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

Abstract

Abstract. We prove a Liouville-type theorem for semilinear parabolic systems of the formin the whole space R N × R. Very recently, Quittner [Math. Ann., DOI 10.1007/s00208-015-1219-7 (2015)] has established an optimal result for m = 2 in dimension N ≤ 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-Véron, we partially improve the results of Quittner in dimensions N ≥ 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 (β 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 Liouvilletype theorem in the half-space R N + × 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.

A Liouville-type theorem for cooperative parabolic systems

“…The aim of this paper is to prove the nonexistence of nontrivial solution of problem (1) in the entire space, such a result is called the Liouville-type theorem. This is a continuation of [16] where the author prove the Liouville-type theorem in the special case s = r for the system of m components…”

“…We stress that the proof of Theorem 1.1 is not straightforward in comparison with that in [16]. The only main difficulty arising in this paper is the presence of different exponents r and s. This makes the Gidas-Spruck and Bidaut-Véron techniques become more delicate.…”

“…[7,10,9]. Recently, the singularity estimates and the Liouville-type theorems have been studied in [15,18,16].…”

“…We note that, by a simple scaling, one can reduce the system (1) to a parabolic system with gradient structure as in Theorem A. More recently, Phan and Souplet [16] have used a different approach to establish a Liouville-type theorem for problem (1) in a larger range of p in dimension N ≥ 3, under the restrictions r = s and a 12 = a 21 .…”

“…Due to the fact that N N −2 < p B when N ≥ 3, our result is a partial improvement of that in Quittner [18] in high dimensions (as seen for system of two components). Our tool for the proof of Theorem 1.1 is a refinement of [16], which is essentially based on Gidas-Spruck technique [11] developed by Bidaut-Véron [4] (see also [5] for elliptic system). This technique consists of nonlinear integral estimates and Bochner formula.…”