Related papers: Special fast diffusion with slow asymptotics. Entr…

We study the large-time behaviour of the solutions $u$ of the evolution equation involving nonlinear diffusion and gradient absorption $\partial_t u - \Delta_p u + |\nabla u|^q=0$. We consider the problem posed for $x\in {\mathbb R}^N $ and…

We study the large-time behaviour of the solutions of the evolution equation involving nonlinear diffusion and gradient absorption, $$ \partial_t u - \Delta_p u + |\nabla u|^q=0 . $$ We consider the problem posed for $x\in \real^N$ and t>0…

We prove the short-time asymptotic formula for the interfaces and local solutions near the interfaces for the nonlinear double degenerate reaction-diffusion equation of turbulent filtration with fast diffusion and strong absorption \[…

We introduce a framework for designing efficient diffusion models for $d$-dimensional symmetric-space Riemannian manifolds, including the torus, sphere, special orthogonal group and unitary group. Existing manifold diffusion models often…

We consider radial solutions to the fast diffusion equation $u_t=\Delta u^m$ on the hyperbolic space $\mathbb{H}^{N}$ for $N \ge 2$, $m\in(m_s,1)$, $m_s=\frac{N-2}{N+2}$. By radial we mean solutions depending only on the geodesic distance…

This work is concerned with homogenization problems for elliptic equations of the type \[ \begin{cases} \mathfrak{L}_{\delta} u_{\delta} + \lambda u_{\delta} = f_{\delta} \qquad \text{in} \;\; D, \\ \qquad \quad \;\, u = 0 \qquad \,…

In this article, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of a parabolic evolution equations on complex plane, in which the diffusion operator has the form \(\overline{\Box}_{\varphi} =…

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\gamma>0$ and $\eta>0$. Suppose either (i) $\alpha\ne 0$ and $\beta=0$ or (ii) $\alpha\in\mathbb{R}$ and $\beta\ne 0$ holds. We will study the elliptic equation $\Delta (f^m/m)+\alpha f+\beta x\cdot\nabla…

We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous medium equation, $\partial_t u +(-\Delta)^{s}(u^m)=0$, and some…

In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem $$ u_t-\Delta u=au\log u+Vu, \ \ u>0 $$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative…

Heat diffusion has been widely used in brain imaging for surface fairing, mesh regularization and cortical data smoothing. Motivated by diffusion wavelets and convolutional neural networks on graphs, we present a new fast and accurate…

We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian…

Let $\Omega\subset\R^n$ be a smooth bounded domain and let $a_1,a_2,\dots,a_{i_0}\in\Omega$, $\widehat{\Omega}=\Omega\setminus\{a_1,a_2,\dots,a_{i_0}\}$ and $\widehat{R^n}=\R^n\setminus\{a_1,a_2,\dots,a_{i_0}\}$. We prove the existence of…

We study the positivity and asymptotic behaviour of nonnegative solutions of a general nonlocal fast diffusion equation, \[\partial_t u + \mathcal{L}\varphi(u) = 0,\] and the interplay between these two properties. Here $\mathcal{L}$ is a…

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\eta>0$, $\eta_0>0$, $\rho_1>0$, $-\frac{\rho_1}{2}<\beta<\frac{m\rho_1}{n-2-nm}$ and $\alpha=\frac{2\beta+\rho_1}{1-m}$. We will prove the existence of radially symmetric solution of the equation…

The object of this paper is the uniqueness for a $d$-dimensional Fokker-Planck type equation with non-homogeneous (possibly degenerated) measurable not necessarily bounded coefficients. We provide an application to the probabilistic…

We study the large time behavior of nonnegative solutions to the Cauchy problem for a fast diffusion equation with critical zero order absorption $$ \partial_{t}u-\Delta u^m+u^q=0 \quad \quad \hbox{in} \ (0,\infty)\times\real^N\, $$ with…

We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =\Delta (u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative…

Let $\Omega$ be a domain in $\mathbb R^N$, where $N \ge 2$ and $\partial\Omega$ is not necessarily bounded. We consider two fast diffusion equations $\partial_t u= \mbox{div}(|\nabla u|^{p-2}{\nabla u})$ and $\partial_t u= \Delta u^{m}$,…

We consider the highly nonlinear and ill-posed inverse problem of determining some general expression $F(x,t,u,\nabla_xu)$ appearing in the diffusion equation $\partial_tu-\Delta_x u+F(x,t,u,\nabla_xu)=0$ on $\Omega\times(0,T)$, with $T>0$…