We study stable blow-up dynamics in the generalized Hartree equation with radial symmetry, a Schrödinger-type equation with a nonlocal, convolution-type nonlinearity: $iu_t+Δu +\left(|x|^{-(d-2)} \ast |u|^{p} \right) |u|^{p-2}u = 0, x \in \mathbb{R}^d$. First, we consider the $L^2$-critical case in dimensions d=3, 4, 5, 6, 7 and obtain that a generic blow-up has a self-similar structure and exhibits not only the square root blowup rate but also the log-log correction (via asymptotic analysis and functional fitting). In this setting we also study blow-up profiles and show that generic blow-up solutions converge to the rescaled $Q$, a ground state solution of the elliptic equation $-ΔQ+Q- \left(|x|^{-(d-2)} \ast |Q|^p \right) |Q|^{p-2} Q =0$.
We also consider the $L^2$-supercritical case in dimensions d=3,4. We derive the profile equation for the self-similar blow-up and establish the existence and local uniqueness of its solutions. As in the NLS $L^2$-supercritical regime, the profile equation exhibits branches of non-oscillating, polynomially decaying (multi-bump) solutions. A numerical scheme of putting constraints into solving the corresponding ODE is applied during the process of finding the multi-bump solutions. Direct numerical simulation of solutions to the generalized Hartree equation by the dynamic rescaling method indicates that the $Q_{1,0}$ is the profile for the stable blow-up. In this supercritical case, we obtain the blow-up rate without any correction. This blow-up happens at the focusing level $10^{-5}$, and thus, numerically observable (unlike the $L^2$-critical case).
In summary, we find that the results are similar to the behavior of stable blowup dynamics in the corresponding NLS settings. Consequently, one may expect that the form of the nonlinearity in the Schrödinger-type equations is not essential in stable blow-up.
In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . In the previous work of the author we know that there exists an stable self-similar blow-up dynamics for slightly $L^2$-supercritical gKdV equations. Such solution can be viewed as solutions with single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time.
In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . We will prove the existence and stability of a blow-up dynamic with self-similar blow-up rate in the energy space $H^1$ and give a specific description of the formation of the singularity near the blow-up time.
We study the blowup behavior for the focusing energy-supercritical semilinear wave equation in 3 space dimensions without symmetry assumptions on the data. We prove the stability of the ODE blowup profile.
In this paper, we are concerned with solutions to the Cauchy problem for Chern-Simons-Schrödinger equations in the mass supercritical case. First we establish the local well-posedness of solutions in the radial space. Then we consider scattering versus blow-up dichotomy for radial data below the ground state threshold.
We consider the focusing cubic wave equation in the energy supercritical case, i.e., in dimensions $d \geq 5$. For this model an explicit nontrivial self-similar blowup solution was recently found by the first and third author in \cite{GlogicSchoerkhuber}. Furthermore, the solution is proven to be co-dimension one stable in $d=7$. In this paper, we study the equation from a numerical point of view. For $d=5$ and $d=7$ in the radial case, we provide evidence that this solution is at the threshold between generic ODE blowup and dispersion. In addition, we investigate the spectral problem that un…
We consider spherically symmetric supercritical focusing wave equations outside a ball. Using mixed analytical and numerical methods, we show that the threshold for blowup is given by a codimension-one stable manifold of the unique static solution with exactly one unstable direction. We analyze in detail the convergence to this critical solution for initial data fine-tuned to the threshold.
We consider the focusing nonlinear Schrödinger equations $i\partial_t u+Δu +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities $p_c<p<(1+\e)p_c$ with $p_c=1+\frac{4}{N}$ and $0<\e\ll 1$. We prove the existence and stability in the energy space $H^1$ of a self similar finite time blow up dynamics and provide a qualitative description of the singularity formation near the blow up time
We study a class of Hartree type equations and prove a quantitative blow up rate for their blow up solutions. This is an analogue of the result by Merle and Raphaël on 3d NLS.
We consider the fractional Schrödinger equations with focusing Hartree type nonlinearity. When the energy is negative, we use a Glassey's virial type argument to show the finite time blow-up of solutions.
A class of equations with exponential nonlinearities on a compact Riemannian surface is considered. More precisely, we study an asymmetric sinh-Gordon problem arising as a mean field equation of the equilibrium turbulence of vortices with variable intensities. We start by performing a blow-up analysis in order to derive some information on the local blow-up masses. As a consequence we get a compactness property in a supercritical range. We next introduce a variational argument based on improved Moser-Trudinger inequalities which yields existence of solutions for any choice of the underlying su…
We study the focusing stochastic nonlinear Schrödinger equation in 1D in the $L^2$-critical and supercritical cases with an additive or multiplicative perturbation driven by space-time white noise. Unlike the deterministic case, the Hamiltonian (or energy) is not conserved in the stochastic setting, nor is the mass (or the $L^2$-norm) conserved in the additive case. Therefore, we investigate the time evolution of these quantities. After that we study the influence of noise on the global behavior of solutions. In particular, we show that the noise may induce blow-up, thus, ceasing the global ex…
We consider the 1D nonlinear Schrödinger equation (NLS) with focusing \emph{point nonlinearity}, $$i\partial_tψ+ \partial_x^2ψ+ δ|ψ|^{p-1}ψ= 0$$ where $δ=δ(x)$ is the delta function supported at the origin. In the $L^2$ supercritical setting $p>3$, we construct self-similar blow-up solutions belonging to the energy space $L_x^\infty \cap \dot H_x^1$. This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic Weber functions and solving the jump condition at $x=0$ imposed by the $δ$ term.…
We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrödinger equation $i\partial_t u + Δu + |u|^{p-1}u=0$ on $\mathbf{R}^d$, close to the mass critical case and for any space dimension $d\ge 1$. These profiles bifurcate from the ground state solitary wave. The argument relies on the classical matched asymptotics method suggested in [Sulem, C.; Sulem, P.-L., The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999] which needs to be applied in a degenerate case due t…
We consider the semiclassical limit of the Hartree equation with a data causing a focusing at a point. We study the asymptotic behavior of phase function associated with the WKB approximation near the caustic when a nonlinearity is supercritical. In this case, it is known that a phase shift occurs in a neighborhood of focusing time in the case of focusing cubic nonlinear Schrödinger equation. Thanks to the smoothness of the nonlocal nonlinearities, we justify the WKB-type approximation of the solution for a data which is larger than in the previous results and is not necessarily well-prepared.…
We consider the Yang-Mills equations in $(1+d)$-dimensional Minkowski spacetime. It is known that in the supercritical case, i.e., for $d \geq 5$, these equations admit closed form equivariant self-similar blowup solutions \cite{BieBiz15}. These solutions are furthermore conjectured to be the universal attractors for generic large equivariant data evolutions. In this paper we partially prove this conjecture. Namely, we show that for all odd $d \geq 5$ the blowup mechanism exhibited by these solutions is stable.
In this paper we prove finite-time blowup of radially symmetric solutions to the quasilinear parabolic-parabolic two-dimensional Keller-Segel system for any positive mass. This is done in case of nonlinear diffusion and also in the case of nonlinear cross-diffusion provided the nonlinear chemosensitivity term is assumed not to decay. Moreover, it is shown that the above-mentioned lack of non-decay assumption is essential with respect to keeping the dichotomy finite-time blowup against boundedness of solutions. Namely, we prove that without the non-decay assumption possible asymptotic behaviour…
We consider the non linear focusing wave equation $\partial_{tt}u-Δu-u|u|^{p-1}=0$ in large dimensions and for radially symmetric data, in the energy supercritical zone for p large enough. We construct finite time blow up solutions that concentrate the soliton profile $u \sim \frac{1}{λ^{\frac{2}{p-1}}}Q (\frac{r} λ)$, $Q$ being a special stationary state of the equation. The blow up is of type II i.e all norms below scaling remain bounded. The scale shrinks at quantized blow up rates: we exhibit a countable family of asymptotic behaviors near blow up time for the scale lambda. For each rate w…
We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation \begin{equation*} iu_{t}=-\frac{1}{2}Δu+\frac{1}{2}V(x)u-|u|^{p-1}u+L_Ωu,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}, \end{equation*} where $N=2$ or $3$ and $V(x)$ is an anisotropic harmonic potential. Here $L_Ω$ is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stabl…
For the focusing cubic wave equation, we find an explicit, non-trivial self-similar blowup solution $u^*_T$, which is defined on the whole space and exists in all supercritical dimensions $d \geq 5$. For $d=7$, we analyze its stability properties without any symmetry assumptions and prove the existence of a set of perturbations which lead to blowup via $u^*_T$ in a backward light cone. Moreover, this set corresponds to a co-dimension one Lipschitz manifold modulo translation symmetries in similarity coordinates.
We consider the energy-supercritical harmonic map heat flow from $\mathbb{R}^d$ into $\mathbb{S}^d$, under an additional assumption of 1-corotational symmetry. We are interested by the 7 dimensional case which is the borderline between the Type I blowup regime. We construct for this problem a stable finite time blowup solution under the condition of corotational symmetry that blows up via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{λ(t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the rate $$λ(t) \sim \frac{\sqrt{(T-t)}}{|\l…
The harmonic map heat flow is a geometric flow well known to produce solutions whose gradient blows up in finite time. A popular model for investigating the blow-up is the heat flow for maps $\mathbb R^{d}\to S^{d}$, restricted to equivariant maps. This model displays a variety of possible blow-up mechanisms, examples include self-similar solutions for $3\le d\le 6$ and a so-called Type II blow-up in the critical dimension $d=2$. Here we present the first constructive example of Type II blow-up in higher dimensions: for each $d\ge7$ we construct a countable family of Type II solutions, each ch…
A Hybrid Physics-Informed Neural Network Framework for Subcritical and Supercritical Dynamics in Multi-Species Chemotaxis
African Institute for Mathematical Sciences, Research and Innovation Centre（非洲数学科学研究所研究与创新中心）；Technical University Berlin（柏林工业大学）；German Aerospace Center (DLR)（德国航空航天中心）
We consider co-rotational wave maps from (1+3)-dimensional Minkowski space into the three-sphere. This model exhibits an explicit blowup solution and we prove the asymptotic nonlinear stability of this solution in the whole space under small perturbations of the initial data. The key ingredient is the introduction of a novel coordinate system that allows one to track the evolution past the blowup time and almost up to the Cauchy horizon of the singularity. As a consequence, we also obtain a result on continuation beyond blowup.
We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree-Fock and Hartree-Fock-Bogoliubov type equations, which describe the evolution of attractive fermionic systems (e. g. white dwarfs). Our main results are twofold: First, we extend the recent blowup result of [Hainzl and Schlein, Comm. Math. Phys. \textbf{287} (2009), 705--714] to Hartree-Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree-Fock-Bogoliubov mode…
We study dynamical properties of blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation \[ i\partial_t u -(-Δ)^s u = -|u|^αu, \quad u(0) = u_0, \quad \text{on } [0,\infty) \times \mathbb{R}^d, \] where $d \geq 2, \frac{d}{2d-1} \leq s <1$, $\frac{4s}{d}<α<\frac{4s}{d-2s}$ and $u_0 \in \dot{H}^{s_{\text{c}}} \cap \dot{H}^s$ is radial with the critical Sobolev exponent $s_{\text{c}}$. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in $\dot{H}^{s_{\text{c}}} \cap \dot{H}^s$. …
We consider the wave equation in space dimension $3$, with an energy-supercritical nonlinearity which can be either focusing or defocusing. For any radial solution of the equation, with positive maximal time of existence $T$, we prove that one of the following holds: (i) the norm of the solution in the critical Sobolev space goes to infinity as $t$ goes to $T$, or (ii) $T$ is infinite and the solution scatters to a linear solution forward in time. We use a variant of the channel of energy method, relying on a generalized $L^p$-energy which is almost conserved by the flow of the radial linear w…
We consider the semilinear heat equation in large dimension $d\geq 11$ $$ \partial_t u =Δu+|u| ^{p-1}u, \ \ p=2q+1, \ \ q\in \mathbb N $$ on a smooth bounded domain $Ω\subset \mathbb R^d$ with Dirichlet boundary condition. In the supercritical range $p\geq p(d)>1+\frac{4}{d-2}$ we prove the existence of a countable family $(u_\ell)_{\ell \in \mathbb N}$ of solutions blowing-up at time $T>0$ with type II blow up: $$ \parallel u_{\ell}(t) \parallel_{L^{\infty}} \sim C (T-t)^{-c_\ell} $$ with blow-up speed $c_\ell>\frac{1}{p-1}$. They concentrate the ground state $Q$ being the only radially and d…
We present a detailed numerical study of solutions to the (generalized) Zakharov-Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the $L^{2}$-subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the $L^2$-critical and supercritical cases, solitons appear to be unstable against both dispersion and blow-up. It is conjectured that blow-up happens in finite time and that blow-up solutions have some resemblance of being self-similar, i.e., the blow-up core forms a rightward moving se…
In this paper, we consider the heat flow for Yang-Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)-$equivariant setting, the Yang-Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove \cite{Wei04}. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar…
We study the semilinear wave equation \[ \partial_t^2 ψ-Δψ=|ψ|^{p-1}ψ\] for $p > 3$ with radial data in three spatial dimensions. There exists an explicit solution which blows up at $t=T>0$ given by \[ ψ^T(t,x)=c_p (T-t)^{-\frac{2}{p-1}} \] where $c_p$ is a suitable constant. We prove that the blow up described by $ψ^T$ is stable in the sense that there exists an open set (in a topology strictly stronger than the energy) of radial initial data that lead to a solution which converges to $ψ^T$ as $t\to T-$ in the backward lightcone of the blow up point $(t,r)=(T,0)$.
We consider the following nonlinear Schrödinger equation with a Hartree nonlinearity: \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u\pm\left(\frac{1}{|x|^{2σ}}\star|u|^2\right)u=0 \] in $\mathbb{R}^N$. We are interested in the existence and behaviour of minimal mass blow-up solutions. Previous studies have shown the existence of minimal mass blow-up solutions with an inverse power potential and investigated the behaviour of the solution. In this paper, we investigate Hartree nonlinearity, which is a nonlinear term similar to the inverse power-type potential in terms of scaling.
We characterize the dynamics of the finite time blow up solutions with minimal mass for the focusing mass critical Hartree equation with $H^1(\mathbb{R}^4)$ data and $L^2(\mathbb{R}^4)$ data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we also analyze the mass concentration phenomenon of such blow up solutions.
This paper studies the non-negative solutions of the Keller-Segel model with a nonlocal nonlinear source in a bounded domain. The competition between the aggregation and the nonlocal reaction term is highlighted: when the growth factor is stronger than the dampening effect, with the help of the nonlocal term, the model admits a classical solution which is uniformly bounded. Moreover, when the growth factor is of the same order compared to the dampening effect, the nonlocal nonlinear exponents can prevent the chemotactic collapse. Global existence of classical solutions is shown for an appropri…
In this paper we prove that the solutions of the isotropic, spatially homogeneous Nordheim equation for bosons, with bounded initial, data blow up in finite time in the $L^\infty$ norm if the values of the energy and particle density are in the range of values where the corresponding equilibria contains a Dirac mass. We also prove that, in the weak solutions, whose initial data are measures with values of particle and energy densities satisfying the previous condition, a Dirac measure at the origin forms in finite time.
We consider the energy super critical nonlinear Schrödinger equation $$i\pa_tu+Δu+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical regime, we construct a family of smooth finite time blow up solutions which become singular via concentration of a universal profile with the so called type II quantized blow up rates. The essential feature of these solutions is that all norms below scaling remain bounded. Our analysis fully revisits the construction of type II blow up solutions for the corresponding he…
We prove finite time supercritical blowup in a parity-time-symmetric system of the two coupled nonlinear Schrödinger (NLS) equations. One of the equations contains gain and the other one contains dissipation such that strengths of the gain and dissipation are equal. We address two cases: in the first model all nonlinear coefficients (i.e. the ones describing self-action and non-linear coupling) correspond to attractive (focusing) nonlinearities, and in the second case the NLS equation with gain has attractive nonlinearity while the NLS equation with dissipation has repulsive (defocusing) nonli…
We study stable blow-up dynamics in the $L^2$-supercritical nonlinear Schrödinger equation in various dimensions. We first investigate the profile equation and extend the result of X.-P. Wang [38] and Budd et al. [4] on the existence and local uniqueness of solutions of the cubic profile equation to other $L^2$-supercritical nonlinearities and dimensions $d \geq 2$. We then numerically observe the multi-bump structure of such solutions, and in particular, exhibit the $Q_{1,0}$ solution, a candidate for the stable blow-up profile. Next, using the dynamic rescaling method, we investigate stable …
This work concerns the semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic (i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any non-scattering solution goes to infinity at the maximal time of existence. This gives a refinement on known results on energy-subcritical and energy-supercritical wave equation, with a unified proof. The proof relies on the channel of energy method, as in arXiv:1204.0031, in weighted scale-invariant Sobolev spaces which were introduced in arXiv:1506.00788. These space…
We establish global existence, scattering for radial solutions to the energy-critical focusing Hartree equation with energy and $\dot{H}^1$ norm less than those of the ground state in $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.
We consider the gravitational collapse for neutron stars in the Hartree-Fock-Bogoliubov theory. We prove that when the number particle becomes large and the gravitational constant is small such that the attractive interaction strength approaches the Chandrasekhar limit mass slowly, the minimizers develop a universal blow-up profile. It is given by the Lane-Emden solution.
In this paper we are interested in the following critical Hartree equation \begin{equation*} \begin{cases} -Δu =\displaystyle{\Big(\int_Ω\frac{u^{2_μ^\ast} (ξ)}{|x-ξ|^μ}dξ\Big)u^{2_μ^\ast-1}}+\varepsilon u ,~~~\text{in}~Ω,\\ u=0,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{on}~\partialΩ, \end{cases} \end{equation*} where $N\geq4$, $0<μ\leq4$, $\varepsilon>0$ is a small parameter, $Ω$ is a bounded domain in $\mathbb{R}^N$, and $2_μ^\ast=\frac{2N-μ}{N-2}$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. By establishing various versions of local Pohozaev ident…
We consider the Cauchy problem for an energy supercritical nonlinear wave equation that arises in $(1+5)$--dimensional Yang--Mills theory. A certain self--similar solution $W_0$ of this model is conjectured to act as an attractor for generic large data evolutions. Assuming mode stability of $W_0$, we prove a weak version of this conjecture, namely that the self--similar solution $W_0$ is (nonlinearly) stable. Phrased differently, we prove that mode stability of $W_0$ implies its nonlinear stability. The fact that this statement is not vacuous follows from careful numerical work by Bizoń and Ch…
We consider the heat flow of corotational harmonic maps from $\mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In particular, we completely avoid using delicate Lyapunov functionals, monotonicity formulas, indirect arguments, or fragile parabolic structure like the maximum principle. As a matter of fact, our approach reduces the nonlinear …
We consider $L^2$-critical focusing nonlinear Schroedinger equations with Hartree type nonlinearity $$i \pr_t u = -\DD u - \big (Φ\ast |u|^2 \big) u \quad {in $\RR^4$},$$ where $Φ(x)$ is a perturbation of the convolution kernel $|x|^{-2}$. Despite the lack of pseudo conformal invariance for this equation, we prove the existence of critical mass finite-time blowup solutions $u(t,x)$ that exhibit the pseudo-conformal blowup rate $$ \| \nabla u(t) \|_{L^2_x} \sim \frac{1}{|t|} \quad {as} \quad t \nearrow 0 . $$ Furthermore, we prove the finite-codimensional stability of this conformal blow up, by…
