Consider a nonlinear Klein-Gordon equation on $X$, a compact Riemannian manifold without boundary, $(\partial_t^2 - \Delta + m^2)w = N(w)$, where $N$ is a smooth non-linearity. If the non-linearity vanishes at order $n + 1$ at zero, and if the Cauchy data are of small size $\epsilon$ in a regular enough Sobolev space, the solution exists over time intervals of length $c/\epsilon^n$, and remains of size $O(\epsilon)$ over that interval of time.

In this review paper, we shall discuss the question of getting a similar existence result and $O(\epsilon)$ bounds over an interval of length larger than $c/\epsilon^n$. We shall present first some recent results concerning general manifolds, where one obtains some slight improvement of the lower bound of the time of existence given by local existence theory, when the mass parameter $m$ avoids a subset of zero measure.

Then, we turn to the question of obtaining lower bounds of the time of existence in $c_N\epsilon^{-N}$ for any $N$. It turns out that this is possible if one works on very special manifolds (Zoll manifolds), with Hamiltonian non linearities, and again takes the parameter $m$ of the equation outside a set of zero measure.

We shall focus first on the semi-linear case, where ideas of proof may be described easily. We shall then discuss with less details the more involved case of quasi-linear equations.

In this contribution to the proceedings of the 11th Mathematical Society of Japan (MSJ) Seasonal Institute (July 2018) we give an overview of some recent work on a mathematical model for small deformations of a spherical membrane. The idea is to consider perturbations to minimisers of a surface geometric energy. The model is obtained from consideration of second order approximations to a perturbed energy. In particular, the considered problems involve particle constraints and surface phase field energies.

We consider the thresholding scheme and explore its connection to De Giorgi's ideas on gradient flows in metric spaces; here applied to mean curvature flow as the steepest descent of the interfacial area. The basis of our analysis is the observation by Esedoğlu and the second author that thresholding can be interpreted as a minimizing movements scheme for an energy that approximates the interfacial area. De Giorgi's framework provides an optimal energy dissipation relation for the scheme in which we pass to the limit to derive a dissipation-based weak formulation of mean curvature flow. Although applicable in the general setting of arbitrary networks, here we restrict ourselves to the case of a single interface, which allows for a compact, self-contained presentation.

This Note discusses a connection between deterministic Lagrangian coherent structures (robust fluid parcels which move coherently in unsteady fluid flows according to a deterministic ordinary differential equation), and the incorporation of noise or stochasticity which leads to the Fokker–Planck equation (a partial differential equation governing a probability density function). The link between these is via a stochastic ordinary differential equation. It is argued that a closer investigation of the stochastic differential equation offers additional insights to both the other approaches, and in particular to uncertainty quantification in Lagrangian coherent structures.

In this paper we re-visit the standard principle that if a geometric nonlinear PDE problem is set up in the moment map picture then the existence of the solution to the PDE should be related to GIT stability in algebraic geometry. We also take up Kähler metrics with constant Cahen–Gutt moment map and an extension of conformally Kähler, Einstein-Maxwell metrics.

The hyperbolic MBO is a threshold dynamic algorithm which approximates interfacial motion by hyperbolic mean curvature flow. We introduce a generalization of this algorithm for imparting damping terms onto the equation of motion. We also construct corresponding numerical methods, and perform numerical tests. We also use our results to show that the generalized hyperbolic MBO is able to approximate motion by the standard mean curvature flow.

Here, we construct spherically symmetric exact analytic solutions to a reaction-diffusion model for tumour growth with a Gompertz-like growth term. This is achieved by making use of a nonclassical symmetry that is valid whenever the nonlinear reaction term and the nonlinear diffusivity satisfy a particular relationship. Example solutions are given in two different cases. In the first, the tumour size and the cell density within the tumour change, while in the second, the cell density changes but the tumour size remains constant.

In this article, we use geometric singular perturbation techniques combined with an action functional approach to study pinned pulse solutions in a three-component FitzHugh–Nagumo model with a bump-type heterogeneity. We show that the model in the original activator-2 inhibitor framework only supports symmetric pinned pulse solutions that are located in the middle of the bump region. In contrast, asymmetric pinning is possible for parameters not in the activator-2 inhibitor framework.

In this survey we introduce a main result in [AH17], which gives an almost rigidity related to positive Ricci curvature via $\operatorname{RCD}$ theory. This is new even for cloased Riemannian manifolds.

In this paper, we survey our recent results [10, 11] with Motohiro Sobajima, which give blow-up results and upper estimates of lifespan for certain small data to semilinear wave equations with the scaling invariant space-dependent or time-dependent damping:

\equation{\tagDW$_{V_0}$} \cases{

\partial_t^2 u - \Delta u + \dfrac{V_0}{|x|} \partial_t u = |u|^{p_V_0}}, \quad (t, x) \in (0, T) \times \mathbb{R}^N,

(u, \partial_tu)|_{t=0} = \varepsilon (u_0, u_1), \quad x \in \mathbb{R}^N,

\equation{\tagDW$_{\mu}$} \cases{

\partial_t^2 v - \Delta v + \dfrac{\mu}{1 + t} \partial_t v = |v|^{p_\mu}}, \quad (t, x) \in (0,T) \times \mathbb{R}^N,

(v, \partial_tv)|_{t=0} = \varepsilon (v_0, v_1), \quad x \in \mathbb{R}^N,

where $N \in \mathbb{N}$, $V_0 \in [0,(N - 1)^2/(N + 1))$, $p_{V_0} \in (N/(N - 1), p_S(N + V_0)]$, $\mu \in [0, (N^2 + N + 2)/(N + 2))$, $p_{\mu} \in (p_F(N), p_S(N + \mu)]$. Here $p_S$ is the Strauss exponent and $p_F$ is the Fujita exponent. And $(u_0, u_1)$ and $(v_0, v_1)$ are pairs of given functions, whose support is compact, and $\varepsilon \gt 0$ is a small parameter.

We are interested in the asymptotic behavior of the solution to the energy critical nonlinear wave equation with a special scale invariant damping. In the present paper, we show that a global solution to this equation goes to the solution to the linear wave equation with the scale invariant damping as time tends to infinity. We also discuss its time order.

We consider a threshold-type algorithm for curvature-dependent motions of hypersurfaces. This algorithm was numerically studied by Kimura–Notsu [11], Esedoḡlu–Ruuth–Tsai [4] and Mohammad–Švadlenka [13], where they used the signed distance function as the level set function for curvature-dependent motions. In this artcle we present the convergence and its optimal rate of this algorithm, according to Ishii–Kimura [9] and Ishii–Izumi [10].

In this paper, we briefly summarize recent results on a generalization of the Lagrangian mean curvature flow in Kähler-Einstein manifolds. In particular, we focus on the geometry of generalized LMCF in a Kähler manifold with positive weighted Ricci-form and exhibit the extended results according to [6].

We consider the Cauchy problem for systems of quasi-linear wave equations in two and three space dimensions. We investigate the small data global existence and the asymptotic behavior of solutions under a condition in between the standard and weak null conditions.

The traveling waves for area-preserving geometric flows will be discussed. For an evolving plane curve, which is governed by the area-preserving curvature flow and has two endpoints moving freely on the $x$-axis with generating constant contact angles, we prove that there exists only a convex traveling wave. In the proof, a boundary value problem for an angle function associated with the Gauss map of the profile curve is analyzed by means of a shooting argument.

We study a variational problem for piecewise-smooth hypersurfaces in the $(n + 1)$-dimensional Euclidean space. An anisotropic energy is the integral of an energy density that depends on the normal at each point over the considered hypersurface, which is a generalization of the area of surfaces. The minimizer of such an energy among all closed hypersurfaces enclosing the same $(n + 1)$-dimensional volume is unique and it is (up to rescaling) so-called the Wulff shape. The Wulff shape and equilibrium hypersurfaces of this energy for volume-preserving variations are not smooth in general. In this article we give recent results on the uniqueness and non-uniqueness for closed equilibria. We also give nontrivial self-similar shrinking solutions of anisotropic mean curvature flow. This article is an announcement of forthcoming papers [10], [13].

We review a recent result, obtained jointly with S. Miao, on the stability properties of the blow up solutions constructed in [5], [10] for the co-rotational Wave Maps problem. This provides a partial answer to a question raised in [8], [10].

In this work we establish a weighted $L^2 - L^2$ estimate for inhomogeneous wave equation in 3-D, by introducing a Morawetz multiplier with weight to the power $s(1 \lt s \lt 2)$, and then integrating on the light cones and $t$ slice. With this weighted $L^2 - L^2$ estimate in hand, we may give a new proof of global existence for small data Cauchy problem of semilinear wave equation with supercritical power in 3-D. What is more, by combining the Huygens' principle for wave equations in 3-D, the global existence for semilinear wave equation with scale invariant damping in 3-D is established.

In this note, we discuss the limit behavior for a fully nonlinear diffusion equation of power type in one space dimension. It turns out that, when the initial value is Lipschitz and convex, the solution converges locally uniformly to a unique limit function that is independent of the time variable as the exponent tends to infinity. We characterize the limit as the minimal solution of an obstacle problem. Such asymptotic behavior is closely related to applications in mathematical models of image denoising and collapsing sandpiles.

We consider the incompressible Navier–Stokes equations with slip boundary conditions in a three-dimensional curved thin domain around a given closed surface. Under suitable assumptions on given data, we establish the convergence on the limit surface of the average in the thin direction of a strong solution to the bulk equations as the width of the curved thin domain tends to zero. Moreover, we characterize the limit as a unique weak solution to limit equations, which are the damped and weighted Navier–Stokes equations on the limit surface.

This paper is a survey article to review the Cauchy problem for the generalized KdV equation with low-degree of nonlinearity. In [5], the local well-posedness for the equation is established in an appropriate class under a non-degenerate condition for the initial data. In this paper, we give a lower bound estimate for the lifespan of the solution under the condition as a consequence of the contraction principle. The lifespan depends on the size and a quantity corresponding to the distance from zero of the initial data.

Several semilinear partial partial differential equations are considered in homogeneous and isotropic spacetimes. The Cauchy problems of the equations are considered in Lebesgue spaces and Sobolev spaces. Dissipative and anti-dissipative properties of the spatial variance on the problems are remarked.

We consider planar networks minimizing the elastic energy, we state an existence and regularity result, and we discuss some geometric properties of minimal configurations. We also consider the evolution of networks by the gradient flow of the energy, and we give a well-posedness result in the case of natural boundary conditions.

This article is concerned with the Cauchy problem for the quadratic nonlinear Schrödinger equation without gauge invariance. By randomizing the initial data, we prove that the Cauchy problem is almost surely locally well-posed in $H^s(\mathbb{R}^d)$ for $d \ge 5$ and $\frac{d-4}{d-3}s_c \lt s \lt s_c$, where $s_c := \frac{d}{2} - 2$ is the scaling critical regularity.

In this paper, we consider total variation flow with values constrained to a Riemannian manifold (constrained total variation flow), which appears in several fields of mathematical sciences. In particular, we consider the constrained total variation flow on the space of piecewise constant functions and construct a numerical scheme based on the minimizing movement scheme. We present a mathematical result on the convergence of our numerical scheme and a result of the numerical experiment. This paper is based on the joint work with Yoshikazu Giga (The University of Tokyo), Kazutoshi Taguchi (ARISE analytics Inc.), and Masaaki Uesaka (Arithmer Inc./The University of Tokyo), and a full version of this paper has been published in [7].

We consider a blow-up curve for the one dimensional wave equation. Merle–Zaag [5] showed that there is a possibility that the blow-up curve for $\partial_t^2 u - \partial_x^2 u = |u|^{p-1}u$ is not differentiable if the sign of the solution changes. To show the result, they use the variational structure of the equation. In this paper, we study the blow-up curve $\partial_t^2 u - \partial_x^2 u = |\partial_t u|^{p-1}\partial_tu $ which does not have the variational structure. We state that the blow-up curve is not differentiable if the initial data is an odd function which satisfy some conditions. Next, we show the key of the proof. This paper is an announcement of our forthcoming paper.

We discuss recent work concerning stability of solutions to linear and nonlinear wave equations on asymptotically flat space times. This is joint work with Jason Metcalfe, Jesus Oliver, and Daniel Tataru.

In this paper, we consider the 1D compressible Euler equation with the damping coefficient $\lambda/(1 + t)^{\mu}$. Under the assumption that $0 \leq \mu \lt 1$ and $\lambda \gt 0$ or $\mu = 1$ and $\lambda \gt 2$, we prove that solutions exist globally in time, if initial data are small $C^1$ perturbation near constant states. In particular, we remove the conditions on the limit $\lim_{|x| \rightarrow \infty} (u (0,x), v (0,x))$, assumed in previous results.

In this paper we study blow-up and lifespan estimate for solutions to the Cauchy problem with small data for semilinear wave equations with scattering damping and negative mass term. We show that the negative mass term will play a dominant role when the decay of its coefficients is not so fast, thus the solutions will blow up in a finite time. What is more, we establish a lifespan estimate from above which is much shorter than the usual one.

Recently a new formalism for perturbations of Maxwell's equations on the background of the Kerr-NUT-(A)dS spacetime was proposed, with which the equations are reduced to a equation of motion of a scalar field that can be solved by separation of variables. In this formalism the differential operators that commute with the operators of the equations of motion, called symmetry operators, played a key role to establish the separable structure. In this work we propose a method to reproduce these commuting symmetry operators in terms of the geometric quantities associated to the hidden symmetry of the background spacetime.

This paper is an announcement of a forthcoming paper [21]. We study axially asymmetric traveling fronts to reaction-diffusion equations with balanced bistable nonlinear terms. We prove such a solution as the balanced limit of a pyramidal traveling front of an imbalanced bistable reaction-diffusion equation by preserving the ratio of a major axis and a minor axis of the cross section to be a constant. When the constant ratio is not one, the limit functions become axially asymmetric traveling fronts to balanced bistable reaction-diffusion equations.

Given any closed rectifiable set of codimension one with locally finite measure, [6] proved an existence result for Brakke's mean curvature flow starting from such set. After reviewing the definition and the result in [6], an outline of the proof is given.

This paper gives a general framework for the computation of several classes of partial differential equations (PDEs) on hypersurfaces. The approach is to work in a sufficiently thin narrow band, surrounding the surface, in the embedding Euclidean space and appropriately extend the differential operators so that the solutions on the narrow band and surface are equivalent. The solutions in the narrow band are equivalent to the solutions on the surface as in the former are constant along surface normal extensions of the latter. Consequently, it is possible to use existing (high order) numerical methods developed on grids in Euclidean space to solve PDEs on surfaces, with narrow bands whose widths are a small constant multiple of uniform Cartesian grid spacing. We apply the formulation to PDEs that originate from variational principles defined on surfaces as well as Hamilton-Jacobi-Bellman equations on surfaces that are derived from optimal control problems. This framework also provides a systematic way for solving PDE's on the unstructured point clouds that are sampled from the surface.

The mathematical theory of isometric embedding is applied to study the notion of quasilocal mass in general relativity. In particular, I shall report some recent progress of quasilocal mass with reference to a cosmological spacetime, such as the de Sitter or the Anti-de Sitter spacetime, or a blackhole spacetime, such as the Schwarzschild spacetime. This article is based on joint work with Po-Ning Chen, Ye-Kai Wang, and Shing-Tung Yau.

We present recent developments in the theory of quasi-periodic solutions to nonlinear PDEs, such as the nonlinear Schrödinger and the nonlinear Klein-Gordon equations. These solutions hold in arbitrary dimensions, and the quasi-periodicity can be either in time or space. The method hinges on multi-scale analysis, harmonic analysis and algebraic analysis.

Simons cone is a minimal cone in $\mathbb{R}^8$, which provides a counterexample to the Bernstein problem. Corresponding objects for the Allen-Cahn equation are saddle solutions. They are conjectured to be minimal in dimension 8. We discuss some progress on this problem, in particular, the existence of minimizing solutions to Allen-Cahn equation in $\mathbb{R}^8$.

This paper presents a general existence and uniqueness result for harmonic maps with prescribed singularities into non-positively curved targets, and surveys a number of applications to general relativity. It is based on a talk delivered by the author at The 11th Mathematical Society of Japan Seasonal Institute, The Role of Metrics in the Theory of Partial Differential Equations.

This is a survey of a novel integral transform and its applications to solving second-order hyperbolic equations in the curved space-times.

In 1917, very shortly after Einstein had announced his master equation on the general relativity and Schwarzschild had discovered the first explicit solution to the equation, H. Weyl characterized the Schwarzschild metric, by a harmonic function. Since then, the solutions to the Einstein equation with a certain set of symmetries are identified with elliptic variational problems, in particular the harmonic map equation. In collaboration with Marcus Khuri, Yukio Matsumoto and Gilbert Weinstein, we constructed a new set of stationary solutions to the 5-dimensional vacuum Einstein equation, which contains non-spherical event horizons. The higher dimensional spacetime exhibit a wider range of topological structures, compared to our 4-dimensional physical spacetime, and those stationary solutions are thus geometrically interesting.

This is an announcement of [3] and [4] which study the global regularity of the 3+1 dimensional Einstein-Klein Gordon system with a $U(1) \times \mathbb{R}$ isometry group. We reduce the Cauchy problem of the Einstein-Klein Gordon system to a 2+1 dimensional system. Next, the energy of this system cannot concentrate near the first possible singularity, and hence is small. Then, we show that the global regularity holds for the reduced 2+1 system with initial data of small energy.