Related papers: Accelerating solutions in integro-differential equ…
We employ a generalization of Einstein's random walk paradigm for diffusion to derive a class of multidimensional degenerate nonlinear parabolic equations in non-divergence form. Specifically, in these equations, the diffusion coefficient…
The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption $\partial_t u - \Delta_p u + |\nabla u|^{p-1} = 0$ in $(0, \infty) \times…
In this paper, we first consider two scalar nonlocal diffusion problems with a free boundary and a fixed boundary. We obtain the global existence, uniqueness and longtime behaviour of solution of these two problems. The spreading-vanishing…
Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and…
In dense neutrino environments, such as provided by core-collapse supernovae or neutron-star mergers, neutrino angular distributions may be unstable to collective flavor conversions, whose outcome remains to be fully understood. These…
We prove optimal estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations in $\mathbb{R}^d$. An important special case is the time-fractional diffusion equation, which has seen much…
Convection-diffusion equations arise in a variety of applications such as particle transport, electromagnetics, and magnetohydrodynamics. Simulation of the convection-dominated regime for these problems, even with high-fidelity techniques,…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
We study a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)\ (x\in [g(t),h(t)])$ with free boundary conditions $g'(t)=-u_x(t,g(t))+\alpha$ and $h'(t)=-u_x(t,g(t))-\alpha$ for some $\alpha>0$. Such problems may be used to describe…
We study large deviations for some non-local parabolic type equations. We show that, under some assumptions on the non-local term, problems defined in a bounded domain converge with an exponential rate to the solution of the problem defined…
In the last three decades, powerful computer-assisted techniques have been developed in order to validate a posteriori numerical solutions of semilinear elliptic problems of the form $\Delta u +f(u,\nabla u) = 0$. By studying a well chosen…
Existence of specific \emph{eternal solutions} in exponential self-similar form to the following quasilinear diffusion equation with strong absorption$$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$posed for…
We are interested in the threshold phenomena for propagation in nonlocal diffusion equations with some compactly supported initial data. In the so-called bistable and ignition cases, we provide the first quantitative estimates for such…
We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy $S(u) := \{X : D u(X) = 0 \}$. Our main Theorem reveals that along…
In theory, diffusion curves promise complex color gradations for infinite-resolution vector graphics. In practice, existing realizations suffer from poor scaling, discretization artifacts, or insufficient support for rich boundary…
We study positive solutions of the super-fast diffusion equation in the whole space with initial data which are unbounded as $|x|\to\infty$. We find an explicit dependence of the slow temporal growth rate of solutions on the initial spatial…
This paper is concerned with the qualitative properties of the solutions of mixed integro-differential equation \begin{equation}\label{eq 1} \left\{ \arraycolsep=1pt \begin{array}{lll} (-\Delta)_x^{\alpha} u+(-\Delta)_y u+u=f(u)\quad \ \…
We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator. More precisely, $$ u_t=\nabla\cdot(u\nabla (-\Delta)^{-s}u), \quad \ 0<s<1. $$ The problem is posed in $\{x\in\ren, t\in…
Discrete diffusion models have emerged as a powerful generative modeling framework for discrete data with successful applications spanning from text generation to image synthesis. However, their deployment faces challenges due to the high…
We investigate the large-time dynamics of solutions of multi-dimensional reaction-diffusion equations with ignition type nonlinearities. We consider solutions which are in some sense locally persistent at large time and initial data which…