Special Session 123: Nonlinear phenomena in elliptic and parabolic equations

Critical norm blow-up rates for the energy supercritical nonlinear heat equation

Tobias Barker University of Bath England Co-Author(s):    Hideyuki Miura and Jin Takahashi

Abstract: We study the behavior of the scaling critical Lebesgue norm for blow-up solutions to the nonlinear heat equation (the Fujita equation). For the energy supercritical nonlinearity, we give estimates of the blow-up rate for the critical norm.

LOW REGULARITY RESULTS FOR DEGENERATE POISSON PROBLEMS

Marta Calanchi Universit\`a degli Studi di Milano Italy Co-Author(s):    M. GROSSI

Abstract: We study the Poisson problem, \[ \begin{cases} -{\rm div}(d^\beta\nabla u)=f&{\rm in}\ \Omega\ u=0&{\rm on}\ \partial\Omega, \end{cases} \] where $\Omega\subset R^N$, $N\ge2$ is a smooth bounded domain, $f$ is a continuous function, $\beta< 1$, and $d(x)=dist(x,\partial\Omega )$. We describe the behaviour of $u$ near $\partial\Omega$ and discuss some of its regularity properties.

Existence of multi-bubbling solutions for a critical Hartree type equation

Daniele Cassani University of Insubria & RISM Italy Co-Author(s):

Abstract: We consider the following equation $$-\Delta u=K(|x`|,x``)\left(|x|^{-\alpha}\ast K(|x`|,x``)|u|^{2_{\alpha}^{\ast}}\right)|u|^{2_{\alpha}^{\ast}-2}u$$ in $\mathbb{R}^N$, where $N\geq 5$ and $2_{\alpha}^{\ast}$ is the so-called upper critical exponent in the Hardy-Littlewood-Sobolev inequality and $K(|x`|, x``)$ with $(x`, x``)\in $\mathbb{R}^2\times $\mathbb{R}^{N-2}$, is bounded and non-negative. Under proper assumptions on the potential function $K$, we obtain the existence of infinitely many solutions for the nonlocal critical equation by using a finite dimensional reduction argument and local Pohozaev identities. It is a remarkable fact that the order $\alpha$ of the Riesz potential influences the existence/non-existence of solutions.

Uniform boundedness of solutions for superlinear heat equations

Yohei Fujishima Shizuoka University Japan Co-Author(s):    Toru Kan

Abstract: We study the $L^\infty$ estimates of radially symmetric solutions for a superlinear heat equation with the Dirichlet boundary condition, and show the uniform boundedness of global solutions. In particular, our results are applicable for nonlinear terms growing extremely faster than the exponential function when the space dimension is greater than 2 and smaller than 10.

Recent progress on the study of solutions to nonlinear Dirac equations

Qi Guo Renmin University of China Peoples Rep of China Co-Author(s):    Pan Chen, Yanheng Ding, Xiaojing Dong, Huayang Wang, Yuanyang Yu

Abstract: In this talk, I will begin with a brief review of physical and geometric background of the Dirac operator, along with the spectral structure and orthogonal decomposition. Then I will introduce the definitions of action and energy ground states of nonlinear Dirac equations and discuss key properties such as existence, nonrelativistic limits, and convergence rates.

A system of superlinear elliptic equations in a cylinder

Yanyan Guo Central China Normal University Peoples Rep of China Co-Author(s):    Bernhard Ruf, Shuangjie Peng

Abstract: This talk concerns with the existence of positive solutions of a coupled semi-linear elliptic system defined in a cylinder. The system couples an equation defined in the whole cylinder $\Omega$ with another equation defined at the bottom of the cylinder $\Omega'\times\{0\}$. We provide a priori $L^\infty$ bounds for all positive solutions of the system when the nonlinear terms satisfy certain growth conditions. Using the a priori bounds and topological arguments, we prove the existence of positive solutions for these particular semi-linear elliptic systems.

Vanishing phenomena for Trudinger-Moser Inequalities with a scale parameter

Masato Hashizume The University of Osaka Japan Co-Author(s):

Abstract: We consider the Trudinger-Moser inequality on a bounded domain with a scale parameter. We investigate the asymptotic behavior of maximizers attaining the inequality, with respect to the parameter. We first show that this behavior depends on the exponent in the Trudinger-Moser functional. When the exponent is large, an energy concentration phenomenon occurs, whereas when it is small, a vanishing phenomenon occurs. We then focus on the vanishing phenomenon and establish the asymptotic expansion of the optimal constant, as well as the profile of maximizers exhibiting vanishing behavior. Furthermore, we show that, after a suitable transformation, the maximizers converge to a maximizer of a $L^2$-normalized variational problem.

Global existence at the critical mass for the two-dimensional fully parabolic Keller--Segel system

Tatsuya Hosono Osaka Metropolitan University Japan Co-Author(s):

Abstract: We study the Cauchy problem for the two-dimensional fully parabolic Keller--Segel system at the critical mass. Global-in-time existence at the critical mass is known under radial symmetry or under additional moment assumptions on the initial data. A common approach in the literature is to impose such conditions, as they provide effective control of the behavior of solutions at spatial infinity. However, these assumptions are extrinsic to the intrinsic scaling structure underlying the critical mass phenomenon. In the absence of moment conditions, classical energy-based methods cease to be effective, and the problem becomes considerably more delicate at the critical threshold due to the degeneracy of the underlying energy structure. In this talk, we establish global-in-time existence for initial data at the critical mass without imposing any additional assumptions. The proof relies on a reconstructed Lyapunov functional and refined dissipative estimates that recover sufficient control of the dynamics in the whole space.

Bifurcating solutions of a Hatree-type elliptic equation

Yoshitsugu Kabeya Osaka Metropolitan University Japan Co-Author(s):    Vitaly Moroz (Swansea University, UK) and So Maeda (Nara Kita Senior High School, Japan)

Abstract: We consider a Hatree-type elliptic equation (having the convolution of the unknown function) and discuss the existence of bifurcating solutions from the trivial solutions.

Fundamental solution and diffusion limits for the heat equation in a half-space with a diffusive dynamical boundary condition

Tatsuki Kawakami Ryukoku University Japan Co-Author(s):    Kazuhiro Ishige, Sho Katayama

Abstract: In this talk, we derive an explicit representation of the fundamental solution to the heat equation in a half-space with a diffusive dynamical boundary condition, and establish sharp pointwise upper and lower bounds. We also investigate qualitative properties of the associated solutions, including precise decay estimates. Furthermore, we analyze the diffusion limits of solutions to the initial--boundary value problem and clarify the role of the diffusive dynamical boundary condition in determining their behavior. This talk is based on joint work with Prof. K. Ishige (University of Tokyo) and Dr. S. Katayama (Keio University).

On self-similar solutions of time-fractional semilinear heat equations

Mizuki Kojima Kanagawa University Japan Co-Author(s):

Abstract: We consider self-similar solutions of semilinear heat equations involving the Caputo type fractional derivative with respect to time. We mainly investigate some properties of radially symmetric self-similar solutions. In particular, the solution can have a singularity at the origin under certain situation, which is specific to the time-fractional equation.

Solvability of the Cauchy problem for the porous medium equation with singular initial data

Nobuhito Miyake Faculty of Mathematics, Kyushu University Japan Co-Author(s):    Kazuhiro Ishige, Ryuichi Sato

Abstract: We consider the solvability of the Cauchy problem for the porous medium equation with a power-type nonlinearity. More precisely, we establish a sufficient condition on the initial data for the existence of solutions in terms of uniformly local Morrey-type spaces. As an application, we identify the optimal singular profile of the initial data for solvability. This talk is based on the joint work with Kazuhiro Ishige (the University of Tokyo) and Ryuichi Sato (Iwate University).

Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance

Yasuhito Miyamoto The University of Tokyo Japan Co-Author(s):    Kotaro Hisa

Abstract: We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider \[ \begin{cases} \partial_tu-\Delta u=f(u), & x\in {\bf R}^N,\ t>0,\ u(x,0)=u_0(x), & x\in {\bf R}^N, \end{cases} \] where $N>2$. We assume that the growth rate of $f$ is less than the Joseph-Lundgren exponent for $N>10$ and it satisfies certain assumptions guaranteeing a positive radial singular stationary solution $u^*$. We prove that if $u_0=u^*$, then the problem has at least two positive solutions, namely $u^*$ and $u(t)$ which satisfies $u(t)\in L_{loc}^{\infty}(0,t_0;L^{\infty}({\bf R}^N))$ for some $t_0>0$ and $$ u(t)\to u^*\quad\text{in}\ L^{\gamma}_{ul}({\bf R}^N)\quad\text{as}\ t\to 0^+ $$ for $1\le \gamma

Ground states of a nonlocal variational problem and Thomas-Fermi limit for the Choquard equation

Vitaly Moroz Swansea University Wales Co-Author(s):    Damiano Greco, Zeng Liu, Yanghong Huang

Abstract: We study nonnegative optimizers of a Gagliardo-Nirenberg type inequality that interpolates between the nonlocal Riesz energy and two $L^q$-norms. A particular case of the equivalent problem has been studied in connection with the Keller-Segel diffusion-aggregation models over the past few decades. The more general case considered in our work arises in the study of the Thomas-Fermi limit regime for the Choquard equations with local repulsion. We establish, for the first time, the optimal ranges of parameters for the validity of the above interpolation inequality, discuss the existence and qualitative properties of the nonnegative maximisers, and in some special cases estimate the optimal constant. In special cases it is known that the maximisers are H\older continuous and compactly supported in a ball. We show that, depending on the regime, maximisers may also be smooth functions supported on the entire space, or discontinuous functions consisting of the characteristic function of a ball and a nonconstant, nonincreasing H\older continuous function supported on the same ball. We use these qualitative properties of the maximisers to justify the validity of the Thomas-Fermi approximations for the Choquard equations with local repulsion. The results are verified numerically through extensive examples.

Polyharmonic Steklov eigenvalue problems

Michiaki Onodera Institute of Science Tokyo Japan Co-Author(s):

Abstract: This talk concerns the spectral properties of polyharmonic Steklov eigenvalue problems on bounded domains. In particular, we will discuss the positivity of the first eigenfunction, which is nontrivial for higher-order elliptic problems due to the lack of a general maximum principle. This talk is based on joint work with Masato Hashizume and Nobuhito Miyake.

An Orlicz space approach to exponential elliptic problems

Federica Sani University of Modena and Reggio Emilia Italy Co-Author(s):    Alberto Boscaggin (University of Torino), Francesca Colasuonno (University of Torino), Benedetta Noris (Politecnico di Milano)

Abstract: In this talk, we present a variational approach to semilinear elliptic problems with exponential nonlinearities in (possibly unbounded) annular domains of $\R^N$. These problems present significant analytical difficulties in dimensions $N \geq 3$, where the classical Sobolev framework is not sufficient to handle nonlinear terms with exponential growth. To overcome these difficulties, we combine tools from Orlicz space theory (which provide a natural functional setting to handle non-polynomial nonlinearities) with nonsmooth critical point methods in the spirit of Szulkin. This approach allows us to recover a suitable variational structure despite the lack of standard compactness properties. Under suitable structural assumptions, we establish the existence of positive solutions that are not radially symmetric, thereby exhibiting symmetry-breaking phenomena for this class of elliptic problems.

A simple proof of attainability for the Sobolev inequality

Megumi Sano Nara Wowen`s University Japan Co-Author(s):

Abstract: We give a simple proof of the existence of a minimizer for the Sobolev inequality. Our proof is based on a representation formula via a cut-off fundamental solution.

Deficit estimates for an entropic form of Gagliardo-Nirenberg inequalities via nonlinear parabolic equations

Takeshi Suguro Kumamoto University Japan Co-Author(s):

Abstract: We consider several types of Gagliardo--Nirenberg inequalities. The functional inequalities can be expressed in terms of the Tsallis entropy, a one-parameter extension of the Boltzmann--Shannon entropy. In this talk, we obtain deficit estimates for the inequalities by using properties of solutions to nonlinear parabolic equations corresponding to the Tsallis entropy.

On Schrodinger equations with critical growth and zero on the boundary of a spectral gap

Cristina Tarsi Universita` degli Studi di Milano Italy Co-Author(s):

Abstract: In this talk we consider a Schr\"{o}dinger equation of the type \begin{equation*} -\Delta u+V(x)u=f(x,u)+\lambda u\quad \text{for}~~x\in\mathbb R^N, N\geq 2 \end{equation*} where $V$ is a periodic and non constant potential and $f$ is a nonlinear term with critical power growth if $N\geq 3$ or critical exponential growth if $N=2$. The spectrum of the self-adjoint operator $S=-\Delta +V$ is purely continuous and may contain gaps. Assuming that $0$ lies on the boundary of a spectral gap, we prove the existence of a solution in $H^2_{loc}(\mathbb R^N)$.

Fokker--Planck equations and n--dimensional Poincar\\`e inequalities for isotropic densities

Elide Terraneo University of Milan Italy Co-Author(s):    G. Furioli (Univ. of Bergamo, Italy), A. Pulvirenti and G. Toscani (Univ. of Pavia, Italy)

Abstract: We consider some n-dimensional functional inequalities of the type of Poincar\`e, with weight, for isotropic probability densities. Unlike previous studies, for example by S.G. Bobkov, and M. Ledoux (Ann. Probab. 37 (2009)), the derivation of these inequalities is strongly related to their connection with the problem of convergence to equilibrium for a class of Fokker--Planck type equations characterized by a variable coefficient of diffusion. This is a result obtained in collaboration with G. Furioli (Univ. of Bergamo, Italy), A. Pulvirenti and G. Toscani (Univ. of Pavia, Italy).

Co-rotating nearly parallel helical vortices of 3D incompressible Euler equations in infinite cylinders

Jie Wan Beijing Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we consider the existence of traveling-rotating helical vortices to 3D incompressible Euler equations in infinite cylinders, whose support sets consist of helical tubes with small cross-sections of radius $ O(\varepsilon) $ and concentrate near several types of co-rotating helical solutions of nearly parallel vortex filaments model as $ \varepsilon\to 0 $, which justify the result in Klein, Majda and Damodaran [1995, JFM] and generalize results in Guerra and Musso [2026, Math Ann]. The key is to find clustered solutions to a semilinear elliptic equations in divergence form \begin{equation*} \begin{cases} -\varepsilon^2\text{div}(K(x)\nabla u)= (u-q|\ln\varepsilon|)^{p}_+,\ \ &x\in \Omega,\ u=0,\ \ &x\in\partial \Omega \end{cases} \end{equation*} for small values of $ \varepsilon $. This is a joint work with Prof. Daomin Cao.