On the Cauchy Problem Associated to a Nonequilibrium Bose-Einstein Condensate of Exciton Polaritons

Abstract

We study a nonequilibrium Gross--Pitaevskii type system recently proposed to model exciton-polariton condensates. The coupled dispersive-dissipative equations present numerous mathematical challenges, and the known previous methods do not seem to apply in a standard way to study the global dynamics and singularity formation. We consider initial data in Sobolev spaces defined on euclidean (Rd) and periodic (T^d) domains and we prove global in time existence results for small data (in all dimensions) with regularity above the algebra structure under some extra hypotheses. By using Strichartz estimates, we obtain global well-posedness in L^2(R) x L^2(R) (with exponential decay in some physical cases), what cannot be applied to higher dimensions. Furthermore, under some physical assumptions, we show the existence of initial data, in both cases R^d or T^d, such that the corresponding solutions blow up in finite or infinite time, with exponential rate. Also, using a self-contained approach we also present an interesting result about the existence of initial data with higher regularity, in periodic domains, such that the corresponding solutions either blow up in finite time or have unbounded Sobolev norms with a vanishing dissipation parameter. Finally, we present some numerical simulations in the cases R^d, d = 1, 2, illustrating our results and motivating new research for the system.