Related papers: Refined asymptotics for the Cauchy problem for the…
We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a subcritical exponent. We show that separable solutions are stable in some suitable sense by finding a class…
We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. These rates depend explicitly on…
This paper is concerned with the Cauchy-Dirichlet problem for fast diffusion equations posed in bounded domains, where every energy solution vanishes in finite time and a suitably rescaled solution converges to an asymptotic profile.…
We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast $p$-Laplacian evolution equation $u_t=\Delta_p u$ on the whole Euclidean space, in the so-called "good fast diffusion range" $\tfrac{2N}{N+1}<p<2$.…
This paper is concerned with a quantitative analysis of asymptotic behaviors of (possibly sign-changing) solutions to the Cauchy-Dirichlet problem for the fast diffusion equation posed on bounded domains with Sobolev subcritical exponents.…
We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameter $1<p<2$ and fractional exponent $s\in (0,1)$. Rather standard theory shows that the Cauchy Problem for data in the…
We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non--linear Fokker--Planck equation, we find a…
We consider the Cauchy problem for the evolutive discrete p-Laplacian in infinite graphs, with initial data decaying at infinity. We prove optimal sup and gradient bounds for nonnegative solutions, when the initial data has finite mass, and…
We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameters $p>2$ and $s\in (0,1)$ (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue $L^q$ spaces…
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 consider the Cauchy problem for one-dimensional p-system with damping of space-dependent coefficient. This system models the compressible flow through porous media in the Lagrangean coordinate. Our concern is an asymptotic behavior of…
We study the large time behavior of solutions to the Cauchy problem for the quasilinear absorption-diffusion equation $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\real^N\times(0,\infty), $$ with exponents $p>m>1$ and $\sigma>0$…
We consider non-negative solutions of the fast diffusion equation $u_t=\Delta u^m$ with $m \in (0,1)$, in the Euclidean space R^d, d?3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to…
We consider the asymptotic behavior of solutions to the convection-diffusion equation: \[ \partial_t u - \mathrm{div}\left(a(x)\nabla u\right) = d\cdot\nabla \left(\left\lvert u\right\rvert ^{q-1}u\right),\ \ x\in\mathbb{R}^n, \ t>0 \] with…
We study the long-time asymptotics of the doubly nonlinear diffusion equation $\rho_t={div}({|\nabla\rho^m|^{p-2}\nabla\rho^m})$ in $\RR^n$, in the range $\frac{n-p}{n(p-1)}<m>\frac{n-p+1}{n(p-1)}$ and $1p\infty$ where the mass of the…
We establish existence, uniqueness as well as quantitative estimates for solutions to the fractional nonlinear diffusion equation, $\partial_t u +{\mathcal L}_{s,p} (u)=0$, where ${\mathcal L}_{s,p}=(-\Delta)_p^s$ is the standard fractional…
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…
In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in velocity-jump processes in several dimensions. Given integers $n,d\ge 1$, let $\mathbf…
We investigate fine global properties of nonnegative, integrable solutions to the Cauchy problem for the Fast Diffusion Equation with weights (WFDE) $u_t=|x|^\gamma\mathrm{div}\left(|x|^{-\beta}\nabla u^m\right)$ posed on…
We study qualitative properties of non-negative solutions to the Cauchy problem for the fast diffusion equation with gradient absorption \partial_t u -\Delta_{p}u+|\nabla u|^{q}=0\quad in\;\; (0,\infty)\times\RR^N, where $N\ge 1$,…