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Special Session 60: Nonlinear Evolution Equations and Related Topics


Existence of distributional solutions to elliptic systems of $p$ -Laplacian type for locally integrable forcing

Goro Akagi

Tohoku University
Japan

Co-Author(s): Hiroki Miyakawa

Abstract:

This talk is concerned with existence and maximal regularity estimates for distributional solutions to degenerate/singular elliptic systems of

$p$ -Laplacian type with absorption and (prescribed) locally integrable forcing posed in unbounded Lipschitz domains. In particular, the forcing terms may not belong to the dual space of an energy space, e.g.,

$W^{1,p}_{\rm loc}$ , which is necessary for the existence of weak (or energy) solutions of class

$W^{1,p}_{\rm loc}$ . The method of a proof relies on both local energy estimates and a relative truncation technique developed by Bul\`{i}\v{c}ek and Schwarzacher (Calc. Var. PDEs in 2016), where the bounded domain case is studied for (globally) integrable forcing.

Boundedness and stabilization in some degenerate parabolic-elliptic-elliptic attraction-repulsion chemotaxis system

Yutaro Chiyo

Tokyo University of Science, Department of Mathematics
Japan

Co-Author(s): Yutaro Chiyo

Abstract:

This talk deals with a degenerate parabolic-elliptic-elliptic attraction-repulsion chemotaxis system. In the repulsion-dominant case, boundedness and stabilization in the nondegenerate version were obtained by C.-Yokota (2022) and C. (2022) by using the repulsion effect. However, in the degenerate case, there is no result obtained by extracting the effects of the attraction and repulsion. The purpose of this talk is to establish boundedness and stabilization in a degenerate system in the repulsion-dominant case.

A threshold type algorithm for fourth order geometric motions

Katsuyuki Ishii

Kobe University
Japan

Co-Author(s):

Abstract:

In this talk we would like to propose a threshold type algorithm for fourth order geometric motions, which are Willmore flows with lower order terms. This type of this algorithm was firstly proposed by Bence, Merriman and Osher to compute mean curvature flows. We use fourth order linear parabolic equations to construct our algorithm and show a consistency result if the initial surface is smooth. This is based on my joint work with Professores Y. Kohsaka, N. Miyake and K, Sakakibara

Non-linear evolution equations with non-local coefficients and smoothing effect

Akisato Kubo

Fujita Health University
Japan

Co-Author(s):

Abstract:

We investigate the global existence in time and asymptotic profile of the solution of some nonlinear evolution equations with strong dissipation and the proliferation term:

$\ w_{tt}= D\Delta w_{t} +\nabla\cdot(\alpha(w_{t})e^{-w}\chi[w]) + \mu(1-w_{t})w_{t},\ \mbox{in}\ {\Omega}\times(0,T)\,$ where

$D, \mu$ are positive constants,

$\alpha(\cdot)$ is an sufficiently smooth function,

$\Omega$ is a bounded domain in

$R^n$ with smooth boundary

$\partial \Omega$ and

$\nu$ is the outer unit normal vector on

$\partial \Omega$ ,

$\chi[w]:=\chi[w](x,t)$ is a non-local term. We will show the existence and asymptotic behaviour of solutions to the initial and zero-Neumann boundary value problem of the equation. We will apply our results to a model of mathematical biology, and we discuss the smoothing effect.

Non-autonomous singular perturbations of semilinear problems with dynamic boundary conditions

Jihoon Lee

Chonnam National University
Korea

Co-Author(s):

Abstract:

In this talk we prove the continuity and the Gromov-Hausdorff stability of the solutions for a class of semilinear problems with dynamic boundary condition of pure reactive and reactive diffusive type. Our approach involves a non-autonomous singular perturbations of a reaction-diffusion equation with large diffusion in all domain and its boundary. This is joint work with P.T.P. Lopes and L. Pires. If you are interested in this topic, please refer to the papers on the following website: http://jlee.jnu.ac.kr/Research.html

A chimera gradient flow approach to chemotaxis systems with indirect signal production

Yoshifumi Mimura

Nihon University
Japan

Co-Author(s):

Abstract:

In this talk, we discuss the existence of time global solutions for chemotaxis systems involving indirect signal generation. In particular, the case involving a degenerate diffusion term is considered from a variational rather than a semigroup approach; for each of the three unknown functions, an approximate solution is constructed by applying the so-called minimizing movement scheme. Since this system of equations is not a gradient flow, the relative compactness of the approximate solutions is not guaranteed, but the presence of Lyapunov functions provides the conditions for the existence of time global solutions and their relative compactness.

Existence of time-fractional gradient flows for nonconvex energies in Hilbert spaces

Yoshihito Nakajima

Tohoku University
Japan

Co-Author(s): Goro Akagi

Abstract:

This talk is concerned with the solvability of time-fractional gradient flow equations for nonconvex energies in Hilbert spaces. Main results consist of local and global (in time) existence of (continuous) strong solutions to time-fractional evolution equations governed by the difference of two subdifferential operators in Hilbert spaces. In contrast with classical evolution equations (with standard time-derivatives), there arise several new difficulties such as lack of chain-rule identity and low regularity of solutions from the subdiffusive nature of the problem. To prove the main results, integral forms of chain-rule formulae for time-fractional derivatives, a Lipschitz perturbation theory for time-fractional gradient flows for convex energies and Gronwall-type lemmas for nonlinear Volterra integral inequalities are developed. These abstract results are also applied to the Cauchy-Dirichlet problem for some

$p$ -Laplace subdiffusion equations with blow-up terms.

Construction of distance functions for topology optimization

Tomoyuki Oka

Fukuoka Institute of Technology
Japan

Co-Author(s): Tomoyuki Oka

Abstract:

The topology optimization problem is a problem that determines the shape and topology of materials that minimize a given energy and attracts attention in industry. However, the obtained shapes are not always manufacturable. In this talk, we construct a distance function, which is one of the geometric features, by employing a solution of an elliptic equation.

Boundedness of solutions to a chemotaxis system with a Robin boundary condition

Yuya Tanaka

Department of Mathematical Sciences, Kwansei Gakuin University
Japan

Co-Author(s): Silvia Frassu, Giuseppe Viglialoro

Abstract:

In studies of chemotaxis system, Neumann boundary conditions are usually assumed. In this talk we deal with a chemotaxis system under a Robin boundary condition and discuss global existence and boundedness of solutions. This is a joint work with Silvia Frassu and Giuseppe Viglialoro.

Optimal control problem of evolution equation governed by hypergraph Laplacian

Shun Uchida

Oita University/Faculty of Science and Technology
Japan

Co-Author(s):

Abstract:

In this talk, we consider some optimal control problem of an ODE governed by the hypergraph Laplacian, which is defined as a subdifferential of a convex function and then is a set-valued operator. In our previous works, we see that this ODE has some properties which resemble those of the PDEs with p-Laplacian. By using methods for a priori estimates, we can assure the existence of the optimal control for a suitable cost function. However, since the hypergraph Laplacian is a set-valued operator, it seems to be difficult to derive the necessary optimality condition for this problem. To cope with this difficulty, we introduce an approximation problem and assure the optimality condition for this. We also discuss the convergence of the condition to that for the original problem.

Stability of non-zero equilibrium states for the viscous conservation laws with delay effect

Yoshihiro Ueda

Kobe University
Japan

Co-Author(s):

Abstract:

In this talk, we consider the stability of the non-zero equilibrium state for the viscous conservation laws with a delay effect. The linear stability is analyzed by using the characteristic equation of the corresponding eigenvalue problem. If our equation does not have a delay effect, the characteristic equation is given by a polynomial equation. On the other hand, if our equation has a delay effect, the characteristic equation becomes a transcendental equation, and it is difficult to analyze it. In this situation, we apply the useful known result concerned with the characteristic equation for the ordinary delay differential equations and try to get the sharp stability condition for the viscous viscous conservation laws with delay.

Weighted Energy-Dissipation approach to semilinear gradient flows with state-dependent dissipation

Riccardo Voso

University of Vienna
Austria

Co-Author(s): Goro Akagi, Ulisse Stefanelli

Abstract:

We investigate the Weighted Energy-Dissipation variational approach to semilinear gradient flows with state-dependent dissipation. A family of parameter-dependent functionals defined over entire trajectories is introduced and proved to admit global minimizers. These global minimizers correspond to solutions of elliptic-in-time regularizations of the limiting causal problem. By passing to the limit in the parameter we show that such global minimizers converge, up to subsequences, to a solution of the gradient flow.

Standing waves for the nonlinear Schr\odinger-Poisson system with a doping profile

Tatsuya Watanabe

Kyoto Sangyo University
Japan

Co-Author(s): Mathieu Colin

Abstract:

In this talk, we consider the nonlinear Schr\odinger-Poisson system with a doping profile, which appears in the study of semi-conductor theory. We are interested in the existence of ground state solutions and their orbital stability. The presence of a doping profile causes several difficulties, such as the proof of the strict sub-additivity and the uniqueness of a maximum point of a fibering map.When the doping profile is a characteristic function supported on a bounded smooth domain, some geometric quantities related to the domain, such as the mean curvature, are responsible for the existence of ground state solutions.