Matteo Bonforte

Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

The goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy–Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities, and rates for nonlinear diffusion equations.

Super and ultracontractive bounds for doubly nonlinear evolution equations

We use logarithmic Sobolev inequalities involving the p-energy functional recently derived in [15], [21] to prove Lp-Lq smoothing and decay properties, of supercontractive and ultracontractive type, for the semigroups associated to doubly nonlinear evolution equations of the form u· = ?p(um) (with m(p - 1) = 1) in an arbitrary euclidean domain, homogeneous Dirichlet boundary conditions being assumed. The bound are of the form ||u(t)||q = C||u0||r? / ts for any r = q I [1,+8) and t > 0 and the exponents s, ? are shown to be the only possible for a bound of such type.

Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

We investigate quantitative properties of nonnegative solutions

Reverse Smoothing Effects, Fine Asymptotics, and Harnack Inequalities for Fast Diffusion Equations

u(x)\ge 0

Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains

u(x)≥0 to the semilinear diffusion equation

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

\mathcal {L}u= f(u)