Related papers: Symmetry Classification for the Nonlinear Heat Con…
The equations of pre-metric electromagnetism are formulated as an exterior differential system on the bundle of exterior differential 2-forms over the spacetime manifold. The general form for the symmetry equations of the system is computed…
In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary…
Let u = {u(t, x), t $\in$ [0, T ], x $\in$ R d } be the solution to the linear stochastic heat equation driven by a fractional noise in time with correlated spatial structure. We study various path properties of the process u with respect…
In this paper, a new nonlinear heat equation is studied that arises as a model of the collective behavior of automated vehicles. The properties of the solutions of this equation are studied by introducing the appropriate notion of a weak…
Typical heat engines exhibit a kind of homotypy: The heat exchanges between a cyclic heat engine and its two heat reservoirs abide by the same function type; the forward and backward flows of an autonomous heat engine also conform to the…
Let $G=(V,E)$ be a locally finite connected weighted graph, and $\Omega$ be an unbounded subset of $V$. Using Rothe's method, we study the existence of solutions for the semilinear heat equation $\partial_tu+|u|^{p-1}\cdot u=\Delta…
We derive a matrix version of Li \& Yau--type estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative sectional curvatures and parallel Ricci tensor, similarly to what R.~Hamilton did…
We derive the form of the asymptotic series, as $t\to +\infty$, for a general solution $h(t)$ of the non-linear differential equation $h(t)^{3}(h''(t)+h'(t))=1$.
We derive an adaptation of Li & Yau estimates for positive solutions of semilinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We then apply these estimates to obtain a Harnack inequality and to discuss…
Partial Differential Equations (PDEs) are widely used for modeling the physical phenomena and analyzing the dynamical behavior of many engineering and physical systems. The heat equation is one of the most well-known PDEs that captures the…
The well-established symmetry relations for linear transport phenomena can not, in general, be applied in the non-linear regime. Here we propose a set of symmetry relations with respect to bias voltage and magnetic field for the non-linear…
We study a basic linear elliptic equation on a lower dimensional rectifiable set $S$ in $\mathbb{R}^N$ with the Neumann boundary data. Set $S$ is a support of a finite Borel measure $\mu$. We will use the measure theoretic tools to…
The suggestion of writing, for some problems, nonlinear state equations not as dx/dt = F(x,u,t), but as dx/dt = [A(t,x)]x + [B(t,x)]u(t), which is more "constructive", is considered supported by arguments related to: the axiomatization of…
The purpose of this paper is to study gradient estimate of Hamilton - Souplet - Zhang type for the general heat equation $$ u_t=\Delta_V u + au\log u+bu $$ on noncompact Riemannian manifolds. As its application, we show a Harnak inequality…
In this paper, we prove sharp gradient estimates for a positive solution to the heat equation $u_t=\Delta u+au\log u$ in complete noncompact Riemannian manifolds. As its application, we show that if $u$ is a positive solution of the…
We study non-equilibrium quantum transport of spin, heat, and charge in diffusive heterostructures including both superconductors and materials with spin-dependent fields, such as textured ferromagnets and spin-orbit coupled materials.…
A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical…
The problem of heat conduction in one-dimensional piecewise homogeneous composite materials is examined by providing an explicit solution of the one-dimensional heat equation in each domain. The location of the interfaces is known, but…
In this paper we study the quenching problem in nonlinear heat equations with power nonlinearities. For nonlinearities of power p<0 and for an open set of slowly varying initial conditions we prove that the solutions will collapse in a…
In this paper we consider the class $\mathcal{A}$ of those solutions $u(x,t)$ to the conjugate heat equation $\frac{d}{dt}u = -\Delta u + Ru$ on compact K\"ahler manifolds $M$ with $c_1 > 0$ (where $g(t)$ changes by the unnormalized…