391 citations · 668 across the 36 of their papers we have counts for
44 papers
Depleting the signal: Analysis of chemotaxis-consumption models -- A survey
Johannes Lankeit, Michael Winkler
We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototy…
Phenotype switching in chemotaxis aggregation models controls the spontaneous emergence of large densities
Kevin J Painter, Michael Winkler
We consider a phenotype-switching chemotaxis model for aggregation, in which a chemotactic population is capable of switching back and forth between a chemotaxing state (performing…
A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion
Michael Winkler
The taxis-type migration-consumption model accounting for signal-dependent motilities, as given by \[ u_t = Δ\big(uϕ(v)\big), v_t = Δv-uv, \qquad (*) \] is considered for suitably…
Refined regularity analysis for a Keller-Segel-consumption system involving signal-dependent motilities
Genglin Li, Michael Winkler
We consider the Keller-Segel-type migration-consumption system involving signal-dependent motilities, $$\left\{ \begin{array}{l} u_t = Δ\big(uϕ(v)\big), \\[1mm] v_t = Δv-uv, \end{a…
Relaxation in a Keller-Segel-consumption system involving signal-dependent motilities
Genglin Li, Michael Winkler
Two relaxation features of the migration-consumption chemotaxis system involving signal-dependent motilities, $$ \left\{ \begin{array}{l} u_t = Δ\big(uϕ(v)\big), \\[1mm] v_t = Δv-u…
Weak solutions to triangular cross diffusion systems modeling chemotaxis with local sensing
Laurent Desvillettes, Philippe Laurençot, Ariane Trescases +1
New estimates and global existence results are provided for a class of systems of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly fea…
Global existence in reaction-diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects
Johannes Lankeit, Michael Winkler
We introduce a generalized concept of solutions for reaction-diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, qu…
Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary
Johannes Lankeit, Michael Winkler
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}…
A double critical mass phenomenon in a no-flux-Dirichlet Keller-Segel system
Jan Fuhrmann, Johannes Lankeit, Michael Winkler
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…
A critical blow-up exponent for flux limitation in a Keller-Segel system
Michael Winkler
The parabolic-elliptic cross-diffusion system \[ \left\{ \begin{array}{l} u_t = Δu - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm] 0 = Δv - μ+ u, \qquad \int_Ωv=0, \qq…
Conditional estimates in three-dimensional chemotaxis-Stokes systems and application to a Keller-Segel-fluid model accounting for gradient-dependent flux limitation
Michael Winkler
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 ma…
Asymptotic stability of spatial homogeneity in a haptotxis model for oncolytic virotherapy
Youshan Tao, Michael Winkler
This work considers a model for oncolytic virotherapy, as given by the reaction-diffusion-taxis system $$ \left\{ \begin{array}{l} u_t = Δu - \nabla \cdot (u\nabla v)-ρuz, \\[1mm]…
Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation
Nancy Rodriguez, Michael Winkler
We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Short et al. (Math. Mod. Meth. Appl. Sci. 18, 2008). The foc…
Existence theory and qualitative analysis for a fully cross-diffusive predator-prey system
Youshan Tao, Michael Winkler
This manuscript considers a Neumann initial-boundary value problem for the predator-prey system $$ \left\{ \begin{array}{l} u_t = D_1 u_{xx} - χ_1 (uv_x)_x + u(λ_1-u+a_1 v), \\[1mm…
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)
Christina Surulescu, Michael Winkler
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…
On the global existence and qualitative behavior of one-dimensional solutions to a model for urban crime
Nancy Rodriguez, Michael Winkler
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} \partia…
Analysis of a one-dimensional forager-exploiter model
Youshan Tao, Michael Winkler
\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)…
Does fluid interaction affect regularity in the three-dimensional Keller-Segel system with saturated sensitivity?
Michael Winkler
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\cd…
Migration-driven benefit in a two-species nutrient taxis system
Piotr Krzyzanowski, Michael Winkler, Dariusz Wrzosek
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…
Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory
Johannes Lankeit, Michael Winkler
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 o…
The fast signal diffusion limit in Keller-Segel(-fluid) systems
Yulan Wang, Michael Winkler, Zhaoyin Xiang
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 \b…
Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions
Philippe Souplet, Michael Winkler
We study the blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel-Patlak system in space dimensions . In view of the biological backg…
A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation
Marek Fila, Michael Winkler
We establish a Gagliardo-Nirenberg-type inequality in for functions which decay fast as . We use this inequality to derive upper bounds for the decay r…
Singular structure formation in a degenerate haptotaxis model involving myopic diffusion
Michael Winkler
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 environmen…