} \nabla u)$.

I am looking for a Morawetz type inequality : in the setting of the linear wave equation, and if $u$ denotes a solution to this pde, it is known that : $$ \int _0^T \int_{\mathbb{R}^d} \frac{|\nabla \!\!\!\! / u|^2}{|x|}\, \mathrm{d}x \, \mathrm{d}t \leqslant C$$ which is the standard Morawetz inequality for the linear wave equation.

I tried to follows the method presented in Tao's book "Nonlinear Dispersive Equations; Local and Global analysis", page 161.

For the record it consists in finding the stress energy tensor $T^{ij}$ : $T^{00}$ must be the energy, and the other quantities satisfied : $$\partial _t T^{00} + \partial _ j T^{0j}=0,$$ and $$\partial T^{0j} + \partial _k T^{jk}=0.$$ Then the method consists in computing : $$\sum _j \int_{\mathbb{R}^d} \frac{x_j}{|x|} T^{0j} \, \mathrm{d}x ,$$ and hoping for good by parts integration, and finaly get a Morawetz type inequliaty.

In my setting, $T^{00}=\frac{(\partial _t u)^2}{2} + \frac{2}{5} |\nabla u|^{5/2}$ so that $T^{0j}=-\partial _t u |\nabla u|^{1/2} \partial _{x_j} u$ but we can not find the $T^{jk}$ corresponding and the the method of Tao doesn't give anything that I could use.

Is there a way to prove a Morawetz type inequality for this equation ?

Thank you.