Special Session 178: Nonlinear Evolution Equations and Related Topics

On the nonsmooth analysis of doubly nonlinear evolution equations of second order

Aras Bacho California Institute of Technology USA Co-Author(s):

Abstract: Existence of strong solutions of an abstract Cauchy problem for a class of doubly nonlinear evolution inclusion of second order is established via a semi-implicit time discretization method. The principal parts of the operators acting on $u$ and $u^{\prime}$ are multi-valued subdifferential operators and are discretized implicitly. A non-variational and non-monotone perturbation acting nonlinearly on $u$ and $u^{\prime}$ is allowed and discretized explicitly in time. The convergence of a variational approximation scheme is established using methods from convex analysis. In addition, it is proven that the solution satisfies an energy-dissipation equality. Applications of the abstract theory to various examples, e.g., a model in visco-elastic-plasticity, are provided.

Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system

Yutaro Chiyo Tokyo University of Science Japan Co-Author(s):    Kazuki Hasegawa, Tomomi Yokota

Abstract: This talk deals with a quasilinear attraction-repulsion chemotaxis system. The purpose of this talk is to show global existence and boundedness of classical solutions by applying energy methods.

Cauchy Problem for a Cubic Novikov Equation

Nilay Duruk Mutlubas Sabanci University Turkey Co-Author(s):

Abstract: The partial differential equation with cubic nonlinearity proposed by Novikov [1] will be discussed in this talk: \begin{equation*} (1-\partial_x^2)u_t=(1+\partial_x)( u^2u_{xx}+ uu_x^2-2u^2u_x). \end{equation*} Initial value problems corresponding to the equation either on the line or on the circle will be presented. The recent results obtained both for local existence and finite time blow-up of solutions will be provided referring to the ideas recently proved in [2,3] for the quadratic nonlinear equation as a member of the integrable family of equations proposed by Novikov. [1] V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., vol. 42, paper 342002, (2009). [2] N. Duruk Mutlubas and I. L. Freire, Existence and uniqueness of periodic pseudospherical surfaces emanating from Cauchy problems, Proc. R. Soc. A., vol. 480, paper 20230670, (2024). [3] N. Duruk Mutlubas and I. L. Freire, Global and blow-up solutions for a non-local integrable equation with applications to geometry, https://doi.org/10.48550/arXiv.2505.12232, (2025).

Asymptotic behavior of nonlinear Robin energies with convex boundary nonlinearities

Kosuke Kita Hokkaido university Japan Co-Author(s):

Abstract: In this talk, we consider boundary energies generated by convex functions. Motivated by recent work by Buttazzo--Ognibene (arXiv:2506.06914) on nonlinear Robin energies with power-type boundary nonlinearities, we study asymptotics for nonlinear Robin energies with more general boundary nonlinearities. While the power case already reveals the Dirichlet and Neumann limiting behaviors, it is natural to ask whether the same mechanism remains valid beyond polynomial growth. Our aim is not only to enlarge the class of admissible nonlinearities, but also to replace arguments tied to explicit power computations with a unified variational framework based on convex duality and Moreau--Yosida regularization. This approach identifies the leading-order asymptotic term through the convex conjugate of the boundary energy and yields quantitative control of the remainder under suitable assumptions. In this way, we recover the classical power-type situation as a special case, while covering genuinely non-power boundary responses that lie outside the scope of explicit model-dependent calculations. These results provide a variational perspective on how Dirichlet- and Neumann-type limits emerge from nonlinear boundary penalization.

Non-linear evolution equations arising from mathematical models in biology with proliferation and re-establishment

Akisato Kubo Fujita Health University Japan Co-Author(s):

Abstract: In this talk we investigate the global existence in time and asymptotic profile of the solution of a nonlinear evolution equation: $$\ w_{tt}= D\Delta w_{t} +\nabla\cdot(\alpha(w_{t})e^{-l(t)}\chi[w]) + \mu(1-w_{t})w_{t}+\beta(1-w_t),\ \mbox{in}\ {\Omega}\times(0,T)\,$$ arising from non-local mathematical models in biology with proliferation and re-establishment, for $l(t)=a+dt, a,d>0,$ where $D, \mu$ are positive constants, $\alpha(\cdot)$ and $ \beta(\cdot)$ are polynomial functions with respect to $\cdot$, $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial \Omega$ and $\nu$ is the outer unit normal vector on $\partial \Omega$, $\chi[w]$ is a non-local term. By making use of the related singular integral operator, we consider the initial and zero-Neumann boundary value problem and derive estimates required for our desired result. We will apply our result to the original non-local mathematical model (by Gerisch and Chaplain), which reflects the mathematical understanding of the biological processes described cell migration in vivo.

Weighted Inertia-Dissipation-Energy approach to doubly nonlinear wave equations

Alice Marveggio University of Bonn Germany Co-Author(s):    Goro Akagi, Verena Boegelein, and Ulisse Stefanelli

Abstract: We discuss a variational approach to doubly nonlinear wave equations of the form $\rho u_{tt} + g(u_t) - \Delta u + f(u)=0$. This approach hinges on the minimization of a parameter-dependent family of uniformly convex functionals over entire trajectories, namely the so-called Weighted Inertia-Dissipation-Energy (WIDE) functionals. We prove that the WIDE functionals admit minimizers and that the corresponding Euler-Lagrange system is solvable in the strong sense. Moreover, we check that the parameter-dependent minimizers converge, up to subsequences, to a solution of the target doubly nonlinear wave equation as the parameter goes to $0$. The analysis relies on specific estimates on the WIDE minimizers, on the decomposition of the subdifferential of the WIDE functional, and on the identification of the nonlinearities in the limit.

Global solvability of the Q-tensor model for nematic liquid crystals

Miho Murata Shizuoka University Japan Co-Author(s):    Daniele Barbera, Yoshihiro Shibata

Abstract: The molecules of nematic liquid crystal flow as in a liquid phase; however, they have an orientation order. The orientation order is described by the symmetric, traceless matrix Q. Beris and Edwards proposed a Q-tensor model, the coupled system of the Navier-Stokes equations and a parabolic-type equation describing the evolution of the order parameter Q, to represent nematic liquid crystal flows. The aim of this talk is to prove the unique existence of the global-in-time solution for small initial data in the maximal regularity class for the Q-tensor model in the half-space $\mathbb R^N_+$, $N \ge 2$. In this talk, we especially discuss the weighted estimates of solutions for the linearized problem to control the higher-order terms of the solutions. This talk is based on joint work with Prof. Yoshihiro Shibata (Waseda University) and Dr. Daniele Barbera (Politecnico di Torino).

Two-material optimal design problem for the heat equation

Tomoyuki Oka Department of Intelligent Mechanical Engineering/Fukuoka Institute of Technology Japan Co-Author(s):    Kei Matsushima

Abstract: In this talk, we consider a two-material optimal design problem for the time-averaged duality pairing between a heat source and the weak solution of an initial-boundary value problem for the heat equation with a two-material diffusion coefficient, under a volume constraint. From the viewpoint of optimality, we focus on the long-time behavior of solutions to the relaxed problem.

A panoramic view of exponential attractors

Stefanie Sonner Radboud University Netherlands Co-Author(s):    Radoslaw Czaja

Abstract: We provide a unifying framework for the construction of exponential attractors for infinite dimensional dynamical systems that allows us to generalize, improve and compare existing methods that are commonly used to construct exponential attractors. For autonomous deterministic systems we formulate necessary and sufficient conditions for the existence of discrete exponential attractors in terms of a covering condition for iterates of the absorbing set under the time evolution of the semigroup. The parameters in the covering condition determine the estimate for the fractal dimension of the exponential attractor and the exponential rate of attraction. We then verify the covering condition for existing approaches to construct exponential attractors where the most general setting concerns quasi-stable semigroups in complete metric spaces. Generalizing previous notions and methods used in the literature on exponential attractors then allows us to compare widely used approaches. To conclude, we mention generalizations of the constructions for non-autonomous and random dynamical systems.

Parametric nonlinear nonhomogeneous singular problems with an indefinite perturbation

VASILE STAICU University of Aveiro Portugal Co-Author(s):    Sergiu Aizicovici, N. S. Papageorgiou and Vasile Staicu

Abstract: { We consider a singular Dirichlet problem driven by a nonlinear nonhomogeneous differential operator and with a reaction involving the competing effects of a parametric singular term and of an indefinite superlinear Caratheodory perturbation. Using variational tools together with truncation and comparison techniques, we prove an existence and multiplicity result which is global in the parameter $\lambda >0$ (a bifurcation-type theorem).}

Long-term behavior in a model for tuberculosis granuloma formation

Yuya Tanaka Department of Mathematical Sciences, Kwansei Gakuin University Japan Co-Author(s):    Masaaki Mizukami

Abstract: This talk deals with a model for tuberculosis granuloma formation. This model was proposed by Feng (J.\ Nonlinear Complex Data Sci.; 2024; 25; no. 1, 19-35), and moreover, global existence of classical/weak solutions was established in 2-/3-dimensional settings by Fuest, Lankeit and Mizukami (Nonlinear Anal.\ Real World Appl.; 2025; 85; Paper No. 104369, 14 pp). However, the long-term behavior of these solutions has remained open. The purpose of this talk is to show asymptotic stability of constant equilibria and grow-up of solutions. This is a joint work with Masaaki Mizukami (Kyoto University of Education).

A doubly nonlinear parabolic equation with nonlinear perturbation under relaxed growth and exponent conditions

Shun Uchida Oita University Japan Co-Author(s):    Shun Uchida

Abstract: In this talk, we consider the initial boundary value problem for a doubly nonlinear parabolic equation with nonlinear perturbation, subject to homogeneous Dirichlet boundary conditions. Our main goal is to relax the growth conditions on the nonlinear term and to reduce the constraints on the exponent range, allowing the results to cover both singular and degenerate cases. The proof relies on an $L^{\infty}$-estimate for a time-discrete problem, obtained in earlier work, combined with the $L^{\infty}$-energy method.

A variational approach to optimal control of gradient flows

Riccardo Voso UTIA, Czech Academy of Sciences Czech Rep Co-Author(s):    Ulisse Stefanelli, Takeshi Fukao

Abstract: We consider a general optimal control problem in the setting of gradient flows. Two approximations of the problem are presented, both relying on the variational reformulation of gradient-flow dynamics via the Weighted-Energy-Dissipation variational approach. This consists in the minimization of global-in-time functionals over trajectories, combined with a limit passage. We show that the original nonpenalized problem and the two successive approximations admits solutions. Moreover, resorting to a $\Gamma$-convergence analysis we show that penalised optimal controls converge to nonpenalized one as the approximation is removed.

Ground state solution for the nonlinear Schr\odinger-Poisson system and its link with the Gagliardo-Nirenberg-Coulomb inequality

Tatsuya Watanabe Kyoto Sangyo University Japan Co-Author(s):    Yu Su

Abstract: In this talk, we consider the nonlinear Schr\odinger-Poisson system. The aim of this talk is to present a complete characterization of the energy ground state solution for a critical strength of the interaction. Firstly, we prove the existence of an optimizer of the Gagliardo-Nirenberg-Coulomb inequality. Secondly, we give a full correspondence between an optimizer, an energy ground state solution and an action ground state solution. Finally, we obtain an explicit formula of the critical strength by using the optimizer and the best constant of the Gagliardo-Nirenberg-Coulomb inequality.