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Register a new userOn uniqueness of multi-bubble blow-up solutions and multi-solitons to $L^2$-critical nonlinear Schrodinger equations
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We are concerned with the focusing $L^2$-critical nonlinear Schrodinger equations in $mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of $K$ pseudo-conformal blow-up solutions particularly with low rate $(T-t)^{0+}$, as $tto T$, $1leq K<infty$. Moreover, we also prove the uniqueness in the energy class of multi-solitons which converge to a sum of $K$ solitary waves with convergence rate $(1/t)^{2+}$, as $tto infty$. The uniqueness class is further enlarged to contain the multi-solitons with even lower convergence rate $(1/t)^{frac 12+}$ in the pseudo-conformal space. The proof is mainly based on the pseudo-conformal invariance and the monotonicity properties of several functionals adapted to the multi-bubble case, the latter is crucial towards the upgradation of the convergence to the fast exponential decay rate.
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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.
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.
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}.
In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schrodinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.
We are concerned with the following nonlinear Schrodinger equation $$-varepsilon^2Delta u+ V(x)u=|u|^{p-2}u,~uin H^1(R^N),$$ where $Ngeq 3$, $2<p<frac{2N}{N-2}$. For $varepsilon$ small enough and a class of $V(x)$, we show the uniqueness of positive multi-bump solutions concentrating at $k$ different critical points of $V(x)$ under certain assumptions on asymptotic behavior of $V(x)$ and its first derivatives near those points. The degeneracy of critical points is allowed in this paper.
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