Related papers: Large time behavior for the heat equation on Carno…
The large time behavior of zero mass solutions to the Cauchy problem for a convection-diffusion equation. We provide conditions on the size and shape of the initial datum such that the large time asymptotics of solutions is given either by…
We consider the optimization of a finite-time Carnot engine characterized by small dissipations. We bound the power with a simple inequality and show that the optimal strategy is to perform small cycles around a given working point, which…
Admissible point transformations between Burgers equations with linear damping and time-dependent coefficients are described and used in order to exhaustively classify Lie symmetries of these equations. Optimal systems of one- and…
The large time behavior of general solutions to a class of quasilinear diffusion equations with a weighted source term $$ \partial_tu=\Delta u^m+\varrho(x)u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ with $m>1$, $1<p<m$ and suitable…
This paper is concerned with the large time behavior of solutions to the Euler-Fourier system with damping in $\mathbb{R}^{d}~(d\geq1)$. A time-weighted energy argument has been developed within the $L^2$ framework to derive the optimal…
In this paper we focus on the initial value problem for a hyperbolic-elliptic coupled system of a radiating gas in multi-dimensional space. By using a time-weighted energy method, we obtain the global existence and optimal decay estimates…
The thermodynamic properties such as the specific heat are uniquely determined by the second moments of the energy distribution for a given ensemble averaging. However for small particle numbers the results depend on the ensemble chosen. We…
The fundamental solution of the heat equation on $R^n$ is known as the heat kernel which is also the transition density of a Brownian motion. Similar statements hold when $\R^n$ is replaced by a Lie group. We briefly demonstrate how the…
Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on $\b R^N$. The definition and properties of these…
The large time behavior of solutions to the following generalized Burgers-Fisher-KPP equation $$ \partial_tu=u_{xx}+k(u^n)_x+u^p-u^q, \quad (x,t)\in\mathbb{R}\times(0,\infty), $$ with $n\geq2$, $p>q\geq1$ and $k\in\mathbb{R}$, is considered…
We propose a novel approach for studying small-time asymptotics of the fractional heat content of $C^2$ non-characteristic domains in Carnot groups. Denoting the sub-Laplacian operator by $\mathcal{L}$, the fractional heat content of a…
Consider the solution $\mathcal{Z}(t,x)$ of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data $\mathcal{Z}(0,x) = \delta(x)$. For any real $p>0$, we obtained detailed…
We prove equivalence between nonnegative distributional solutions of the fractional heat equation and caloric functions, i.e., functions satisfying the mean value property with respect to the space-time isotropic $\alpha$-stable process. We…
We analyze the asymptotic behaviour of the heat kernel defined by a stochastically perturbed geodesic flow on the cotangent bundle of a Riemannian manifold for small time and small diffusion parameter. This extends WKB-type methods to a…
This paper considers the hyperbolic-parabolic coupled system, arising from the generalized thermoelastic coupled system, in the whole space $\mathbb{R}^n$. We study some qualitative properties for an energy term by diagonalization…
In an attempt to generalize the Hamilton's principle, an action functional is proposed which, unlike the standard version of the principle, accounts properly for all initial data and the possible presence of dissipation. To this end, the…
We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with H\"{o}lder continuous coefficients. The kernels appearing in the integrals are supported on the level and…
We consider an approximate solution to the heat equation which consists of the derivatives of heat kernel. Some conditions in the initial value, under which the approximation converges to the solution of the heat equation or diverges when…
By using ideas and strong results borrowed from the classical moment problem, we show how -under very general conditions- a discrete number of values of the spectral zeta function (associated generically with a non-decreasing sequence of…
We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption $$\partial\_t u - \Delta\_{p}u + |\nabla u|^{q\_*} = 0 \quad \hbox{in} (0,\infty)\times\mathbb{R}^N\ ,$$ for…