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M. Marras

Publications and source records attributed to M. Marras.

5 recordsLinked to original sources
arXiv2026

H\"older estimates of weak solutions to chemotaxis systems of fast diffusion type

We study a quasilinear chemotaxis system of singular type, where the diffusion operator is given by $\Delta u^m$ with $0<m<1$, corresponding to the fast diffusion regime, and where the chemotactic drift is nonlinear. Since H\"older continuity constitutes the optimal regularity class for weak solutions to the porous medium equation, we establish analogous regularity results for bounded solutions of parabolic--parabolic chemotaxis systems in this setting. The proof is based on a refined De Giorgi--Di Benedetto iteration scheme adapted to the coupled structure of the system. These results advance the understanding of the fine regularity properties of chemotaxis models with nonlinear diffusion, and demonstrate that the interplay between singular diffusion and aggregation exhibits a regularizing mechanism consistent with the porous medium paradigm.

math.AP↗
arXiv2022

Behaviour in time of solutions to fourth-order parabolic systems with time dependent coefficients

This paper deals with a class of initial-boundary value problems for nonlinear fourth order parabolic systems with time dependent coefficients in a bounded domain $Ω\subset \mathbb{R}^N, N\geq 2$. Introducing suitable conditions on the source terms, we obtain a time interval $[0,T],$ where the solution remains bounded by deriving a lower bound $T$ of $t^*$. Moreover, we establish conditions on the shape of the spatial domain and on data sufficient to guarantee that the solution blows up in finite time $t^*$, deriving an upper bound for $t^*$.

math.AP↗
arXiv2022

Blow-up phenomena for a chemotaxis system with flux limitation

In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system \begin{equation*} \left\{ \begin{array}{l} \begin{aligned} &u_t = Δu - \nabla(u f(|\nabla v|^2 )\nabla v), \\[6pt] &0= Δv -μ+ u , \quad \int_Ωv =0, \ \ μ:= \frac 1 {|Ω|} \int_Ω u dx, \\[6pt] &u(x,0)= u_0(x), \end{aligned} \end{array} \right. \end{equation*} in $Ω\times (0,\infty)$, with $Ω$ a ball in $\mathbb{R}^N$, $N\geq 3$ under homogeneous Neumann boundary conditions and $f(ξ) = (1+ ξ)^{-α}$, $0<α< \frac{N-2}{2(N-1)}$, which describes gradient-dependent limitation of cross diffusion fluxes. Under conditions on $f$ and initial data, we prove that a solution which blows up in finite time in $L^\infty$-norm, blows up also in $L^p$-norm for some $p>1$. Moreover, a lower bound of blow-up time is derived. \vskip.2truecm \noindent{\bf AMS Subject Classification }{Primary: 35B44; Secondary: 35Q92, 92C17.} \vskip.2truecm \noindent{\bf Key Words:} finite-time blow-up; chemotaxis.

math.AP↗
arXiv2017

Boundedness in a fully parabolic chemotaxis system with nonlinear diffusion and sensitivity, and logistic source

In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{ t}=\nabla \cdot ((u+1)^{m-1} \nabla u-(u+1)^αχ(v)\nabla v) + ku-μu^2 & x\in Ω, t>0, \\ v_{t} = Δv-vu & x\in Ω, t>0,\\ \end{cases} \end{equation*} $Ω$ being a bounded and smooth domain of $\mathbb{R}^n$, $n\geq 1$, and where $m,k \in \mathbb{R}$, $μ>0$ and $α< \frac{m+1}{2}$. For any $v\geq 0$ the chemotactic sensitivity function is assumed to behave as the prototype $χ(v) = \frac{χ_0}{(1+av)^2}$, with $a\geq 0$ and $χ_0>0$. We prove that for nonnegative and sufficiently regular initial data $u(x,0)$ and $v(x,0),$ the corresponding initial-boundary value problem admits a global bounded classical solution provided $μ$ is large enough.

math.DS↗