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Johannes Lankeit

Publications and source records attributed to Johannes Lankeit.

At least 37 records · Page 2Linked to original sources
arXiv2021

Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic-elliptic Keller-Segel systems

We study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In $n$-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + λu - μu^{1+κ}, \\ 0 = Δv - \frac1{|Ω|} \int_Ωu + u \end{cases} \tag{JL} \end{align*} and \begin{align*} \begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u - u\nabla v) + λu - μu^{1+κ}, \\ 0 = Δv - v + u, \end{cases}\tag{PE} \end{align*} where $λ$ and $μ$ are given spatially radial nonnegative functions and $m, κ> 0$ are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for $m,κ$ leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant $λ, μ> 0$, we find that there are initial data which lead to blow-up in (JL) if \begin{alignat*}{2} 0 \leq κ&< \min\left\{\frac{1}{2}, \frac{n - 2}{n} - (m-1)_+ \right\}&&\qquad\text{if } m\in\left[\frac{2}{n},\frac{2n-2}{n}\right)\\ \text{ or }\quad 0 \leq κ&<\min\left\{\frac{1}{2},\frac{n-1}n-\frac{m}2\right\} &&\qquad \text{if } m\in\left(0,\frac{2}{n}\right), \end{alignat*} and in (PE) if $m \in [1, \frac{2n-2}{n})$ and \begin{align*} 0 \leq κ< \min\left\{\frac{(m-1) n + 1}{2(n-1)}, \frac{n - 2 - (m-1) n}{n(n-1)} \right\}. \end{align*}

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arXiv2021

Finite-time blow-up in the three-dimensional fully parabolic attraction-dominated attraction-repulsion chemotaxis system

We show that the attraction-repulsion chemotaxis system \begin{equation*} \begin{cases} u_t = Δu - χ\nabla\cdot(u\nabla v_1) + ξ\nabla\cdot(u\nabla v_2)\\ \partial_t v_1 = Δv_1 - βv_1 + αu \\ \partial_t v_2 = Δv_2 - δv_2 + γu, \end{cases} \end{equation*} posed with homogeneous Neumann boundary conditions in bounded domains $Ω=B_R \subset \mathbb{R}^3$, $R>0$, admits radially symmetric solutions which blow-up in finite time if it is attraction-dominated in the sense that $χα-ξγ>0$.

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arXiv2021

Generalized solutions to a chemotaxis-Navier-Stokes system with arbitrary superlinear degradation

In this work, we study a chemotaxis-Navier-Stokes model in a two-dimensional setting as below, \begin{eqnarray} \left\{ \begin{array}{llll} \displaystyle n_{t}+\mathbf{u}\cdot\nabla n=Δn-\nabla \cdot(n\nabla c)+f(n), &&x\inΩ,\,t>0,\\ \displaystyle c_{t}+\mathbf{u}\cdot\nabla c=Δc - c+ n, &&x\inΩ,\,t>0,\\ \displaystyle \mathbf{u}_{t}+κ(\mathbf{u}\cdot\nabla)\mathbf{u}=Δ\mathbf{u} +\nabla P+ n\nablaϕ, &&x\inΩ,\,t>0,\\ \displaystyle \nabla\cdot\mathbf{u}=0,&&x\inΩ,\,t>0.\\ \end{array} \right. \end{eqnarray} Motivated by a recent work due to Winkler, we aim at investigating generalized solvability for the model the without imposing a critical superlinear exponent restriction on the logistic source function $f$. Specifically, it is proven in the present work that there exists a triple of integrable functions $(n,c,\mathbf{u})$ solving the system globally in a generalized sense provided that $f\in C^1([0,\infty))$ satisfies $f(0)\ge0$ and $f(n)\le rn-μn^γ$ ($n\ge0$) with any $γ>1$. Our result indicates that persistent Dirac-type singularities can be ruled out in our model under the aforementioned mild assumption on $f$. After giving the existence result for the system, we also show that the generalized solution exhibits eventual smoothness as long as $μ/r$ is sufficiently large.

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arXiv2021

Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

The chemotaxis system \begin{align*} u_t &= Δu - \nabla \cdot (u\nabla v), \\ v_t &= Δv - uv, \end{align*} is considered under the boundary conditions $\frac{\partial u}{\partialν}- u\frac{\partial v}{\partialν}=0$ and $v=v_\star$ on $\partialΩ$, where $Ω\subset\mathbb{R}^n$ is a ball and $v_\star$ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case $n=2$, while global weak solutions are constructed when $n\in \{3,4,5\}$. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.

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arXiv2021

A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system

Derived from a biophysical model for the motion of a crawling cell, the system \[(*)~\begin{cases}u_t=Δu-\nabla\cdot(u\nabla v)\\0=Δv-kv+u\end{cases}\] is investigated in a finite domain $Ω\subset\mathbb{R}^n$, $n\geq2$, with $k\geq0$. While a comprehensive literature is available for cases with $(*)$ describing chemotaxis systems and hence being accompanied by homogeneous Neumann-type boundary conditions, the presently considered modeling context, besides yet requiring the flux $\partial_νu-u\partial_νv$ to vanish on $\partialΩ$, inherently involves homogeneous Dirichlet conditions for the attractant $v$, which in the current setting corresponds to the cell's cytoskeleton being free of pressure at the boundary. This modification in the boundary setting is shown to go along with a substantial change with respect to the potential to support the emergence of singular structures: It is, inter alia, revealed that in contexts of radial solutions in balls there exist two critical mass levels, distinct from each other whenever $k>0$ or $n\geq3$, that separate ranges within which (i) all solutions are global in time and remain bounded, (ii) both global bounded and exploding solutions exist, or (iii) all nontrivial solutions blow up in finite time. While critical mass phenomena distinguishing between regimes of type (i) and (ii) belong to the well-understood characteristics of $(*)$ when posed under classical no-flux boundary conditions in planar domains, the discovery of a distinct secondary critical mass level related to the occurrence of (iii) seems to have no nearby precedent. In the planar case with the domain being a disk, the analytical results are supplemented with some numerical illustrations, and it is discussed how the findings can be interpreted biophysically for the situation of a cell on a flat substrate.

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arXiv2020

Long-term behaviour in a parabolic-elliptic chemotaxis-consumption model

Global existence and boundedness of classical solutions of the chemotaxis--consumption system \begin{align*} n_t &= Δn - \nabla \cdot (n \nabla c), \\ 0 &= Δc - nc, \end{align*} under no-flux boundary conditions for $n$ and Robin-type boundary conditions \[ \partial_ν c = (γ-c) g \] for $c$ (with $γ>0$ and $C^{1+β}(\partialΩ) \ni g > 0$ for some $β\in(0,1)$) are established in bounded domains $Ω\subset\mathbb{R}^{N}$, $N\ge 1$. Under a smallness condition on $γ$, moreover, we show convergence to the stationary solution.

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arXiv2020

Immediate smoothing and global solutions for initial data in $L^1\times W^{1,2}$ in a Keller-Segel system with logistic terms in 2D

This article deals with the logistic Keller-Segel model \[ \begin{cases} u_t = Δu - χ\nabla\cdot(u\nabla v) + κu - μu^2, \\ \\ v_t = Δv - v + u \end{cases} \] in bounded two-dimensional domains (with homogeneous Neumann boundary conditions and for parameters $χ, κ\in \mathbb{R}$ and $μ>0$), and shows that any nonnegative initial data $(u_0,v_0)\in L^1\times W^{1,2}$ lead to global solutions that are smooth in $\barΩ\times(0,\infty)$.

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arXiv2019

A Gagliardo-Nirenberg type inequality for rapidly decaying functions

We improve the Gagliardo-Nirenberg inequality \[ \|φ\|_{L^q(\mathbb{R}^n)} \le C \|\nablaφ\|_{L^r(\mathbb{R}^n)} \mathcal{L}^{-(\frac 1q - \frac{n-r}{rn})} (\|\nablaφ\|_{L^r(\mathbb{R}^n)}), \] $r=2$, $0<q<\frac{rn}{(n-r)_+}$, $\mathcal{L}$ generalizing $\mathcal{L}(s)=\ln^{-1}\frac 2s$ for $0<s<1$, from [M. Fila and M. Winkler: A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation, Adv. Math., 357 (2019), https://doi.org/10.1016/j.aim.2019.106823] for rapidly decaying functions ($φ\in W^{1,r}(\mathbb{R}^n)\setminus\{0\}$ with finite $K=\int_{\mathbb{R}^n} \mathcal{L}(φ)$) by specifying the dependence of $C$ on $K$ and by allowing arbitrary $r\ge1$.

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arXiv2019

Lack of smoothing for bounded solutions of a semilinear parabolic equation

We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all $t>0$. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as $t\to\infty$.

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arXiv2019

Continuation beyond interior gradient blow-up in a semilinear parabolic equation

It is known that there is a class of semilinear parabolic equations for which interior gradient blow-up (in finite time) occurs for some solutions. We construct a continuation of such solutions after gradient blow-up. This continuation is global in time and we give an example when it never becomes a classical solution again.

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arXiv2019

Stationary solutions to a chemotaxis-consumption model with realistic boundary conditions

Previous studies of chemotaxis models with consumption of the chemoattractant (with or without fluid) have not been successful in explaining pattern formation even in the simplest form of concentration near the boundary, which had been experimentally observed. Following the suggestions that the main reason for that is usage of inappropriate boundary conditions, in this article we study solutions to the stationary chemotaxis system \[ \begin{cases} 0 = Δn - \nabla\cdot(n\nabla c) \\ 0 = Δc - nc \end{cases} \] in bounded domains $Ω\subset\mathbb{R}^N$, $N\ge 1$, under no-flux boundary conditions for $n$ and the physically meaningful condition \[ \partial_ν c = (γ-c)g \] on $c$, with given parameter $γ>0$ and $g\in C^{1+β}(Ω)$ satisfying $g\ge 0$, $g \not\equiv 0$ on $\partial Ω$. We prove existence and uniqueness of solutions for any given mass $\int_Ωn > 0$. These solutions are non-constant.

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arXiv2018

Stabilization in the Keller--Segel system with signal-dependent sensitivity

This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{align*} &u_t = Δu - χ\nabla \cdot (uS(v)\nabla v), &v_t = Δv - v + u, \end{align*} where $χ>0$ and $S$ is a given function generalizing the sensitivity $S(s)=\frac{1}{(a+s)^{k}}$, $k>1$, $a\ge 0$, and shows exponential convergence of global classical solutions under an additional smallness condition condition for $χ>0$.

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arXiv2018

Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory

We consider the degenerate parabolic equation with nonlocal source given by \[ u_t=uΔu + u \int_{\mathbb{R}^n} |\nabla u|^2, \] which has been proposed as model for the evolution of the density distribution of frequencies with which different strategies are pursued in a population obeying the rules of replicator dynamics in a continuous infinite-dimensional setting. Firstly, for all positive initial data $u_0\in C^0(\mathbb{R}^n)$ satisfying $u_0\in L^p(\mathbb{R}^n)$ for some $p\in (0,1)$ as well as $\int_{\mathbb{R}^n} u_0=1$, the corresponding Cauchy problem in $\mathbb{R}^n$ is seen to possess a global positive classical solution with the property that $\int_{\mathbb{R}^n} u(\cdot,t)=1$ for all $t>0$. Thereafter, the main purpose of this work consists in revealing a dependence of the large time behavior of these solutions on the spatial decay of the initial data in a direction that seems unexpected when viewed against the background of known behavior in large classes of scalar parabolic problems. In fact, it is shown that all considered solutions asymptotically decay with respect to their spatial $H^1$ norm, so that \[ \mathcal{E}(t):=\int_0^t \int_{\mathbb{R}^n} |\nabla u(\cdot,t)|^2, \qquad t>0, \] always grows in a significantly sublinear manner in that \begin{equation} \frac{\mathcal{E}(t)}{t} \to 0 \qquad \mbox{as } t\to\infty; \qquad (*) \end{equation} the precise growth rate of $\mathcal{E}$, however, depends on the initial data in such a way that fast decay rates of $u_0$ enforce rapid growth of $\mathcal{E}$. To this end, examples of algebraical and certain exponential types of initial decay are detailed, inter alia generating logarithmic and arbitrary sublinear algebraic growth rates of $\mathcal{E}$, and moreover indicating that (*) is essentially optimal.

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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

A Keller-Segel-fluid system with singular sensitivity: Generalized solutions

In bounded smooth domains $Ω\subset\mathbb{R}^N$, $N\in\{2,3\}$, we consider the Keller-Segel-Stokes system \begin{align*} n_t + u\cdot \nabla n &= Δn - χ\nabla \cdot(\frac{n}{c}\nabla c),\\ c_t + u\cdot \nabla c &= Δc - c + n,\\ u_t &= Δu + \nabla P + n\nabla ϕ, \qquad \nabla \cdot u=0, \end{align*} and prove global existence of generalized solutions if \[ χ<\begin{cases} \infty,&N=2,\\ \frac{5}{3},&N=3. \end{cases} \] These solutions are such that blow-up into a persistent Dirac-type singularity is excluded.

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arXiv2018

On the global generalized solvability of a chemotaxis model with signal absorption and logistic growth terms

Introducing a suitable solution concept, we show that in bounded smooth domains $Ω\subset \mathbb{R}^n$, $n\ge 1$, the initial boundary value problem for the chemotaxis system \begin{align*} u_t&=Δu -χ\nabla\cdot\left(\frac{u}{v}\nabla v\right)+κu -μu^2,\\ v_t&=Δv -uv, \end{align*} with homogeneous Neumann boundary conditions and widely arbitrary initial data has a generalized global solution for any $μ, κ, χ>0$.

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