In this talk we are going to discuss some results concerning the questions on the
regularity of solutions for the following degenerate equations:
?pu + u = f in Q =
;
@u
@
(0.1) = 0 on @
(0; T);
and
ut ? pu = f in Q = U (0; T);
@u
@
(0.2) = 0 on @U (0; T);
u(:; 0) = u0 in U;
where p denotes the p-Laplacian with p > 2.
These problems are somehow connected to the equations modelling certain phenomena
where the dissipation is nonlinear, such as the solidi cation processes of certain polymers.
Anyway, in what regards the mathematical challenges to obtain further regularity to the
gradient of the solutions of (0.1) or (0.2), these problems have been attracting attention from
the community in the past year. For instance, the main diculty of problems (0.1) or (0.2)
arise from the lack of information on the derivatives of order greater than one of the solutions.
Our strategy is to investigate the relation of such gradients to the so-called Sobolev Spaces
of Fractional order.
In short, we intent to recall some past results on this subject and also to show some recent
contributions by the author on the subject.
