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Giuseppe Viglialoro

Publications and source records attributed to Giuseppe Viglialoro.

32 records · Page 2Linked to original sources
arXiv2021

On some sharper boundedness conditions in the higher-dimensional chemotaxis-consumption model

For the classical zero-flux chemotaxis-consumption model \begin{equation*} u_t= Δu - χ\nabla \cdot (u \nabla v) \quad \textrm{and}\quad v_t=Δv- uv, \quad \text{ with } (x,t)\in Ω\times (0,T_{max}), \end{equation*} $Ω$ being a bounded and smooth domain of $\mathbb{R}^n$, $n\geq 3$, $χ$ some positive number and $T_{max} \in (0,\infty]$, the following was established in a paper by Tao: for every sufficiently regular initial data $u(x,0)=u_0(x)\geq 0$ and $v(x,0)=v_0(x) \geq 0$, there is $χ(\lVert v_0 \rVert_{L^\infty(Ω)})$ such that for all $0<χ\leq χ(\lVert v_0 \rVert_{L^\infty(Ω)})$, the initial-boundary value problem has a unique classical solution in $Ω\times (0,\infty)$ which is bounded. In this paper, whenever $n\geq 5$, we obtain the same claim for larger values of the constant $χ(\|v_0\|_{L^{\infty}(Ω)})$.

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arXiv2021

Remarks on two connected papers about Keller-Segel systems with nonlinear production

These notes aim to provide a deeper insight on the specifics of two articles dealing with chemotaxis models with nonlinear production. More precisely, we are referring to the papers "Boundedness of solutions to a quasilinear parabolic-parabolic chemotaxis model with nonlinear signal production" by X. Tao, S. Zhou and M. Ding [J. Math. Anal. Appl. 474:1 (2019) 733-747] and "Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation" by S. Frassu and G. Viglialoro [Acta Appl. Math. 171:1 (2021), 19]. These works, independently published in these last years, present results leaving open room for further improvement. Indeed, in the first a gap in the proof of the main claim appears, whereas the cornerstone assumption in the second is not sharp. In these pages we give a more complete picture to the relative underlying comprehension.

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arXiv2020

Boundedness in a chemotaxis system with consumed chemoattractant and produced chemorepellent

We study this zero-flux attraction-repulsion chemotaxis model, with linear and superlinear production $g$ for the chemorepellent and sublinear rate $f$ for the chemoattractant: \begin{equation}\label{problem_abstract} \tag{$\Diamond$} \begin{cases} u_t= Δu - χ\nabla \cdot (u \nabla v)+ξ\nabla \cdot (u \nabla w) & \text{ in } Ω\times (0,T_{max}),\\ v_t=Δv-f(u)v & \text{ in } Ω\times (0,T_{max}),\\ 0= Δw - δw + g(u)& \text{ in } Ω\times (0,T_{max}). %u(x,0)=u_0(x), \; v(x,0)=v_0(x) & x \in \barΩ. \end{cases} \end{equation} In this problem, $Ω$ is a bounded and smooth domain of $\R^n$, for $n\geq 1$, $χ,ξ,δ>0$, $f(u)$ and $g(u)$ reasonably regular functions generalizing the prototypes $f(u)=K u^α$ and $g(u)=γu^l$, with $K,γ>0$ and proper $ α, l>0$. Once it is indicated that any sufficiently smooth $u(x,0)=u_0(x)\geq 0$ and $v(x,0)=v_0(x)\geq 0$ produce a unique classical and nonnegative solution $(u,v,w)$ to \eqref{problem_abstract}, which is defined in $Ω\times (0,T_{max})$, we establish that for any such $(u_0,v_0)$, the life span $\TM=\infty$ and $u, v$ and $w$ are uniformly bounded in $Ω\times (0,\infty)$, (i) for $l=1$, $n\in \{1,2\}$, $α\in (0,\frac{1}{2}+\frac{1}{n})\cap (0,1)$ and any $ξ>0$, (ii) for $l=1$, $n\geq 3$, $α\in (0,\frac{1}{2}+\frac{1}{n})$ and $ξ$ larger than a quantity depending on $χ\lVert v_0 \rVert_{L^\infty(Ω)}$, (iii) for $l>1$ any $ξ>0$, and in any dimensional settings. Finally, an indicative analysis about the effect by logistic and repulsive actions on chemotactic phenomena is proposed by comparing the results herein derived for the linear production case with those in \cite{LankeitWangConsumptLogistic}.

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arXiv2020

Boundedness for a fully parabolic Keller-Segel model with sublinear segregation and superlinear aggregation

This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such production and chemoattractant, we establish that the related initial-boundary value problem has a unique classical solution which is uniformly bounded in time. To be precise, we study this zero-flux problem \begin{equation}\label{problem_abstract} \tag{$\Diamond$} \begin{cases} u_t= Δu - \nabla \cdot (f(u) \nabla v) & \text{ in } Ω\times (0,T_{max}),\\ v_t=Δv-v+g(u) & \text{ in } Ω\times (0,T_{max}),\\ \end{cases} \end{equation} where $Ω$ is a bounded and smooth domain of $\mathbb{R}^n$, for $n\geq 2$, and $f(u)$ and $g(u)$ are reasonably regular functions generalizing, respectively, the prototypes $f(u)=u^α$ and $g(u)=u^l$, with proper $α, l>0$. After having shown that any sufficiently smooth $ u(x,0)=u_0(x)\geq 0, \, v(x,0)=v_0(x)\geq 0$ emanate a unique classical and nonnegative solution $(u,v)$ to problem \eqref{problem_abstract}, which is defined on $Ω\times (0,T_{max})$ with $T_{max}$ denoting the maximum time of existence, we establish that for any $l\in (0,\frac{2}{n})$ and $\frac{2}{n}\leq α<1+\frac{1}{n}-\frac{l}{2}$, $T_{max}=\infty$ and $u$ and $v$ are actually uniformly bounded in time. This paper is in line with the contribution by Horstmann and Winkler, moreover, extends the result by Liu and Tao. Indeed, in the first work it is proved that for $g(u)=u$ the value $α=\frac{2}{n}$ represents the critical blow-up exponent to the model, whereas in the second, for $f(u)=u$, corresponding to $α=1$, boundedness of solutions is shown under the assumption $0<l<\frac{2}{n}.$

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arXiv2020

On the solvability of a two-dimensional Ventcel problem with variable coefficients

This paper deals with the following mixed boundary value problem \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} -Δu = f &\mbox{in $Ω$,} \\ u = φ&\mbox{on $Γ_{\! D}$,} \\ u_ν- a_2 \, Δ_{τ\,} u + a_0 \, u = g &\mbox{on $Γ_{\! ν}$,} \end{cases} \end{equation} where $Ω$ is some bounded domain of $\mathbb{R}^2$ with $\partial Ω=Γ_{\!D}\cup Γ_{\! ν}$, $ν$ indicating the normal unit vector to $Γ_{\! ν}$ and $Δ_τ$ the Laplace--Beltrami operator along~$Γ_{\! ν}$. Additionally, $f(\bf x)$, $φ(\bf x)$, $a_2(\bf x)$, $a_0(\bf x)$ and $g(\bf x)$ are convenient functions defined on $Ω$, $Γ_{\!D}$ and $Γ_{\! ν}$, and ${\bf x} = (x,y)$ denotes a two-dimensional array. Under suitable assumptions on the data, we first give the definition of a weak solution $u$ to the problem and then we prove that it is uniquely solvable. Further, we consider a particular case of \eqref{ProblemAbstract} arising in real-world applications: we discuss the resulting model and provide an explicit solution.

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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}.

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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}

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arXiv2019

A refined criterion and lower bounds for the blow--up time in a parabolic--elliptic chemotaxis system with nonlinear diffusion

This paper deals with unbounded solutions to the following zero--flux chemotaxis system \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} % about u u_t=\nabla \cdot [(u+α)^{m_1-1} \nabla u-χu(u+α)^{m_2-2} \nabla v] & (x,t) \in Ω\times (0,T_{max}), \\[1mm] % about v 0=Δv-M+u & (x,t) \in Ω\times (0,T_{max}), \end{cases} \end{equation} where $α>0$, $Ω$ is a smooth and bounded domain of $\mathbb{R}^n$, with $n\geq 1$, $t\in (0, T_{max})$, where $T_{max}$ the blow-up time, and $m_1,m_2$ real numbers. Given a sufficiently smooth initial data $u_0:=u(x,0)\geq 0$ and set $M:=\frac{1}{|Ω|}\int_Ωu_0(x)\,dx$, from the literature it is known that under a proper interplay between the above parameters $m_1,m_2$ and the extra condition $\int_Ωv(x,t)dx=0$, system \eqref{ProblemAbstract} possesses for any $χ>0$ a unique classical solution which becomes unbounded at $t\nearrow T_{max}$. In this investigation we first show that for $p_0>\frac{n}{2}(m_2-m_1)$ any blowing up classical solution in $L^\infty(Ω)$--norm blows up also in $L^{p_0}(Ω)$--norm. Then we estimate the blow--up time $T_{max}$ providing a lower bound $T$.

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arXiv2019

Explicit lower bound of blow--up time for an attraction--repulsion chemotaxis system

In this paper we study classical solutions to the zero--flux attraction--repulsion chemotaxis--system \begin{equation}\label{ProblemAbstract} \tag{$\Diamond$} \begin{cases} u_{ t}=Δu -χ\nabla \cdot (u\nabla v)+ξ\nabla \cdot (u\nabla w) & \textrm{in }Ω\times (0,t^*), \\ 0=Δv+αu-βv & \textrm{in } Ω\times (0,t^*),\\ 0=Δw+γu-δw & \textrm{in } Ω\times (0,t^*),\\ \end{cases} \end{equation} where $Ω$ is a smooth and bounded domain of $\mathbb{R}^2$, $t^*$ is the blow--up time and $α,β,γ,δ,χ,ξ$ are positive real numbers. From the literature it is known that under a proper interplay between the above parameters and suitable smallness assumptions on the initial data $u({\bf x},0)=u_0\in C^0(\barΩ)$, system \eqref{ProblemAbstract} has a unique classical solution which becomes unbounded as $t\nearrow t^*$. The main result of this investigation is to provide an explicit lower bound for $t^*$ estimated in terms of $\int_Ωu_0^2 d{\bf x}$ and attained by means of well--established techniques based on ordinary differential inequalities.

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arXiv2018

Solvability of a Keller-Segel system with signal-dependent sensitivity and essentially sublinear production

In this paper we consider the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{t}=Δu-\nabla \cdot (u χ(v)\nabla v) & \textrm{in}\quad Ω\times (0,\infty), \\ 0=Δv-v+g(u) & \textrm{in}\quad Ω\times (0,\infty),\\ \end{equation*} in a smooth and bounded domain $Ω$ of $\mathbb{R}^2$. The chemotactic sensitivity $χ$ is a general nonnegative function from $C^1((0,\infty))$ whilst $g$, the production of the chemical signal $v$, belongs to $C^1([0,\infty))$ and satisfies $λ_1\leq g(s)\leq λ_2(1+s)^β$, for all $s\geq 0$, $0\leqβ\leq \frac{1}{2}$ and $0<λ_1\leq λ_2.$ It is established that no chemotactic collapse for the cell distribution $u$ occurs in the sense that any arbitrary nonnegative and sufficiently regular initial data $u(x,0)$ emanates a unique pair of global and uniformly bounded functions $(u,v)$ which classically solve the corresponding initial-boundary value problem. Finally, we illustrate the range of dynamics present within the chemotaxis system by means of numerical simulations.

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arXiv2018

Global existence and boundedness of solutions to a chemotaxis-consumption model with singular sensitivity

In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_t=Δu -χ\nabla \cdot (\frac{u}{v} \nabla v) \\ v_t=Δv-f(u)v \end{cases} \end{equation*} in a smooth and bounded domain $Ω$ of $\mathbb{R}^2$, with $χ>0$ and $f\in C^1(\mathbb{R})$ essentially behaving like $u^β$, $0<β<1$. Precisely for $χ<1$ and any sufficiently regular initial data $u(x,0)\geq 0$ and $v(x,0)>0$ on $\barΩ$, we show the existence of global classical solutions. Moreover, if additionally $m:=\int_Ωu(x,0)$ is sufficiently small, then also their boundedness is achieved.

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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.

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