A singular growth phenomenon in a Keller–Segel–type parabolic system involving density‐suppressed motilities

Yulan Wang and Michael Winkler

Mathematische Nachrichten, 2024, vol. 297, issue 6, 2353-2364

Abstract: A no‐flux initial‐boundary value problem for εuεt=Δ(uεvε−α),vεt=Δvε−vε+uε(★)$$\begin{equation*} \hspace*{92pt} \def\eqcellsep{&}\begin{array}{cc} \left\{ \def\eqcellsep{&}\begin{array}{ll} \varepsilon {u}_{\varepsilon t}& =\Delta ({u}_{\varepsilon}{v}_{\varepsilon}^{-\alpha}),\\ {v}_{\varepsilon t}& =\Delta{v}_{\varepsilon}-{v}_{\varepsilon}+{u}_{\varepsilon}\end{array} \right.\qquad \qquad(\star ) \end{array} \end{equation*}$$is considered in a ball Ω⊂Rn$\Omega \subset \mathbb {R}^n$, where n≥3$n\ge 3$ and ε>0$\varepsilon >0$. Under the assumption that α>nn−2$\alpha >\frac{n}{n-2}$, it is shown that for each m>0$m>0$, there exist T>0$T>0$ and a positive v0∈W1,∞(Ω)$v_0\in W^{1,\infty }(\Omega)$ with the property that whenever u0∈W1,∞(Ω)$u_0\in W^{1,\infty }(\Omega)$ is nonnegative with ∫Ωu0=m$\int _\Omega u_0=m$, the global solutions to (★$\star$) emanating from the initial data (u0,v0)$(u_0,v_0)$ have the property that lim supε↘0supt∈(0,T)∥uε(·,t)∥Lp(Ω)=∞for allp>n2.$$\begin{eqnarray*} \hspace*{160pt}\limsup _{\varepsilon \searrow 0} \sup _{t\in (0,T)} \Vert u_\varepsilon (\cdot,t)\Vert _{L^p(\Omega)} = \infty \qquad \mbox{for all } p>\frac{n}{2}. \end{eqnarray*}$$

Date: 2024
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