A Liouville-type theorem for the $3$-dimensional parabolic Gross-Pitaevskii and related systems

We prove a Liouville-type theorem for semilinear parabolic systems of the form $${\partial_t u_i}-\Delta u_i =\sum_{j=1}^{m}\beta_{ij} u_i^ru_j^{r+1}, \quad i=1,2,...,m$$ in the whole space ${\mathbb R}^N\times {\mathbb R}$. Very recently, Quittner [{\em Math. Ann.}, DOI 10.1007/s00208-015-1219-7 (2015)] has established an optimal result for $m=2$ in dimension $N\leq 2$, and partial results in higher dimensions in the range $p< N/(N-2)$. By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-V\'eron, we partially improve the results of Quittner in dimensions $N\geq 3$. In particular, our results solve the important case of the parabolic Gross-Pitaevskii system -- i.e. the cubic case $r=1$ -- in space dimension $N=3$, for any symmetric $(m,m)$-matrix $(\beta_{ij})$ with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouville-type theorem in the half-space ${\mathbb R}^N_+\times {\mathbb R}$. As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.