Condensation and Non-condensation in the Quasilinear 4-wave Kinetic Equation

We study finite-time condensation at zero frequency for the isotropic quasilinear 4-wave kinetic equation in three dimensions. With Schrodinger dispersion and a product interaction coefficient with exponent $\beta \in (0,3\mathrm{/}4]$, we construct global measure solutions $(g_t{)}_{t\ge 0}$ for every nonnegative initial measure $g_0$ with finite wave action and energy. We then establish the sharp threshold $\beta =1\mathrm{/}4$ for finite-time condensation of the constructed solutions. For $\beta \in [1\mathrm{/}4,3\mathrm{/}4]$, the constructed solutions satisfy $g_t(\{0\})=g_0(\{0\})$ for all $t\ge 0$. For $\beta \in (0,1\mathrm{/}4)$, every constructed solution with $(g_0{)}_{\mathrm{a}\mathrm{c}}\mathrm{\ne }0$ satisfies $\mathrm{\parallel }{g}_t-{\mathcal{N}}_0{\delta }_0{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}\le C{\mathcal{N}}_0(1+{\mathcal{N}}_0^{2}t{)}^{-(1-4\beta )\mathrm{/}40}$ for all $t\ge 0$, where ${\mathcal{N}}_0=g_0([0,\mathrm{\infty }))$ and $C=C(\beta ,g_0\mathrm{/}{\mathcal{N}}_0)>0$. In particular, for every $\zeta \in (0,1)$, there exists $T_{\zeta }<\mathrm{\infty }$ such that $g_t(\{0\})\ge (1-\zeta ){\mathcal{N}}_0$ for all $t\ge T_{\zeta }$.