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Register a new userA note on log-log blow up solutions for stochastic nonlinear Schrodinger equations
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In this short note, we present a construction for the log-log blow up solutions to focusing mass-critical stochastic nonlinear Schroidnger equations with multiplicative noises. The solution is understood in the sense of controlled rough path as in cite{SZ20}.
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We are concerned with the multi-bubble blow-up solutions to rough nonlinear Schrodinger equations in the focusing mass-critical case. In both dimensions one and two, we construct the finite time multi-bubble solutions, which concentrate at $K$ distinct points, $1leq K<infty$, and behave asymptotically like a sum of pseudo-conformal blow-up solutions in the pseudo-conformal space $Sigma$ near the blow-up time. The upper bound of the asymptotic behavior is closely related to the flatness of noise at blow-up points. Moreover, we prove the conditional uniqueness of multi-bubble solutions in the case where the asymptotic behavior in the energy space $H^1$ is of the order $(T-t)^{3+zeta}$, $zeta>0$. These results are also obtained for nonlinear Schrodinger equations with lower order perturbations, particularly, in the absence of the classical pseudo-conformal symmetry and the conversation law of energy. The existence results are applicable to the canonical deterministic nonlinear Schrodinger equation and complement the previous work [43]. The conditional uniqueness results are new in both the stochastic and deterministic case.
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We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.
We consider a dispersive equation of Schr{o}dinger type with a non-linearity slightly larger than cubic by a logarithmic factor. This equation is supposed to be an effective model for stable two dimensional quantum droplets with LHY correction. Mathematically, it is seen to be mass supercritical and energy subcritical with a sign-indefinite nonlinearity. For the corresponding initial value problem, we prove global in-time existence of strong solutions in the energy space. Furthermore, we prove the existence and uniqueness (up to symmetries) of nonlinear ground states and the orbital stability of the set of energy minimizers. We also show that for the corresponding model in 1D a stronger stability result is available.
In this note, we give an alternative proof of the theorem on soliton selection for small energy solutions of nonlinear Schrodinger equations (NLS) which we studied in Anal. PDE 8 (2015), 1289-1349 and more recently in Annals of PDE (2021) 7:16. As in in the latter paper we use the notion of Refined Profile, with the difference that here we do not modify the modulation coordinates and we do not search for Darboux coordinates. This shortens considerably the proof.
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