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

Abstract

We prove a Liouville-type theorem for semilinear parabolic systems of the form $$\begin{aligned} {\partial _t u_i}-\Delta u_i =\sum _{j=1}^{m}\beta _{ij} u_i^ru_j^{r+1}, \quad i=1,2,\ldots ,m \end{aligned}$$ in the whole space \({\mathbb R}^N\times {\mathbb R}\). Very recently, Quittner (Math Ann. 364, 269–292, 2016) has established an optimal result for \(m=2\) in dimension \(N\le 2\), 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-Veron, we partially improve the results of Quittner in dimensions \(N\ge 3\). 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.