609 research outputs found
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on bounded domains
We investigate quantitative properties of the nonnegative solutions
to the nonlinear fractional diffusion equation, , posed in a bounded domain, with for . As we use one of the most common
definitions of the fractional Laplacian , , in a bounded
domain with zero Dirichlet boundary conditions. We consider a general class of
very weak solutions of the equation, and obtain a priori estimates in the form
of smoothing effects, absolute upper bounds, lower bounds, and Harnack
inequalities. We also investigate the boundary behaviour and we obtain sharp
estimates from above and below. The standard Laplacian case or the linear
case are recovered as limits. The method is quite general, suitable to be
applied to a number of similar problems
On a singular heat equation with dynamic boundary conditions
In this paper we analyze a nonlinear parabolic equation characterized by a
singular diffusion term describing very fast diffusion effects. The equation is
settled in a smooth bounded three-dimensional domain and complemented with a
general boundary condition of dynamic type. This type of condition prescribes
some kind of mass conservation; hence extinction effects are not expected for
solutions that emanate from strictly positive initial data. Our main results
regard existence of weak solutions, instantaneous regularization properties,
long-time behavior, and, under special conditions, uniqueness.Comment: 23 page
Quantitative Local Bounds for Subcritical Semilinear Elliptic Equations
The purpose of this paper is to prove local upper and lower bounds for weak
solutions of semilinear elliptic equations of the form , with
, defined on bounded domains of \RR^d, , without
reference to the boundary behaviour. We give an explicit expression for all the
involved constants. As a consequence, we obtain local Harnack inequalities with
explicit constant, as well as gradient bounds.Comment: 2 figure
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