arXiv subjects

Michael Winkler

Publications and source records attributed to Michael Winkler.

At least 55 records · Page 3Linked to original sources

arXiv2020

Conditional estimates in three-dimensional chemotaxis-Stokes systems and application to a Keller-Segel-fluid model accounting for gradient-dependent flux limitation

This manuscript deals with the three-dimensional version of a flux-limited Keller-Segel system coupled to the incompressible Stokes equations through transport and buoyancy. The main goal consists in verifying that within a certain parameter regime, known as being optimal therefor in some fluid-free simplification, a feature of blow-up prevention by suitably strong flux limitation persists also in the framework of the considered full chemotaxis-fluid system. To achieve this, as a secondary objective of possibly independent interest the manuscript separately establishes some conditional bounds for fluid fields and taxis gradients in a fairly general setting. The application thereof to the specific problem under consideration thereafter facilitates the derivation of a result on global existence of bounded classical solutions for widely arbitrary initial data actually, indeed within the entire and essentially optimal parameter range, through an argument which appears to be signficantly condensed when compared to reasonings pursued in previous related works.

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arXiv2020

Asymptotic stability of spatial homogeneity in a haptotxis model for oncolytic virotherapy

This work considers a model for oncolytic virotherapy, as given by the reaction-diffusion-taxis system $$ \left\{ \begin{array}{l} u_t = Δu - \nabla \cdot (u\nabla v)-ρuz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = D_w Δw - w + uz, \\[1mm] z_t = D_z Δz - z - uz + βw, \end{array} \right. $$ in a smoothly bounded domain $Ω\subset\mathbb{R}^2$, with parameters $D_w>0, D_z>0, β>0$ and $ρ\ge 0$.\\ % Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if $β<1$, whereas whenever $β>1$ and $\frac{1}{|Ω|}\int_Ω u(\cdot,0)>\frac{1}{β-1}$, infinite-time blow-up occurs at least in the particular case when $ρ=0$.\abs % In order to provide an appropriate complement to this, the present work reveals that for any $ρ\ge 0$ and arbitrary $β>0$, at each prescribed level $γ\in (0,\frac{1}{(β-1)_+})$ one can identify an $L^\infty$-neighborhood of the homogeneous distribution $(u,v,w,z)\equiv (γ,0,0,0)$ within which all initial data lead to globally bounded solutions that stabilize toward the constant equilibrium $(u_\infty,0,0,0)$ with some $u_\infty>0$.

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arXiv2020

Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation

We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Short et al. (Math. Mod. Meth. Appl. Sci. 18, 2008). The focus here is on the question how far a certain nonlinear enhancement in the random diffusion of criminal agents may exert visible relaxation effects. Specifically, in the context of the system \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot (u^{m-1} \nabla u) - χ\nabla \cdot \Big(\frac{u}{v} \nabla v \Big) - uv + B_1(x,t), \\[1mm] v_t = Δv +uv - v + B_2(x,t), \end{array} \right. \end{eqnarray*} it is shown that whenever $χ>0$ and the given nonnegative source terms $B_1$ and $B_2$ are sufficiently regular, the assumption \begin{eqnarray*} m>\frac{3}{2} \end{eqnarray*} is sufficient to ensure that a corresponding Neumann-type initial-boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is seen to be globally bounded if both $B_1$ and $B_2$ are bounded and $\liminf_{t\to\infty} \int_ΩB_2(\cdot,t)$ is positive. This is supplemented by numerical evidence which, besides illustrating associated smoothing effects in particular situations of sharply structured initial data in the presence of such porous medium type diffusion mechanisms, indicates a significant tendency toward support of singular structures in the linear diffusion case $m=1$.

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arXiv2020

Existence theory and qualitative analysis for a fully cross-diffusive predator-prey system

This manuscript considers a Neumann initial-boundary value problem for the predator-prey system $$ \left\{ \begin{array}{l} u_t = D_1 u_{xx} - χ_1 (uv_x)_x + u(λ_1-u+a_1 v), \\[1mm] v_t = D_2 v_{xx} + χ_2 (vu_x)_x + v(λ_2-v-a_2 u), \end{array} \right. \qquad \qquad (\star) $$ in an open bounded interval $Ω$ as the spatial domain, where for $i\in\{1,2\}$ the parameters $D_i, a_i, λ_i$ and $χ_i$ are positive. Due to the simultaneous appearance of two mutually interacting taxis-type cross-diffusive mechanisms, one of which even being attractive, it seems unclear how far a solution theory can be built upon classical results on parabolic evolution problems. In order to nevertheless create an analytical setup capable of providing global existence results as well as detailed information on qualitative behavior, this work pursues a strategy via parabolic regularization, in the course of which ($\star$) is approximated by means of certain fourth-order problems involving degenerate diffusion operators of thin film type. During the design thereof, a major challenge is related to the ambition to retain consistency with some fundamental entropy-like structures formally associated with ($\star$); in particular, this will motivate the construction of an approximation scheme including two free parameters which will finally be fixed in different ways, depending on the size of $λ_2$ relative to $a_2 λ_1$.

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arXiv2019

Does indirectness of signal production reduce the explosion-supporting potential in chemotaxis-haptotaxis systems? Global classical solvability in a class of models for cancer invasion (and more)

We propose and study a class of parabolic-ODE models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumor microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature consituting a core difference to both classical Keller-Segel chemotaxis systems and Chaplain-Lolas type chemotaxis-haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

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arXiv2019

On the global existence and qualitative behavior of one-dimensional solutions to a model for urban crime

We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system \begin{eqnarray*} \left\{ \begin{array}{ll} u_t = u_{xx} - χ\big(\frac{u}{v} \partial_x u \big)_x - uv +B_1(x,t), \qquad & x\in Ω, \ t>0, \\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \qquad & x\in Ω, \ t>0, \end{array} \right. \end{eqnarray*} which was introduced by Short et al. in [Short2008] with $χ=2$ to describe the dynamics of urban crime In bounded intervals $Ω\subset\mathbb{R}$ and with prescribed suitably regular nonnegative functions $B_1$ and $B_2$, we first prove the existence of global classical solutions for any choice of $χ>0$ and all reasonably regular nonnegative initial data. We next address the issue of determining the qualitative behavior of solutions under appropriate assumptions on the asymptotic properties of $B_1$ and $B_2$. Indeed, for arbitrary $χ>0$ we obtain boundedness of the solutions given strict positivity of the average of $B_2$ over the domain; moreover, it is seen that imposing a mild decay assumption on $B_1$ implies that $u$ must decay to zero in the long-term limit. Our final result, valid for all $χ\in\left(0,\frac{\sqrt{6\sqrt{3}+9}}{2}\right),$ which contains the relevant value $χ=2$, states that under the above decay assumption on $B_1$, if furthermore $B_2$ appropriately stabilizes to a nontrivial function $B_{2,\infty}$, then $(u,v)$ approaches the limit $(0,v_\infty)$, where $v_\infty$ denotes the solution of \begin{eqnarray*} \left\{ \begin{array}{l} -\partial_{xx}v_\infty + v_\infty = B_{2,\infty}, \qquad x\in Ω, \\[1mm] \partial_x v_{\infty}=0, \qquad x\in\partialΩ. \end{array} \right. \end{eqnarray*}

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arXiv2019

Analysis of a one-dimensional forager-exploiter model

\begin{abstract} \noindent % We consider the one-dimensional parabolic system The system \bas \left\{ \begin{array}{l} u_t= u_{xx} - χ_1 (uw_x)_x, \\[1mm] v_t = v_{xx} - χ_2 (vu_x)_x, \\[1mm] w_t = dw_{xx} - λ(u+v)w - μw + r, \end{array} \right. \eas % that has been proposed as a model to describe social interactions within mixed forager-exploiter groups. is considered in a bounded real interval, with positive parameters $χ_1,χ_2,d,λ$ and $μ$, and with $r \ge 0$. Proposed to describe social interactions within mixed forager-exploiter groups, this model extends classical one-species chemotaxis-consumption systems by additionally accounting for a second axis mechanism coupled to the first in a consecutive manner. \abs % It is firstly shown that for all suitably regular initial data $(u_0, v_0, w_0)$, an associated Neumann-type initial-boundary value problem possesses a globally defined bounded classical solution. Moreover, it is asserted that this solution stabilizes to a spatially homogeneous equilibrium at an exponential rate under a smallness condition on $\min\{\io u_0, \io v_0\}$ that appears to be consistent with predictions obtained from formal stability analysis.\abs

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arXiv2018

Does fluid interaction affect regularity in the three-dimensional Keller-Segel system with saturated sensitivity?

A class of Keller-Segel-Stokes systems generalizing the prototype \[ \left\{ \begin{array}{rcl} n_t + u\cdot\nabla n &=& Δn - \nabla \cdot \Big(n(n+1)^{-α}\nabla c\Big), c_t + u\cdot\nabla c &=& Δc-c+n, u_t +\nabla P &=& Δu + n \nabla ϕ+ f(x,t), \qquad \nabla\cdot u =0, \end{array} \right. \qquad \qquad (\star) \] is considered in a bounded domain $Ω\subset R^3$, where $ϕ$ and $f$ are given sufficiently smooth functions such that $f$ is bounded in $Ω\times (0,\infty)$. It is shown that under the condition that \[ α>\frac{1}{3}, \] for all sufficiently regular initial data a corresponding Neumann-Neumann-Dirichlet initial-boundary value problem possesses a global bounded classical solution. This extends previous findings asserting a similar conclusion only under the stronger assumption $α>\frac{1}{2}$. In view of known results on the existence of exploding solutions when $α<\frac{1}{3}$, this indicates that with regard to the occurrence of blow-up the criticality of the decay rate $\frac{1}{3}$, as previously found for the fluid-free counterpart of ($\star$), remains essentially unaffected by fluid interaction of the type considered here.

math.AP↗

arXiv2018

Migration-driven benefit in a two-species nutrient taxis system

A model describing the competition of two species for a common nutrient is studied. It is assumed that one of the competitors is motionless while the other has the ability to move upwards gradients of the nutrient density. It is proved that under suitable assumptions on the initial data, in the long time perspective the ability to move turns out to be a crucial feature providing competitive advantage irrespectively of a possible difference between the species with regard to their rates of proliferation and nutrient uptake.

math.AP↗

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

The fast signal diffusion limit in Keller-Segel(-fluid) systems

This paper deals with convergence of solutions to a class of parabolic Keller-Segel systems, possibly coupled to the (Navier-)Stokes equations in the framework of the full model \begin{eqnarray*} \left\{ \begin{array}{lcl} \, \, \partial_t n_ε + u_ε \cdot \nabla n_ε &=& Δn_ε - \nabla \cdot \Big( n_ε S(x, n_ε, c_ε)\cdot\nabla c_ε\Big) + f(x, n_ε, c_ε), \\[1mm] ε\partial_t c_ε + u_ε\cdot\nabla c_ε &=& Δc_ε - c_ε + n_ε , \\[1mm] \,\,\partial_t u_ε + κ(u_ε\cdot\nabla) u_ε &=& Δu_ε + \nabla P_ε + n_ε \nablaϕ, \qquad \nabla\cdot u_ε=0 \end{array} \right. \end{eqnarray*} to solutions of the parabolic-elliptic counterpart formally obtained on taking $ε\searrow 0$. In smoothly bounded physical domains $Ω\subset {\mathbb R}^{N}$ with $N\ge 1$, and under appropriate assumptions on the model ingredients, we shall first derive a general result which asserts certain strong and pointwise convergence properties whenever asserting that supposedly present bounds on $\nabla c_ε$ and $u_ε$ are bounded in $L^λ((0,T);L^q(Ω))$ and in $L^\infty((0,T);L^r(Ω))$, respectively, for some $λ\in (2,\infty]$, $q>N$ and $r>\max\{2,N\}$ such that $\frac{1}λ+\frac{N}{2q}<\frac{1}{2}$. To our best knowledge, this seems to be the first rigorous mathematical result on a fast signal diffusion limit in a chemotaxis-fluid system.

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arXiv2018

Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions $n\ge 3$

We study the blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in space dimensions $n\ge 3$. In view of the biological background of this system and of its mass conservation property, blowup is usually interpreted as a phenomenon of concentration or aggregation of the bacterial population. Understanding the asymptotic behavior of solutions at the blowup time is thus meaningful for the interpretation of the model. Under mild assumptions on the initial data, for $n\ge 3$, we show that the final profile satisfies $C_1|x|^{-2}\le u(x,T)\le C_2|x|^{-2}$, with convergence in $L^1$ as $t\to T$. This is in sharp contrast with the two-dimensional case, where solutions are known to concentrate to a Dirac mass at the origin (plus an integrable part). We also obtain refined space-time estimates of the form $u(x,t)\le C(T-t+|x|^2)^{-1}$ for type~I blowup solutions. Previous work had shown that radial, self-similar blowup solutions (which satisfy the above estimates) exist in dimensions $n\ge 3$ and do not exist in dimension $2$. Our results thus reveal that the final profile displayed by these special solutions actually corresponds to a much more general phenomenon.

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arXiv2017

A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation

We establish a Gagliardo-Nirenberg-type inequality in $\mathbb{R}^n$ for functions which decay fast as $|x|\to\infty$. We use this inequality to derive upper bounds for the decay rates of solutions of a degenerate parabolic equation. Moreover, we show that these upper bounds, hence also the Gagliardo-Nirenberg-type inequality, are sharp in an appropriate sense.

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arXiv2017

A Singular Differential Equation Stemming from an Optimal Control Problem in Financial Economics

We consider the ordinary differential equation $x^2 u'' = axu'+bu-c(u'-1)^2, x\in (0,x_0)$, with $a\in\mathbb{R}, b\in\mathbb{R}$, $c>0$ and the singular initial condition $u(0)=0$, which in financial economics describes optimal disposal of an asset in a market with liquidity effects. It is shown in the paper that if $a+b < 0$ then no solutions exist, whereas if $a+b\ge0$ then there are infinitely many solutions with indistinguishable asymptotics near 0. Moreover, it is proved that in the latter case there is precisely one solution $u$ corresponding to the choice $x_0=\infty$ which is such that $0 \le u(x)\le x$ for all $x>0$, and that this solution is strictly increasing and concave.

math.OC↗

arXiv2017

Singular structure formation in a degenerate haptotaxis model involving myopic diffusion

We consider the system \[ u_t=\big(d(x)u\big)_{xx} - \big(d(x)uw_x\big)_x, \quad w_t=-ug(w), \] which arises as a simple model for haptotactic migration in heterogeneous environments, such as typically occurring in the invasive dynamics of glioma. A particular focus is on situations when the diffusion herein is degenerate in the sense that the zero set of $d$ is not empty. It is shown that if such possibly present degeneracies are sufficiently mild in the sense that \[ \int_Ω\frac{1}{d}<\infty, \] then under appropriate assumptions on the initial data a corresponding initial-boundary value problem, posed under no-flux boundary conditions in a bounded open real interval $Ω$, possesses at least one globally defined generalized solution. Moreover, despite such degeneracies the considered myopic diffusion mechanism is seen to asymptotically determine the solution behavior in the sense that for some constant $μ_\infty>0$, the obtained solution satisfies \[ u(\cdot,t)\rightharpoonup \frac{μ_\infty}{d} \ \mbox{in } L^1(Ω) \quad \mbox{and} \quad w(\cdot,t) \to 0 \ \mbox{in } L^\infty(Ω) \quad \mbox{as } t\to\infty, \qquad (\star) \] and that hence in the degenerate case the solution component $u$ stabilizes toward a state involving infinite densities, which is in good accordance with experimentally observed phenomena of cell aggregation. Finally, under slightly stronger hypotheses inter alia requiring that $\frac{1}{d}$ belong to $L\log L(Ω)$, a substantial effect of diffusion is shown to appear already immediately by proving that for a.e.~$t>0$, the quantity $\ln (du(\cdot,t))$ is bounded in $Ω$. In degenerate situations, this particularly implies that the blow-up phenomena expressed in ($\star$) in fact occur instantaneously.

math.AP↗

arXiv2017

Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

A class of chemotaxis-Stokes systems generalizing the prototype \[\left\{ \begin{array}{rcl} n_t + u\cdot\nabla n &=& \nabla \cdot \big(n^{m-1}\nabla n\big) - \nabla \cdot \big(n\nabla c\big), c_t + u\cdot\nabla c &=& Δc-nc, u_t +\nabla P &=& Δu + n \nabla ϕ, \qquad \nabla\cdot u =0, \end{array} \right. \] is considered in bounded convex three-dimensional domains, where $ϕ\in W^{2,\infty}(Ω)$ is given. The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that \[ m>\frac{9}{8}. \qquad (\star) \] Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state $(\frac{1}{|Ω|} \int_Ωn_0,0,0)$ in the large time limit. This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than ($\star$).

math.AP↗

arXiv2017

Critical mass for infinite-time aggregation in a chemotaxis model with indirect signal production

We study the Neumann initial-boundary problem for the chemotaxis system $$ \left\{\begin{array}{ll} u_t= Δu - \nabla \cdot (u\nabla v), & x\in Ω, \, t>0, 0=Δv - μ(t)+w, & x\in Ω, \, t>0, τw_t + δw = u, & x\in Ω, \, t>0, \end{array} \right. \qquad \qquad (\star) $$ in the unit disk $Ω:=B_1(0)\subset \R^2$, where $δ\ge 0$ and $τ>0$ are given parameters and $μ(t):=\mint_Ωw(x,t)dx$, $t>0$. It is shown that this problem exhibits a novel type of critical mass phenomenon with regard to the formation of singularities, which drastically differs from the well-known threshold property of the classical Keller-Segel system, as obtained upon formally taking $τ\to 0$, in that it refers to blow-up in infinite time rather than in finite time: Specifically, it is first proved that for any sufficiently regular nonnegative initial data $u_0$ and $w_0$, ($\star$) possesses a unique global classical solution. In particular, this shows that in sharp contrast to classical Keller-Segel-type systems reflecting immediate signal secretion by the cells themselves, the indirect mechanism of signal production in ($\star$) entirely rules out any occurrence of blow-up in finite time. However, within the framework of radially symmetric solutions it is next proved that whenever $δ>0$ and $\io u_0<8πδ$, the solution remains uniformly bounded, whereas for any choice of $δ\ge 0$ and $m>8πδ$, one can find initial data such that $\io u_0=m$, and such that for the corresponding solution we have \bas \|u(\cdot,t)\|_{L^\infty(Ω)} \to \infty \qquad \mbox{as} t\to\infty.

math.AP↗

arXiv2017

A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: Global solvability for large nonradial data

The chemotaxis system \[ \left\{ \begin{array}{l} u_t = Δu - χ\nabla \cdot (\frac{u}{v}\nabla v), v_t=Δv - v+u, \end{array} \right. \] is considered in a bounded domain $Ω\subset \mathbb{R}^n$ with smooth boundary, where $χ>0$. An apparently novel type of generalized solution framework is introduced within which an extension of previously known ranges for the key parameter $χ$ with regard to global solvability is achieved. In particular, it is shown that under the hypothesis that\[ χ< \left\{ \begin{array}{ll} \infty \qquad & \mbox{if } n=2, \sqrt{8} \qquad & \mbox{if } n=3, \frac{n}{n-2} \qquad & \mbox{if } n\ge 4, \end{array} \right. \] for all initial data satisfying suitable assumptions on regularity and positivity, an associated no-flux initial-boundary value problem admits a globally defined generalized solution. This solution inter alia has the property that \[ u\in L^1_{loc}(\barΩ\times [0,\infty)). \]

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