Francesco Petitta

Asymptotic behavior of solutions for linear parabolic equations with general measure data

Abstract In this Note we deal with the asymptotic behavior as t tends to infinity of solutions for linear parabolic equations whose model is { u t − Δ u = μ in ( 0 , T ) × Ω , u ( 0 , x ) = u 0 in Ω , where μ is a general, possibly singular, Radon measure which does not depend on time, and u 0 ∈ L 1 ( Ω ) . We prove that the duality solution, which exists and is unique, converges to the duality solution (as introduced by Stampacchia (1965)) of the associated elliptic problem. To cite this article: F. Petitta, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

A duality approach to the fractional Laplacian with measure data

We describe a duality method to prove both existence and uniqueness of solutions to nonlocal problems like ( ) s v = in R N ; with vanishing conditions at innity. Here is a bounded Radon measure whose support is compactly contained in R N , N 2, and () s is the fractional Laplace operator of order s2 (1=2; 1).

On singular elliptic equations with measure sources

We prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is \begin{eqnarray} \begin{cases} \dys -\Delta u = \frac{f(x)}{u^{\gamma}} +\mu & \text{in}\ \Omega,\\[2mm] u=0 &\text{on}\ \partial\Omega,\\[2mm] u>0 &\text{on}\ \Omega, \end{cases} \end{eqnarray} where Ω is an open bounded subset of ℝ N . Here γ > 0, f is a nonnegative function on Ω , and μ is a nonnegative bounded Radon measure on Ω .

Local estimates for parabolic equations with nonlinear gradient terms

In this paper we deal with local estimates for parabolic problems in

Some remarks on the duality method for integro-differential equations with measure data

{\mathbb{R}^N}

Optimal waiting time bounds for some flux-saturated diffusion equations

with absorbing first order terms, whose model is

A non-existence result for nonlinear parabolic equations with singular measures as data

\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.