The dynamics of globally minimizing orbits of Lagrangian systems can be studied using the Barrier function, as Mather first did, or using the pairs of weak KAM solutions introduced by Fathi. The central observation of the present paper is that Fathi weak KAM pairs are precisely the admissible pairs for the Kantorovich problem dual to the Monge transportation problem with the Barrier function as cost. We exploit this observation to recover several relations between the Barrier functions and the set of weak KAM pairs in an axiomatic and elementary way.

We shall discuss the derivation, via a scaling limit starting from large-scale interacting systems, of a certain class of nonlinear partial differential equations, especially with singular structures, such as the Stefan problem, the free boundary problem of elliptic type, an evolutionary variational inequality and a stochastic partial differential equation with reflection. A family of independent Brownian particles is first taken up as a simple example to explain the idea behind the scaling limit. Then, we survey several results concerning two kinds of model: the interacting particle systems on $\mathbb{Z}^d$ called lattice gases and the $\nabla \varphi$ interface model, which is a microscopic system for interfaces separating two distinct phases. The entropy method, which plays an essential role in the proof of the hydrodynamic limit, is explained in brief.

We study some PDE's of motion of elastic wires. These PDE's naturally require differential geometry. We mainly treat closed curves.

We review some recent work concerning an ill-posed forward–backward parabolic equation, which arises e.g. in the theory of phase transitions. Some new results are also presented, concerning local existence and uniqueness of solutions within a certain class of physical interest, and a hint of their proofs is given.

In this note we summarize some of our results on surface superconductivity and nucleation of smectics, with emphasis on analogies between superconductors and liquid crystals.

We describe the proof of mass quantization of collapse in the simplified system of chemotaxis [31], and study the blowup rate in connection with the free energy transmission (emergence).

We construct explicit solutions of the anisotropic motion of closed polygonal plane curves by a power of crystalline cuvature, in the case where the initial curves are nonconvex and the power is less than one: The solutions develop degenerate pinching singularities of a "whisker"-type and a split-type in finite time, and do not become convex polygons. Moreover, in the splitting case, we conjecture degenerate pinching rate from numerical experiments.

A one-dimensional semi-linear elliptic system with constraint to the growth of its solutions at respective infinities is discussed. This system appears in a rescaling limit of a competition-diffusion system which describes very strong inter-specific competition between two biological species. A necessary and sufficient condition for the existence of solutions is characterized in terms of the constraint at respective infinities. Also the uniqueness of a solution and several asymptotic estimates of its derivatives at infinities are stated. Moreover similar problems are discussed for an associated inhomogeneous linear elliptic system with constraint to the growth of its solutions at infinities.

We consider the Gierer-Meinhardt system on a rectangle in two-dimensional space. This system is considered to generate some spiky pattern for a wide range of parameters. On a rectangle, this system also has a stripe solution, which is known to be unstable from numerical results. In this paper, we show the instability of the stripe pattern in a mathematically rigorous manner by using the SLEP method.

The change of structures of radial solutions to elliptic equations is discussed. Especially, by some perturbation on a "potential function", the non-existence of rapidly decaying solutions fails and the existence of that one is ensured. Also, in another case, the uniqueness of rapidly decaying solutions breaks and multiple rapidly decaying ones appear. Some insights to the number of solutions are also presented.

In this survey, we consider a generalized Lotka-Volterra competition model with diffusion, and discuss the bifurcation structure of positive stationary solutions for the model. To do this, the comparison principle, the bifurcation theory, and the numerical verification are employed.

We consider a diffusion-chemotaxis-growth system which models pattern formation by a bacterial colony. In the case of spatial dimension one we prove that the initial-boundary value problem for the system has a unique solution on the entire time interval $(0, +\infty)$, and that the solution remains bounded.

In this paper we prove the existence of standing waves for the nonlinear Schrödinger equation with double power nonlinearity and harmonic potential. The nonlinearity of our problem does not satisfy the global Ambrosetti-Rabinowitz condition. Therefore, in general, it seems difficult to obtain a boundedness of Palais-Smale sequence for the associated functional. We overcome this by the compactness argument.

It is well known that the Hamiltonian formalism plays an central role in classical mechanics. In this survey, we show that the Hamiltonian formalism is useful for studying pattern formation problems in reaction-diffusion systems. Although they are not derived from the Newton principle (the motion law), the notion of gradient/skew-gradient structure enables us to use the Hamiltonian formalism for their study. This structure was originally introduced by [14], and formulated in more abstract fashion by [6]. We explain usefulness of the gradient/skew-gradient structure through the linear stability analysis of standing pulse solutions and spatially periodic patterns in reaction-diffusion systems of activator-inhibitor type.

On the unit sphere $S^2$ given a zone with periodically undulating boundaries, we consider periodic rotating waves (curves) in this zone which are driven by geodesic curvature. We state without proof the existence of periodic rotating waves. Then we study how the average rotating speed depends on the geometry of the boundaries, give the estimate of this average speed by using its homogenization limit.

We consider the parabolic equation and the associated stationary equation with non-local term. We prove the global existence of a solution for parabolic equation and discuss the property of $\omega$-limit set.

We study the existence of self-similar solutions for semilinear heat equations by making use of the methods for semilinear elliptic equations. In particular, via the variational approach, we show the existence of the second solution, which implies the non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data.

We consider a reaction-diffusion system consisting of an activator and an inhibitor which models biological pattern formation. A complete description of the entire dynamics of the kinetic system, i.e., the system without diffusion terms, is given. In particular, the $\alpha$-limit sets and the $\omega$-limit sets of all trajectories are determined.

We consider a variational problem posed by Tadjbakhsh and Odeh to describe the shape of an elastic ring in the plane under uniform pressure. Regarding the ring as a smooth closed curve, the Euler-Lagrange equation reduces to a second order ordinary differential equation for the curvature with the periodic boundary condition. The asymptotic form of solutions is presented as the external pressure tends to infinity. This is done by studying a singular perturbation problem for the Euler-Lagrange equation.

A reaction-diffusion-advection system of equations: Mikhailov-Hildebrand-Ertl model [4] in $\mathbb{R}^2$ is treated, and the global solutions are constructed. For the local existence, semigroup method is used, and by constructing a priori estimates the global existence is proved.

In this paper, we describe blowup solutions to so called Keller-Segel system and its simplified system. Keller-Segel system was introduced to describe the aggregation of cellular slime molds.

We want to investigate blowup solutions to Keller-Segel system. However, it is difficult for us. Then, we investigate solutions to a simplified system of Keller-Segel system, since we can expect that the structure of solutions to the simplified system is similar to the one to Keller-Segel system.

In this paper, we will describe some results for solutions to the simplified system, our conjecture for solutions to Keller-Segel system and the relation between those results and our conjecture.

Considered is a reaction-diffusion system consisting of an activator and an inhibitor which was proposed by Gierer and Meinhardt to model biological pattern formation. We prove that the initial-boundary value problem for the activator-inhibitor system has a unique solution for all $t \gt 0$ if the production rate of the activator is well-controlled by the inhibitor. Moreover, we prove that the solution stays in a bounded region if the source term for the activator becomes positive somewhere. We consider also how the source term for the activator affects the shape of stationary solutions in one spatial dimension.

We study the blow up phenomena of least energy solutions to some semilinear elliptic boundary value problem $(P_{\varepsilon, a})$ below on domains of a manifold which has a metric pointwise conformal to the Euclidean metric. Typical examples of our problem are set on domains of spaces of constant positive or negative curvature. It is known that the least energy solutions concentrate at one point in the domain as a parameter involved tends to 0. We characterize the location of concentration point of the least energy solutions as the maximum point of some function, defined by the coefficient function, the conformal factor and the (Euclidean) Robin function on the domain.

We consider the limiting system to a chemotaxis model equation with growth term by using singular limit analysis, which describes aggregation of biological individuals. The conditions to the linearized stability of symmetric localized stationary solutions of this system in $\mathbf{R}^3$ is shown.