Special Session 131: Recent progress on singularities formations of some evolution partial differential equations

Blow-Up Dynamics for the L^2 critical case of the 2D Zakharov-Kuznetsov equation

Francisc Bozgan NYUAD United Arab Emirates Co-Author(s):    Tej-Eddine Ghoul, Nader Masmoudi

Abstract: We investigate the blow-up dynamics for the $L^2$ critical two-dimensional Zakharov-Kuznetsov equation with initial data $u_0$ slightly exceeding the mass of the soliton solution $Q$, which satisfies $-\Delta Q + Q - Q^3 = 0$. Employing methodologies analogous to those used in the study of the gKdV equation of Martel, Merle and Raphael, we categorize the behavior of the solution into three outcomes: asymptotic stability, finite-time blow-up, or divergence from the soliton`s vicinity. The universal blow-up behavior that we find is slightly different from the conjecture of Klein, Roudenko and Stoilov, by deriving a non-trivial, computationally determinable constant for the blow-up rate, dependent on the two-dimensional soliton`s behavior. The construction of blow-up solution involves the bubbling of the solitary wave which ensures that it is stable.

Singularity of the 2d Keller-Segel system formed by the collision of two collapsing solitons in interaction

Charles Collot CY Cergy Paris Universite France Co-Author(s):    Tej-Eddine Ghoul, Nader Masmoudi, Van Tien Nguyen

Abstract: The two-dimensional Keller-Segel system admits finite time blowup solutions, which is the case if the initial density has a total mass greater than 8pi and a finite second moment. Several constructive examples of such solutions have been obtained, where for all of them a perturbed stationary state undergoes scale instability and collapses at a point, resulting in a 8pi-mass concentration. It was conjectured that singular solutions concentrating simultaneously more than one solitons could exist. We construct rigorously such a new blowup mechanism, where two stationary states are simultaneously collapsing and colliding, resulting in a 16pi-mass concentration at a single blowup point, and with a new blowup rate which corresponds to the formal prediction by Seki, Sugiyama and Velazquez. We develop for the first time a robust framework to construct rigorously such blowup solutions involving simultaneously the non-radial collision and concentration of several solitons, which we expect to find applications to other evolution problems.

Nonlinear wave equations in Cosmology: Some results, but mostly open problems

Jean-Pierre Eckmann University of Geneva Switzerland Co-Author(s):    Farbod Hassani, Hatem Za`ag

Abstract: In certain cosmological models (effective field theories) one encounters a non-linear wave equation of the form $$ u_{tt}=\alpha u_{xx} + \beta (u_x)^2 $$ with $\alpha>0$ and $\beta>0$ in $\ge 1$ dimension. While cosmologists believed that solutions stay bounded for large enough $\alpha$, it has been known for some time that nontrivial initial conditions lead to divergence in finite time. After explaining some of these results, I will focus also on the scaling of the diverging solutions. (For the experts: In the cosmological context, one is not allowed to scale the initial condition, since it is given by background conditions.)

Stable self-similar blowup for the Keller-Segel model in three dimensions

Irfan Glogic Bielefeld University Germany Co-Author(s):    Birgit Schorkhuber

Abstract: We consider the three-dimensional parabolic-elliptic Keller-Segel model for bacterial chemotaxis. From the work of Brenner et al. in 1999, it is known that this model admits an explicit radial imploding self-similar solution. We prove the nonlinear radial asymptotic stability of this blowup profile. For this, we develop a stability analysis framework that applies to a large class of semilinear parabolic equations. In particular, we outline a robust technique to treat the underlying spectral problems. This is joint work with Birgit Schorkhuber (Innsbruck).

Blow-up Dynamics in Coupled Wave Systems with Tricomi Effects and Scale-Invariant Damping

Makram Hamouda Imam Abdlrahman Bin Faisal University Saudi Arabia Co-Author(s):    M.F. Ben Hassen, M. A Hamza,

Abstract: We present some results on the blow-up of systems of semilinear coupled waves with scale-invariant damping and time-derivative nonlinearities, examining various scenarios that include mass terms and time-dependent propagation speeds. A key novelty lies in a more refined characterization of the blow-up region, with a particular focus on the impact of the Tricomi term, which significantly alters the dynamics of the system. These findings relate to the well-known Glassey exponent. From a numerical perspective (Lattice Boltzmann methods and PINNs), we explore a tentative of blow-up time detection for some toy models. The determination of the threshold between blow-up and global existence regions is an interesting problem, but here we intend to provide some numerical insights for proving the conjectures on the critical nonlinearity exponent.

The blow-up rate for some nonlinear evolution equations in the log non-scaling invariance case

Mohamed Ali Hamza Imam Abdulrahman Bin Faisal University Saudi Arabia Co-Author(s):    Hatem Zaag

Abstract: In this talk, we discuss some evolution equations with logarithmic nonlinearity in the subconformal case. We show that the blow-up rate of any singular solution to the problem is like the ODE solution associated. In other terms, all blow-up solutions in the subconformal range are Type I solutions. This will constitute a good start in proving that the scale invariance property is not crucial in deriving the blow-up rate.

Blow-up phenomena in an integrable system with a singular integral and its application to traffic flow

Kohei Higashi Musashino University Japan Co-Author(s):    Kohei Higashi

Abstract: We investigate blow-up phenomena in an integrable system with a singular integral, which is described by the equation $u_t = -2A u_x T u_x - V u_x + D u_{xx}$. Here, $ T $ is a singular integral operator with a $\coth$-type kernel, which incorporates nonlocal effects. $ A $, $V $, and $ D $ are constants. By utilizing exact solutions, we examine four aspects: (1) determining the conditions under which blow-up occurs; (2) identifying the locations of blow-up points; (3) analyzing the form of the blow-up solution; and (4) exploring the system`s behavior after blow-up. Furthermore, we apply these findings to a traffic flow model, where blow-up corresponds to complete congestion in high-density regions.

Mathematical and Numerical Studies on Blow-up Rate of Solutions to Some Quasilinear Parabolic Equation

Tetsuya Ishiwata Shibaura Institute of Technology Japan Co-Author(s):    Koichi Anada, Takeo Ushijima

Abstract: In this talk, we discuss the blow-up rate of solutions to some quasilinear parabolic equations. In particular, we focus on type II blow-up solutions and show the mathematical results and the numerical observations. We also introduce our numerical estimation method for blow-up rate using the scale invariance.

Hyperbolic inequalities in an exterior domain: A general blow-up result for degenerate hyperbolic inequalities in an exterior domain

Mokhtar KIRANE Khalifa University United Arab Emirates Co-Author(s):    M. Jleli, M. Kirane, B. Samet

Abstract: We consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

Blow-up phenomena in one-dimensional derivative nonlinear wave equations

Jie Liu New York University Abu Dhabi United Arab Emirates Co-Author(s):    Tej-eddine Ghoul, Jie Liu, Nader Masmoudi

Abstract: In this talk, we construct explicit smooth blow-up solutions for the one-dimensional derivative nonlinear wave equation, for which no smooth self-similar solutions exist. Additionally, we establish the stability of these blow-up solutions.

Representation formulas for eigenvalues and eigenfunctions concerning a phase-field model

Tatsuki Mori Musashino University Japan Co-Author(s):    Tatsuki Mori, Yasuhito Miyamoto, Sohei Tasaki, Tohru Tsujikawa, Shoji Yotsutani

Abstract: We have been investigating the global bifurcation diagrams of stationary solutions for a phase field model proposed by Fix and Caginalp in a one-dimensional case. It has recently been shown that there exists a secondary bifurcation with a symmetry-breaking phenomenon from a branch consisting of symmetric solutions in the case where the total enthalpy equals zero. In this talk, we determine the stability/instability of all symmetric solutions and asymmetric solutions near the secondary bifurcation point. Moreover, we show representation formulas for all eigenvalues and eigenfunctions for the linearized eigenvalue problem around the symmetric solutions.

Blowup solutions to the complex Ginzburg-Landau equation

Van Tien Nguyen National Taiwan University Taiwan Co-Author(s):    Jiajie Chen, Thomas Y. Hou, Yixuan Wang

Abstract: We develop a so-called generalized dynamical rescaling method to study singularity formation in the complex Ginzburg-Landau equation (CGL). This innovative technique enables us to capture all relevant symmetries of the problem, allowing us to directly demonstrate a full stability of constructed blowup solutions. One of the advantages of our approach is its ability to circumvent spectral decomposition, which is often complex for problems involving non-self- adjoint operators. Additionally, the (CGL) system lacks a variational structure, making standard energy-type methods difficult to apply. By employing the amplitude-phase representation, we establish a robust analysis framework that enforces vanishing conditions through a carefully chosen normalization and utilizes weighted energy estimates.

Construction of type I-Log blowup for the Keller-Segel system in dimensions $3$ and $4$

Nejla Nouaili CEREMADE Univesite Paris Dauphine PSL France Co-Author(s):    Van Tien Nguyen and Hatem Zaag

Abstract: We construct finite time blowup solutions to the parabolic-elliptic Keller-Segel system \[\pa_t u = \Delta u - \nabla \cdot (u \nabla \mathcal{K}_u), \quad -\Delta \mathcal{K}_u = u \quad \textup{in}\;\; \mathbb{R}^d,\; d = 3,4,\] and derive the final blowup profile \[ u(r,T) \sim c_d \frac{|\log r|^\frac{d-2}{d}}{r^2} \quad \textup{as}\;\; r \to 0, \;\; c_d > 0.\] To our knowledge this provides a new blowup solution for the Keller-Segel system, rigorously answering a question by Brenner, Constantin, Kadanoff, Schenkel, and Venkataramani (Nonlinearity, 1999).

Orbital stability for the vortex pair of the Gross-Pitaevskii equation

Eliot Pacherie CNRS & Cergy University France Co-Author(s):

Abstract: The Gross-Pitaevskii equation admits small speed travelling wave solutions behaving like two well-separated vortices. We show that this solution is orbitally stable for the natural metric of the energy space. The proof is based on two main ingredients : a nonlinear coercivity result for small perturbations of the travelling waves, which generalizes a similar result for a single vortex, as well as a new formulation of the momentum using the coaera formula. This is a joint work with Philippe Gravejat and Frederic Valet

A priori estimates of solutions of local and nonlocal superlinear parabolic problems

Pavol Quittner Comenius University, Bratislava Slovak Rep Co-Author(s):

Abstract: We consider a priori estimates of possibly sign-changing solutions to superlinear parabolic problems and their applications (blow-up rates, continuity of the blow-up time, existence of nontrivial steady states etc.). Our estimates are based on energy, interpolation and bootstrap arguments. We first discuss known results on local problems and then provide new results for problems with nonlocal nonlinearities or nonlocal differential operators. In particular, we deal with nonlocal nonlinearities occuring in the Choquard equation or the Schr\odinger-Poisson-Slater problem, and we also consider problems involving the fractional Laplacian.

The lifespan of classical solutions of one dimensional wave equations with semilinear terms of the spatial derivative

Takiko Sasaki Musashino Unversity Japan Co-Author(s):    Shu Takamatsu, Hiroyuki Takamura

Abstract: This talk is devoted to the lifespan estimates of small classical solutions of the initial value problems for one dimensional wave equations with semilinear terms of the spatial derivative of the unknown function. It is natural that the result is same as the one for semilinear terms of the timederivative. But there are so many differences among their proofs. Moreover, it is meaningful to study this problem in the sense that it may help us to investigate its blow-up boundary in the near future.

General theory and its optimality for nonlinear wave equations in one apace dimension

Hiroyuki Takamura Tohoku University Japan Co-Author(s):    T.Sasaki, K.Morisawa, R.Kido, S.Takamatsu

Abstract: In this talk, the recent progress on nonlinear wave equations in one space dimension will be presented. More precisely, the so-called combined effect plays a key role in the analysis on model equations which improves the general theory for nonlinear wave equations, expected complete more than 30 years ago.

Life-span of solutions for some nonlinear parabolic problems

Slim Tayachi University of Tunis El Manar Tunisia Co-Author(s):

Abstract: In this talk, we present lower and upper bound estimates for the maximal existence time of solutions to the nonlinear heat equation, and solutions to a nonlinear parabolic system. We improve and extend some known results by considering a large class of initial data. Part of the results presented in this talk are from joint work with Fred B. Weissler.

A Blow-up theorem for discrete semilinear wave equation

Tetsuji Tokihiro Musashino University Japan Co-Author(s):    Kohei Higashi, Keisuke Matsuya, Takiko Sasaki, Ryosuke Tsubota

Abstract: In this talk, we evaluate lifespan of the solution of the equation obtained by discretising a semilinear wave equation with a power-type nonlinear term. The discrete equation was first proposed by Matsuya(2013) and has the following form. \begin{align*} u_n^{t+1}+u_n^{t-1}&=\dfrac{4 v_n^t}{2-\delta^2v_n^t|v_n^t|^{p-2}}\quad (n \in \mathbb{Z}^d,\, t \in \mathbb{Z}_{\ge 0})\ v_n^t&:=\frac{1}{2 d}\sum_{i=1}^d\left(u^t_{n+e_i}+u^t_{n-e_i}\right) \end{align*} In continuous limit, this discrete equation turns to the semiliniar wave equation: \begin{align*} u_{tt}=\Delta u +|u|^p \end{align*} This semiinear wave equation is known to explode if the exponent $p$ appearing in the nonlinear term is smaller than a certain value when the initial conditions are sufficiently small, and the discrete equation has been proved to have similar behaviour to the original wave equation. We show that the discrete equation also has similar lifesapn to that of the semilinear wave equation. \[5pt] %\vspace*{5pt} \textbf{Reference}: \ Keisuke Matsuya (2013), A blow-up theorem for a discrete semilinear wave equation, Journal of Difference Equations and Applications, 19:3, 457-465

Inhomogeneous exterior problems for the semilinear critical wave equationsCritical exponents for the quasilinear heat equations with combined nonlinearities

Berikbol T. Torebek Ghent University Belgium Co-Author(s):

Abstract: This work studies the global behavior of solutions to the quasilinear inhomogeneous parabolic equation with combined nonlinearities. Firstly, we focus on an interesting phenomenon of discontinuity of the Fujita-type critical exponents. In particular, we will fill the gap in the results of Jleli-Samet-Souplet (Proc Am Math Soc 148:2579-2593, 2020) for the critical case. We are also interested in the influence of the forcing term on the critical behavior of the considered problem, so we will define the second critical exponent in the sense of Lee-Ni, depending on the forcing term.

Blow-up of solutions of semilinear wave equations in Friedmann-Lemaitre-Robertson-Walker spacetime

Yuta Wakasugi Hiroshima University Japan Co-Author(s):    Kimitoshi Tsutaya

Abstract: Consider semilinear wave equations in the spatially flat Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes. We show some blow-up results obtained in recent years, including decelerating and accelerating expansion cases and power-type and derivative-type nonlinearities. Our approach is based on a generalized Kato`s lemma on ordinary differential inequalities and the test function method.

On optimal blowup stability for wave equations

David Wallauch-Hajdin EPFL Switzerland Co-Author(s):

Abstract: This talk reports on the recent development of a unified approach to derive Strichartz estimates for radial wave equations with self-similar potentials in similarity variables.To illustrate the usefulness of these estimates we will derive an optimal blowup stability result for a nonlinear wave equation.

Critical and subcritical blow-up for the nonlocal shadow limit of the Gierer-Meinhardt system

Hatem Zaag CNRS and Universite Sorbonne Paris Nord France Co-Author(s):

Abstract: The Gierer-Meinhardt system is a model for pattern formation based on Turing`s mechanism. Under some conditions, it reduces to a scalar heat equation, with a nonlinearity showing a pure power divided by some non-local term. Depending on parameters, that equation shows two different types of blow-up behavior: - in some subcritical range of parameters, the non-local term converges to a positive constant, leading to some blow-up behavior similar to the classical semilinear heat equation, with power nonlinearity ; - in the critical case, the non-local term converges to infinity, weakening the effect of the pure power nonlinearity. This leads to a new type of blow-up behavior, unknown in earlier literature. In this talk, we construct examples for the two types of behaviors, and give their blow-up profiles. Our method happens to be a non-trivial adaptation of the classical construction method for the semilinear heat equation.