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

作者：Quoc Hung Phan; Souplet Philippe^*

来源：Mathematische Annalen, 2016, 366(3-4): 1561-1585.

DOI：10.1007/s00208-016-1368-3

We prove a Liouville-type theorem for semilinear parabolic systems of the form in the whole space . Very recently, Quittner (Math Ann. 364, 269-292, 2016) has established an optimal result for in dimension , and partial results in higher dimensions in the range . By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-V,ron, we partially improve the results of Quittner in dimensions . In particular, our results solve the important case of the parabolic Gross-Pitaevskii system-i.e. the cubic case -in space dimension , for any symmetric (m, m)-matrix 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 . 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.

• 出版日期2016-12