A determination of the blowup solutions to the focusing, quintic NLS with mass equal to the mass of the soliton (Q6589458)

The author considers the 1D, focusing, and mass-critical nonlinear Schrödinger equation: \(iu_{t}+u_{xx}+\left\vert u\right\vert ^{4}u=0\), with the initial condition: \(u(0,x)=u_{0}(x)\in L^{2}(\mathbb{R})\). He observes that this is a special case of the Hamiltonian problem: \(iu_{t}+u_{xx}+\left\vert u\right\vert ^{p-1}u=0\), \(u(0,x)=u_{0}(x)\), \(p>1\), and if \(u(t,x)\) is a solution to this last equation, then \(v(t,x)=\lambda ^{2/(p-1)}u(\lambda ^{2}t,\lambda x)\) is also a solution to this last equation. This last expression is called a scaling symmetry. He recalls that for any \(u_{0}\in L^{2}\), there exists \(T(u_{0})>0\) such that the initial value problem is locally well-posed on the interval .\((-T,T)\), and if \(\left\Vert u_{0}\right\Vert _{L^{2}}\) is small, the initial value problem is globally well-posed, and the solution scatters both forward and backward in time. He also recalls the possible blow up property of this solution. \N\NThe first main result of the paper proves that the only symmetric solutions to the above initial value problem with mass \(\left\Vert u_{0}\right\Vert _{L^{2}}=\left\Vert Q\right\Vert _{L^{2}}\) that blow up forward in time are the soliton solutions \(e^{-i\theta }e^{i\lambda ^{2}t}\lambda ^{1/2}Q(\lambda x)\), \(\lambda >0\), \(\theta \in \mathbb{R}\), and the pseudoconformal transformation of the soliton solution \(\frac{1}{(T-t)^{1/2}} e^{i\theta }\exp [\frac{ix^{2}}{4(t-T)}]\exp[\frac{i\lambda ^{2}}{t-T}]Q( \frac{\lambda x}{T-t})\), \(\lambda >0\), \(\theta ,T\in \mathbb{R}\), \(t<T\). Here \(Q(x)=(\frac{3}{\cosh(2x)^{2}})^{1/4}\). For the proof, the author mainly chooses appropriate functions \(\lambda (t)\) and \(\gamma (t)\) such that \( \left\Vert \lambda (t)^{-1/2}e^{-i\gamma (t)}u(t,\frac{x}{\lambda (t)} )-Q(x)\right\Vert _{L^{2}}\leq \eta ^{\ast }\ll 1\), for all \(t>0\), and \( (\epsilon ,Q_{x})=(i\epsilon ,Q_{x})=(\epsilon ,Q^{3})=(i\epsilon ,Q^{3})=0\) , where \(\epsilon (t,x)=\lambda (t)^{-1/2}e^{-i\gamma (t)}u(t,\frac{x}{ \lambda (t)})-Q(x)\). He proves that the infimum \(\inf_{\lambda >0,\gamma \in \mathbb{R}}\left\Vert u_{0}(x)-e^{i\gamma }\lambda ^{1/2}Q(\lambda x)\right\Vert _{L^{2}}\) is achieved. The main step of the proof consists to verify that for some small, fixed constant \(\eta ^{\ast }\ll 1\), if \(u\) is a symmetric solution to the initial value problem on the maximal interval of existence \(I\subset \mathbb{R}\), such that \(u\) blows up forward in time, and \(\sup_{t\in \lbrack 0,\sup (I))}\inf_{\lambda >0,\gamma \in \mathbb{R} }\left\Vert e^{i\gamma }\lambda ^{1/2}u(t,\lambda x)-Q(x)\right\Vert _{L^{2}} \), then \(u\) is a soliton solution or a pseudoconformal transformation of a soliton. He uses the weak sequential convergence result of \textit{C. Fan} in [Int. Math. Res. Not. 2021, No. 7, 4864--4906 (2021;Zbl 1476.35236)], that he extends and an upper semicontinuity property of the quantity \(\inf_{\lambda >0,\gamma \in \mathbb{R}}\left\Vert e^{i\gamma }\lambda ^{1/2}u(t,\lambda x)-Q(x)\right\Vert _{L^{2}}\) as a function of time for any \(t\in I\). This quantity is also continuous in time if it is small. He then introduces a decomposition of a symmetric solution close to \(Q\), up to rescaling and multiplication by a modulus one constant, using a result by \textit{Y. Martel} and \textit{F. Merle} in [Ann. Math. (2) 155, No. 1, 235--280 (2002; Zbl 1005.35081)], that he extends. He proves a long-time Strichartz estimate for the solution \(u\) and an almost conservation of the energy, from which he derives a frequency-localized Morawetz estimate. He proves monotonicity properties of the function \(\lambda (t)\). \N\NThe second main result of the paper gives the expression of the only solutions to the initial value problem with mass \(\left\Vert u_{0}\right\Vert _{L^{2}}=\left\Vert Q\right\Vert _{L^{2}}\) that blow up forward in time.