arXiv subjects

Matteo Bonforte

Publications and source records attributed to Matteo Bonforte.

39 records · Page 3Linked to original sources

arXiv2009

Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=Δ(u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for $x\in\RR^d$, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold $(\RR^d,{\bf g})$, with a metric ${\bf g}$ which is conformal to the standard $\RR^d$ metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker--Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of $m$.

math.AP↗

arXiv2008

Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations

We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =Δ(u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative positivity and boundedness estimates for locally defined solutions in domains of the form $[0,T]\times\RR^d$. They combine into forms of new Harnack inequalities that are typical of fast diffusion equations. Such results are new for low $m$ in the so-called very fast diffusion range, precisely for all $m\le m_c=(d-2)/d.$ The boundedness statements are true even for $m\le 0$, while the positivity ones cannot be true in that range.

math.AP↗

arXiv2007

Asymptotics of the fast diffusion equation via entropy estimates

We consider non-negative solutions of the fast diffusion equation $u_t=Δu^m$ 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 $t\to\infty$ for $m\ge m_c=(d-2)/d$, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if $m\ge m_c$, or close enough to the extinction time if m < mc. Such results are new in the range $m\le m_c$ where previous approaches fail. In the range mc < m < 1 we improve on known results.

math.AP↗