Ricerc@Sapienza

In this paper we describe a new method to derive different type of decay estimates for solutions of evolution equations which allow to describe the asymptotic behavior of the solutions both in presence or absence of "immediate" regularizing properties. Moreover, we give various examples of applications some of which new and dealing with a class of nonlinear problems with degenerate coercivity.

In this paper we study the regularity, the uniqueness and the asymptotic behavior of the solutions to a class of nonlinear operators in dependence of the summability properties of the datum f and of the initial datum $u_0$. The case of only summable data f and $u_0$ is allowed. We prove that these equations satisfy surprising regularization phenomena. Moreover we prove estimates (depending continuously from the data) that for zero datum f become well known decay (or ultracontractive) estimates.

We study the behavior on time of weak solutions to the non-stationary motion of an incompressible fluid with shear rate dependent viscosity in bounded domains when the initial velocity $u_0 in L^2$.
Our estimates show the different behavior of the solution as the growth condition of the stress tensor varies. In the "dilatant" or "shear thickening" case we prove that the decay rate does not depend on $u_0$, then our estimates also apply for irregular initial velocity.