arXiv subjects

Mario Fuest

Publications and source records attributed to Mario Fuest.

29 records · Page 2Linked to original sources

arXiv2021

Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity

Systems of the type $$\begin{cases} u_t = \nabla \cdot (D_1(u) \nabla u - S_1(u) \nabla v) + f_1(u, v),\\ v_t = \nabla \cdot (D_2(v) \nabla v + S_2(v) \nabla u) + f_2(u, v) \end{cases} \qquad (\star)$$ can be used to model pursuit-evasion relationships between predators and prey. Apart from local kinetics given by $f_1$ and $f_2$, the key components in this system are the taxis terms $-\nabla \cdot (S_1(u) \nabla v)$ and $+\nabla \cdot (S_2(v) \nabla u)$; that is, the species are not only assumed to move around randomly in space but are also able to partially direct their movement depending on the nearby presence of the other species. In the present article, we construct global weak solutions of ($\star$) for certain prototypical nonlinear functions $D_i$, $S_i$ and $f_i$, $i \in \{1, 2\}$. To that end, we first make use of a fourth-order regularization to obtain global solutions to approximate systems and then rely on an entropy-like identity associated with ($\star$) for obtaining various a~priori estimates.

math.AP↗

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*}

math.AP↗

arXiv2020

Global solutions near homogeneous steady states in a multi-dimensional population model with both predator- and prey-taxis

We study the system \begin{align*}\label{prob:star} \tag{$\star$} \begin{cases} u_t = D_1 Δu - χ_1 \nabla \cdot (u \nabla v) + u(λ_1 - μ_1 u + a_1 v) \\ v_t = D_2 Δv + χ_2 \nabla \cdot (v \nabla u) + v(λ_2 - μ_2 v - a_2 u) \end{cases} \end{align*} (inter alia) for $D_1, D_2, χ_1, χ_2, λ_1, λ_2, μ_1, μ_2, a_1, a_2 > 0$ in smooth, bounded domains $Ω\subset \mathbb R^n$, $n \in \{1, 2, 3\}$. Without any further restrictions on these parameters, we prove that there exists a constant stable steady state $(u_\star, v_\star) \in [0, \infty)^2$, meaning that there is $\varepsilon > 0$ such that, if $u_0, v_0 \in W^{2, 2}(Ω)$ are nonnegative with $\partial_νu_0 = \partial_νv_0 = 0$ in the sense of traces and \begin{align*} \|u_0 - u_\star\|_{W^{2,2}(Ω)} + \|v_0 - v_\star\|_{W^{2,2}(Ω)} < \varepsilon, \end{align*} then there exists a global classical solution $(u, v)$ of \eqref{prob:star} with initial data $u_0, v_0$ converging to $(u_\star, v_\star)$ in $W^{2, 2}(Ω)$. Moreover, the convergence rate is exponential, except for the case $λ_2 μ_1 = λ_1 a_2$, where it is is only algebraical.

math.AP↗

arXiv2020

On the optimality of upper estimates near blow-up in quasilinear Keller--Segel systems

Solutions $(u, v)$ to the chemotaxis system \begin{align*} \begin{cases} u_t = \nabla \cdot ( (u+1)^{m-1} \nabla u - u (u+1)^{q-1} \nabla v), \\ τv_t = Δv - v + u \end{cases} \end{align*} in a ball $Ω\subset \mathbb R^n$, $n \ge 2$, wherein $m, q \in \mathbb R$ and $τ\in \{0, 1\}$ are given parameters with $m - q > -1$, cannot blow up in finite time provided $u$ is uniformly-in-time bounded in $L^p(Ω)$ for some $p > p_0 := \frac n2 (1 - (m - q))$. For radially symmetric solutions, we show that, if $u$ is only bounded in $L^{p_0}(Ω)$ and the technical condition $m > \frac{n-2 p_0}{n}$ is fulfilled, then, for any $α> \frac{n}{p_0}$, there is $C > 0$ with \begin{align*} u(x, t) \leq C |x|^{-α} \qquad \text{for all $x \in Ω$ and $t \in (0, T_{\max})$}, \end{align*} $T_{\max} \in (0, \infty]$ denoting the maximal existence time. This is essentially optimal in the sense that, if this estimate held for any $α< \frac{n}{p_0}$, then $u$ would already be bounded in $L^{p}(Ω)$ for some $p > p_0$. Moreover, we also give certain upper estimates for chemotaxis systems with nonlinear signal production, even without any additional boundedness assumptions on $u$. The proof is mainly based on deriving pointwise gradient estimates for solutions of the Poisson or heat equation with a source term uniformly-in-time bounded in $L^{p_0}(Ω)$.

math.AP↗

arXiv2020

Approaching optimality in blow-up results for Keller-Segel systems with logistic-type dampening

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system \begin{align}\label{prob:star}\tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + λu - μu^κ, \\\\ 0 = Δv - \overline m(t) + u, \quad \overline m(t) = \frac1{|Ω|} \int_Ωu(\cdot, t) \end{cases} \end{align} in smooth bounded domains $Ω\subset \mathbb R^n$, $n \ge 1$, are known to be global-in-time if $λ\geq 0$, $μ> 0$ and $κ> 2$. In the present work, we show that the exponent $κ= 2$ is actually critical in the four- and higher dimensional setting. More precisely, if \begin{alignat*}{3} \qquad n &\geq 4, &&\quad κ\in (1, 2) \quad &&\text{and} \quad μ> 0 \\\\ \text{or}\qquad n &\geq 5, &&\quad κ= 2 \quad &&\text{and} \quad μ\in \left(0, \frac{n-4}{n}\right), \end{alignat*} for balls $Ω\subset \mathbb R^n$ and parameters $λ\geq 0$, $m_0 > 0$, we construct a nonnegative initial datum $u_0 \in C^0(\overline Ω)$ with $\int_Ωu_0 = m_0$ for which the corresponding solution $(u, v)$ of \eqref{prob:star} blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for $κ\in (1, \frac32)$ (and $λ\geq 0$, $μ> 0$). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function $w(s, t) = \int_0^{\sqrt[n]{s}} ρ^{n-1} u(ρ, t) \,\mathrm dρ$ fulfills the estimate $w_s \le \frac{w}{s}$. Using this information, we then obtain finite-time blow-up of $u$ by showing that for suitably chosen initial data, $s_0$ and $γ$, the function $ϕ(t) = \int_0^{s_0} s^{-γ} (s_0 - s) w(s, t)$ cannot exist globally.

math.AP↗

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.

math.AP↗

arXiv2020

When do Keller-Segel systems with heterogeneous logistic sources admit generalized solutions?

We construct global generalized solutions to the chemotaxis system \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + λ(x) u - μ(x) u^κ,\\ v_t = Δv - v + u \end{cases} \end{align*} in smooth, bounded domains $Ω\subset \mathbb R^n$, $n \geq 2$, for certain choices of $λ, μ$ and $κ$. Here, inter alia, the selections $μ(x) = |x|^α$ with $α< 2$ and $κ= 2$as well as $μ\equiv μ_1 > 0$ and $κ> \min\{\frac{2n-2}{n}, \frac{2n+4}{n+4}\}$ are admissible (in both cases for any sufficiently smooth $λ$). While the former case appears to be novel in general, in the two- and three-dimensional setting, the latter improves on a recent result by Winkler (Adv. Nonlinear Anal. 9 (2019), no. 1, 526-566), where the condition $κ> \frac{2n+4}{n+4}$ has been imposed. In particular, for $n = 2$, our result shows that taking any $κ> 1$ suffices to exclude the possibility of collapse into a persistent Dirac distribution.

math.AP↗

arXiv2019

Blow-up profiles in quasilinear fully parabolic Keller--Segel systems

We examine finite-time blow-up solutions $(u, v)$ to \begin{align} \label{prob:star} \tag{$\star$} \begin{cases} u_t = \nabla \cdot (D(u, v) \nabla u - S(u, v) \nabla v), v_t = Δv - v + u \end{cases} \end{align} in a ball $Ω\subset \mathbb R^n$, $n \ge 2$, where $D$ and $S$ generalize the functions \begin{align*} D(u, v) = (u+1)^{m-1} \quad \text{and} \quad S(u, v) = u (u+1)^{q-1} \end{align*} with $m, q \in \mathbb R$. We show that if $m \gt \frac{n-2}{n}$ as well as $m-q \gt -\frac1n$ and $(u, v)$ is a nonnegative, radially symmetric classical solution to \eqref{prob:star} blowing up at $T_{\textrm{max}} \lt \infty$, then there exists a so-called blow-up profile $U \colon Ω\setminus \{0\} \to [0, \infty)$ satisfying \begin{align*} u(\cdot, t) \to U \quad \text{in $C_{\textrm{loc}}^2(\bar Ω\setminus \{0\})$ as $t \nearrow T_{\textrm{max}}$}. \end{align*} Moreover, for all $α\gt n$ with \begin{align*} α\gt \frac{n(n-1)}{(m-q)n + 1} \end{align*} we can find $C \gt 0$ such that \begin{align*} U(x) \le C |x|^{-α} \end{align*} for all $x \in Ω$.

math.AP↗

arXiv2019

Finite-time blow-up in a two-dimensional Keller--Segel system with an environmental dependent logistic source

The Neumann initial-boundary problem for the chemotaxis system \begin{align} \label{prob:abstract} \tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) + κ(|x|) u - μ(|x|) u^p, \\ 0 = Δv - \frac{m(t)}{|Ω|} + u, \quad m(t) := \int_Ωu(\cdot, t) \end{cases} \end{align} is studied in a ball $Ω= B_R(0) \subset \mathbb R^2$, $R \gt 0$ for $p \ge 1$ and sufficiently smooth functions $κ, μ: [0, R] \rightarrow [0, \infty)$. We prove that whenever $μ', -κ' \ge 0$ as well as $μ(s) \le μ_1 s^{2p-2}$ for all $s \in [0, R]$ and some $μ_1 \gt 0$ then for all $m_0 \gt 8 π$ there exists $u_0 \in C^0(\overline Ω)$ with $\int_Ωu_0 = m_0$ and a solution $(u, v)$ to \eqref{prob:abstract} with initial datum $u_0$ blowing up in finite time. If in addition $κ\equiv 0$ then all solutions with initial mass smaller than $8 π$ are global in time, displaying a certain critical mass phenomenon. On the other hand, if $p \gt 2$, we show that for all $μ$ satisfying $μ(s) \ge μ_1 s^{p-2-\varepsilon}$ for all $s \in [0, R]$ and some $μ_1, \varepsilon \gt 0$ the system \eqref{prob:abstract} admits a global classical solution for each initial datum $0 \le u_0 \in C^0(\overline Ω)$

math.AP↗

arXiv2019

Analysis of a chemotaxis model with indirect signal absorption

We consider the chemotaxis model \begin{align*} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v), \\ v_t = Δv - vw, \\ w_t = -δw + u \end{cases} \end{align*} in smooth, bounded domains $Ω\subset \mathbb R^n$, $n \in \mathbb N$, where $δ\gt 0$ is a given parameter. If either $n \le 2$ or $\|v_0\|_{L^\infty(Ω)} \le \frac1{3n}$ we show the existence of a unique global classical solution $(u, v, w)$ and convergence of $(u(\cdot, t), v(\cdot, t), w(\cdot, t))$ towards a spatially constant equilibrium, as $t \to \infty$. The proof of global existence for the case $n \le 2$ relies on a bootstrap procedure. As a starting point we derive a functional inequality for a functional being sublinear in $u$, which appears to be novel in this context.

math.AP↗

arXiv2018

Boundedness enforced by mildly saturated conversion in a chemotaxis-May-Nowak model for virus infection

We study the system \begin{align*} \label{prob:star} \tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u \nabla v) - u - f(u) w + κ, \\ v_t = Δv - v + f(u) w, \\ w_t = Δw - w + v, \end{cases} \end{align*} which models the virus dynamics in an early stage of an HIV infection, in a smooth, bounded domain $Ω\subset \mathbb R^n, n \in \mathbb N,$ for a parameter $κ\ge 0$ and a given function $f \in C^1([0, \infty))$ satisfying $f \ge 0$, $f(0) = 0$ and $f(s) \le K_f s^α$ for all $s \ge 1$, some $K_f \gt 0$ and $α\in \mathbb R$. We prove that whenever \begin{align*} α\lt \frac2n, \end{align*} solutions to \eqref{prob:star} exist globally and are bounded. The proof mainly relies on smoothing estimates for the Neumann heat semigroup and (in the case $α\gt 1$) on a functional inequality. Furthermore, we provide some indication why the exponent $\frac2n$ could be essentially optimal.

math.AP↗