Related papers: Exotic heat PDE's
We prove a version of the Arezzo-Pacard-Singer blow-up theorem in the setting of Poincar\'e type metrics. We apply this to give new examples of extremal Poincar\'e type metrics. A key feature is an additional obstruction which has no…
This paper shows how the enclosure method which was originally introduced for elliptic equations can be applied to inverse initial boundary value problems for parabolic equations. For the purpose a prototype of inverse initial boundary…
This is to review some recent progress in PDE. The emphasis is on (energy) supercritical nonlinear Schr\"odinger equations. The methods are applicable to other nonlinear equations.
A general recipe is developed for the study of rigid body dynamics in terms of Poincar\'e surfaces of section. A section condition is chosen which captures every trajectory on a given energy surface. The possible topological types of the…
We discuss some of the key ideas of Perelman's proof of Poincar\'e's conjecture via the Hamilton program of using the Ricci flow, from the perspective of the modern theory of nonlinear partial differential equations.
The main objective of this addendum to the mentioned article by Park is to provide some remarks on bifurcation theories for nonlinear partial differential equations (PDE) and their applications to fluid dynamics problems. We only wish to…
We investigate how exotic differential structures may reveal themselves in particle physics. The analysis is based on the A. Connes' construction of the standard model. It is shown that, if one of the copies of the spacetime manifold is…
A class of inverse problems for a heat equation with involution perturbation is considered using four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence…
In this paper, we discuss the Poincar\'{e} series of Kac-Moody Lie algebras, especially for indefinite type. Firstly, we compute the Poincar\'{e} series of certain indefinite Kac-Moody Lie algebras whose Cartan matrices have the same type…
Mixed superposition rules are, in short, a method to describe the general solutions of a time-dependent system of first-order differential equations, a so-called Lie system, in terms of particular solutions of other ones. This article is…
We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler-Poincar\'e equations on Lie groups and homogeneous spaces. Orbit…
There are several techniques in classical case for some PDEs, involving the concept of entropy to show convergence of solutions to a steady state. In this work we deal with the $p$-adic scattering equation and we try to adapt these methods…
This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…
We present an Artificial Student, "Artie," for engineering science disciplines in which the mathematical model is a partial differential equation (PDE); Artie considers here the particular case of steady heat conduction. Artie accepts…
This paper is the second part of the papers in the same title. In this paper, we prove a conjecture of Achar-Henderson, which asserts that the Poincare polynomials of the intersection cohomology complex associated to the closure of…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…
In this expository work we discuss the asymptotic behaviour of the solutions of the classical heat equation posed in the whole Euclidean space. After an introductory review of the main facts on the existence and properties of solutions, we…
A systematic study of non-trivial cubic extensions of the four-dimensional Poincar\'e algebra is undertaken. Explicit examples are given with various techniques (Young tableau, characters etc).
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
In this paper, under Acquistpace-Terreni conditions, we make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of weighted pseudo almost periodic solutions to some classes of nonautonomous…