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 investigate on the uniqueness property of the solutions to a parabolic problem with the data and the coefficient of zero order term only summable functions. Despite all this lack of regularity and without using any notion of entropy or renormalized solutions or the property to be a solution obtained by approximations we prove an uniqueness result. Then, we study the behavior in time of the unique solution.

We study a parabolic equation with the data and the coefficient of zero order term only summable functions. Despite all this lack of regularity we prove that there exists a solution which becomes immediately bounded. Moreover, we study the asymptotic behavior of this solution in the autonomous case showing that the constructed solution tends to the associate stationary solution.

In this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as p → ∞ {p oinfty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in ω or for data f (that do not change sign in ω) possibly vanishing in a set of positive measure.