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

ABSTRACT

In this paper, we consider the zero-flux chemotaxis system $\begin{array}{rl}{u}_t& =\mathrm{\Delta }u-\mathrm{\nabla }\cdot \left(u\chi \right(v\left)\mathrm{\nabla }v\right)\mathrm{in}\mathrm{\Omega }\times (0,\mathrm{\infty }),\\ 0& =\mathrm{\Delta }v-v+g\left(u\right)\mathrm{in}\mathrm{\Omega }\times (0,\mathrm{\infty }),\end{array}$ in a smooth and bounded domain Ω of ${\mathbb{R}}^2$. The chemotactic sensitivity χ is a general nonnegative function from $C^1\left(\right(0,\mathrm{\infty }\left)\right)$ while g, the production of the chemical signal v, belongs to $C^1\left(\right[0,\mathrm{\infty }\left)\right)$ and satisfies ${\lambda }_1\le g\left(s\right)\le {\lambda }_2(1+s{)}^{\beta }$, for all $s\ge 0$, $0\le \beta <1$ and $0<{\lambda }_1\le {\lambda }_2.$ 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.

KEYWORDS:

• Nonlinear parabolic systems
• chemotaxis
• global existence
• boundedness

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35A01
• 35K55
• 35Q92
• 92C17

Acknowledgments

GV is the member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Giuseppe Viglialoro  http://orcid.org/0000-0002-2994-4123

Thomas E. Woolley  http://orcid.org/0000-0001-6225-5365

Additional information

Funding

GV is partially supported by the research project Integro-differential Equations and Non-Local Problems, funded by Fondazione di Sardegna (2017).