8,730 research outputs found
Symmetrization for Linear and Nonlinear Fractional Parabolic Equations of Porous Medium Type
We establish symmetrization results for the solutions of the linear
fractional diffusion equation and
itselliptic counterpart , , using the
concept of comparison of concentrations. The results extend to the nonlinear
version, , but only when
A:\re_+\to\re_+ is a concave function. In the elliptic case, complete
symmetrization results are proved for \ when
is a convex nonnegative function for with , and partial
results when is concave. Remarkable counterexamples are constructed for the
parabolic equation when is convex, resp. for the elliptic equation when
is concave. Such counterexamples do not exist in the standard diffusion case
.Comment: 42 pages, 1 figur
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
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