arXiv subjects

Nicola Pintus

Publications and source records attributed to Nicola Pintus.

3 recordsLinked to original sources

arXiv2019

Blow-up prevention by sublinear production in a n-dimensional attraction-repulsion chemotaxis system

In this paper we study a zero-flux attraction-repulsion che\-mo\-taxis-system. We show that despite any mutual interplay between the repulsive and attractive coefficients from the corresponding chemo-sensitivities, even less any restriction on their own sizes, if the production rate of that chemical signal responsible of the cellular coalescence is sublinear, then any initial data emanate a unique global classical solution, which is as well bounded. Further, in a remark of the manuscript, we also address an open question given in \cite{Viglialoro2019RepulsionAttraction}.

math.AP↗

arXiv2019

On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions

In this paper we analyze the porous medium equation \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} %\begin{cases} u_t=Δu^m + a\io u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad \textrm{in}\quad Ω\times I,%\\ %u_ν-g(u)=0 & \textrm{on}\; \partial Ω, t>0,\\ %u({\bf x},0)=u_0({\bf x})&{\bf x} \in Ω,\\ %\end{cases} \end{equation} where $Ω$ is a bounded and smooth domain of $\R^N$, with $N\geq 1$, and $I= [0,t^*)$ is the maximal interval of existence for $u$. The constants $a,b,c$ are positive, $m,p,q$ proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of $u$. Under some hypothesis on the data, including intrinsic relations between $m,p$ and $q$, and assuming that for some positive and sufficiently regular function $u_0(\nx)$ the Initial Boundary Value Problem (IBVP) associated to \eqref{ProblemAbstract} possesses a positive classical solution $u=u(\nx,t)$ on $Ω\times I$: \begin{itemize} \item [$\triangleright$] when $p>q$ and in 2- and 3-dimensional domains, we determine a \textit{lower bound of} $t^*$ for those $u$ becoming unbounded in $L^{m(p-1)}(Ω)$ at such $t^*$; \item [$\triangleright$] when $p<q$ and in $N$-dimensional settings, we establish a \textit{global existence criterion} for $u$. \end{itemize}

math.AP↗

arXiv2018

Properties of solutions to porous medium problems with different sources and boundary conditions

In this paper we study nonnegative and classical solutions $u=u(\nx,t)$ to porous medium problems of the type \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} u_t=Δu^m + g(u,|\nabla u|) & {\bf x} \in Ω, t\in I,\\ %u_ν+hu=0 & \textrm{on}\; \partial Ω, t>0,\\ u({\bf x},0)=u_0({\bf x})&{\bf x} \in Ω,\\ \end{cases} \end{equation} where $Ω$ is a bounded and smooth domain of $\R^N$, with $N\geq 1$, $I=(0,t^*)$ is the maximal interval of existence of $u$, $m>1$ and $u_0(\nx)$ is a nonngative and sufficiently regular function. The problem is equipped with different boundary conditions and depending on such boundary conditions as well as on the expression of the source $g$, global existence and blow-up criteria for solutions to \eqref{ProblemAbstract} are established. Additionally, in the three dimensional setting and when blow-up occurs, lower bounds for the blow-up time $t^*$ are also derived.

math.AP↗