arXiv subjects
Publications and source records attributed to Francesco Petitta.
In this paper we give a suitable notion of entropy solution of parabolic $p-$laplacian type equations with $1\leq p<2$ which blows up at the boundary of the domain. We prove existence and uniqueness of this type of solutions when the initial data is locally integrable (for $1<p<2$) or integrable (for $p=1$; i.e the Total Variation Flow case).
In this paper we study existence and nonexistence of solutions for a Dirichlet boundary value problem whose model is $$ \begin{cases} -\sum_{m=1}^{\infty} a_m Δu^m= f&\text{in}\ Ω\newline u=0 & \text{on}\ \partialΩ\,, \end{cases} $$ where $Ω$ is a bounded domain of $\mathbb{R}^N$, $a_m$ is a sequence of nonnegative real numbers, and $f$ is in $L^q(Ω)$, $q>\frac{N}{2}$.
Let $Ω\subseteq \mathbb{R}^N$ a bounded open set, $N\geq 2$, and let $p>1$; we prove existence of a renormalized solution for parabolic problems whose model is $$ \begin{cases} u_{t}-Δ_{p} u=μ& \text{in}\ (0,T)\timesΩ,\newline u(0,x)=u_0 & \text{in}\ Ω,\newline u(t,x)=0 &\text{on}\ (0,T)\times\partialΩ, \end{cases} $$ where $T>0$ is any positive constant, $μ\in M(Q)$ is a any measure with bounded variation over $Q=(0,T)\timesΩ$, and $u_o\in L^1(Ω)$, and $-Δ_{p} u=-{\rm div} (|\nabla u|^{p-2}\nabla u )$ is the usual $p$-laplacian.
In this paper we deal with the asymptotic behavior as $t$ tends to infinity of solutions for linear parabolic equations whose model is $$ \begin{cases} u_{t}-Δu = μ& \text{in}\ (0,T)\timesΩ,\\[0.7 ex] u(0,x)=u_0 & \text{in}\ Ω, \end{cases} $$ where $μ$ is a general, possibly singular, Radon measure which does not depend on time, and $u_0\in L^{1}(Ω)$. We prove that the duality solution, which exists and is unique, converges to the duality solution (as introduced by G. Stampacchia) of the associated elliptic problem.