$L^\infty$-regularity thresholds for a generalized {G}ross--{P}itaevskii equation: dispersive blow-up vs. nonlinear stability

We study a generalized Gross-Pitaevskii equation arising in the mean-field description of exciton-polariton Bose-Einstein condensates. The dynamics close to macroscopic wave functions $r_0 e^{iω_0t+ik_0x}$ is governed by a stabilizing relaxation mechanism without suppressing dispersive blow-up. We show that this competition yields $L^\infty$-regularity thresholds. More precisely, we construct solutions for which arbitrarily small perturbations from $C^4_b$ can develop finite time blow-up in $C^3_b$ whereas perturbations from $C^{4+\eta}_b$ for every $\eta>0$, satisfy a global-in-time $C^3_b$-stability estimate. The stability analysis is carried out in modulation spaces $M^k_{\infty,1}$.