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

Research paper by **Silvia Frassu, Giuseppe Viglialoro**

Indexed on: **19 May '20**Published on: **16 May '20**Published in: **arXiv - Mathematics - Analysis of PDEs**

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#### Abstract

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=
\Delta u - \nabla \cdot (f(u) \nabla v) & \text{ in } \Omega \times
(0,T_{max}),\\ v_t=\Delta v-v+g(u) & \text{ in } \Omega \times (0,T_{max}),\\
\end{cases} \end{equation} where $\Omega$ 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 $f(u)=u^\alpha$ and
$g(u)=u^l$, with proper $\alpha, l>0$. 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 $\Omega \times (0,T_{max})$ with $T_{max}$ denoting the
maximum time of existence, we establish that for any $l\in (0,\frac{2}{n})$ and
$\frac{2}{n}\leq \alpha<1+\frac{1}{n}-\frac{l}{2}$, $T_{max}=\infty$ 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 $\alpha=\frac{2}{n}$ represents the critical
blow-up exponent to the model, whereas in the second, for $f(u)=u$,
corresponding to $\alpha=1$, boundedness of solutions is shown under the
assumption $0<l<\frac{2}{n}.$