Related papers: Sets of unique continuation for heat equation
We find a geometrical method of analysing the singularities of a plane nodal curve. The main results will be used in a forthcoming paper on geometric Plucker formulas for such curves. Plane nodal curves, that is plane curves having at most…
A topological shape analysis is proposed and utilized to learn concepts that reflect shape commonalities. Our approach is two-fold: i) a spatial topology analysis of point cloud segment constellations within objects. Therein constellations…
We are concerned with the nodal set of solutions to equations of the form \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- >…
In this article, we study the set of all solutions of linear differential equations using Hurwitz series. We first obtain explicit recursive expressions for solutions of such equations and study the group of differential automorphisms of…
We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued "random-field" solutions to these SPDEs…
Persistent Homology is a powerful tool in Topological Data Analysis (TDA) to capture topological properties of data succinctly at different spatial resolutions. For graphical data, shape, and structure of the neighborhood of individual data…
The nuclear many-body problem at the limits of stability is considered in the framework of the Continuum Shell Model that allows a unified description of intrinsic structure and reactions. Technical details behind the method are highlighted…
In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time…
We present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d-dimensional sphere, where d>1. The asymptotic profile and blowup rate of these solutions are the same as those…
We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolves under the Ricci flow. These upper bounds supplement the well-known global…
This paper presents a complete analysis of the observability property of heat equations with time-dependent real analytic memory kernels. More precisely, we characterize the geometry of the space-time measurable observation sets ensuring…
In this monograph, two sets of parabolic differential equations are studied, each with nonlinear medium response. The equations are generally referred to as "Newell-Whitehead-Segel equation," which model a wide variety of nonlinear…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
We consider unique continuation properties of solutions to a family of evolution equations. Our interest is mainly on nonlinear non-local models. This class contains the Benjamin-Ono, the Intermediate Long Wave, the Camassa-Holm, the…
We consider localized soliton-like solutions in the presence of a stable scalar condensate background. By the analogy with classical mechanics, it can be shown that there may exist solutions of the nonlinear equations of motion that…
In this paper we prove existence and uniqueness of solutions to a nonlocal parabolic problem which generalizes the electric heating problem of a conducting body.
New localized structured solutions for the three-dimensional linear diffusion (heat) equation are presented. These new solutions are written in terms of Airy functions and either Gaussian or Bessel functions. They accelerate along their…
Heating and heat conduction in molecular junctions are considered within a general NEGF formalism. We obtain a unified description of heating in current carrying molecular junctions as well as the electron and phonon contributions to the…
A solution of the heat equation with a distribution-valued potential is constructed by regularization. When the potential is the generalized derivative of a H\"{o}lder continuous function, regularity of the resulting solution is in line…
In this work, we obtain the numerical temperature field to a thermally developing fluid flow inside parallel plates problem with a quantum computing method. The physical problem deals with the heat transfer of a steady state,…