Cross-Constrained Variational Problem for a Davey-Stewartson System∗

This paper concerns the sharp threshold of blowup and global existence of the solution as well as the strong instability of standing wave eu(x) to the system iφt +Δφ+ a|φ|p−1φ+ bE1(|φ|)φ = 0 (DS) in R , where a > 0, b > 0, 1 ≤ p < N+2 (N−2)+ , N ∈ {2, 3} and u is a ground state. First, by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp threshold for global existence and blowup of the solution to the Cauchy problem for (DS) provided 1+ 4 N ≤ p < N+2 (N−2)+ . Secondly, by using the scaling argument, we show that how small the initial data are for the global solutions to exist. Finally, we prove the strong instability of the standing waves with finite time blow up by combining the former results, which partially answer the open problem proposed in [16,Remark 8].