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Michael Winkler

Publications and source records attributed to Michael Winkler.

At least 19 recordsLinked to original sources
arXiv2026

Global smooth behavior in Kuznetsov and Westervelt type viscous wave equations: A unifying approach covering $W^{1,q}$-small initial data

In a smoothly bounded domain $\Om\subset\R^n$ with $n\geq 1$ and $a>0$, we consider an initial-boundary value problem for the general viscous wave equation \bas h(u,u_t) u_{tt} = \Del u_t + a\Del u + f(u,u_t,\na u,\na u_t) \eas which appears in models of nonlinear acoustics wave propagation; well-established equations of Kuznetsov and Westervelt type form particular examples.\abs % While the existing literature offers extensive results on global solutions for sufficiently small initial data $(u_{0}, u_{0t})=(u, u_{t})|_{t=0}$ in second- and higher-order Sobolev spaces it appears to remain open how far global solvability can be established under smallness conditions involving only first-order Sobolev spaces. The present manuscript addresses this question by proving the existence of global classical solutions together with exponential decay of the pair $(u,u_{t})$ in $ W^{1,r}\times W^{1,p}$-Sobolev spaces whenever the nonlinearities $h$ and $f$ are sufficiently smooth and are such that $h(0,0)>0$ as well as $f(0,0,0,0)=0$ and $\na f(0,0,0,0)=0$.

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arXiv2026

Preventing $L^p$ blow-up by local anisotropy of signal production in the Keller-Segel system with strongly differing diffusion rates

In a smoothly bounded domain $\Omega\subset R^n$, $n\le 5$, the manuscript considers the variant of the Keller-Segel system given by \[ \left\{ \begin{array}{l} u_t = D \Delta u - \nabla \cdot (u\nabla v), \\[1mm] v_t = d \Delta v + \nabla \cdot (u\nabla v) - v + u, \end{array} \right. \] which involves an additional contribution $\nabla \cdot (u\nabla v)$ to the chemoattractant evolution, in line with refined modeling literature reflecting an anisotropic correction to the isotropic signal production term $+u$ in the classical Keller-Segel model. It is shown that for arbitrary $D>0$ and $d>0$ and any nonnegative intial data from $W^{1,\infty}(\Omega)\times W^{1, \infty}(\Omega)$, an associated Neumann problem admits a global weak solution $(u,v)$ which, inter alia, satisfies \[ \sup_{t \in (0,\infty)\setminus N} \int_\Omega e^{u^\alpha(\cdot,t)} < \infty \] with some $\alpha>0$ and some null set $N\subset (0,\infty)$.

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arXiv2026

Refined temporal asymptotics near blow-up points in the planar Keller-Segel system

For the Keller-Segel system \[ \left\{\, \begin{aligned} u_t &= \Delta u - \nabla \cdot ( u \nabla v ), \\ v_t &= \Delta v - v + u \end{aligned} \right. \tag{$\star$} \] posed in a planar domain $\Omega$ with Neumann boundary conditions, the existence of classical solutions blowing up at some finite time $T$ has long been established. In fact, it has been shown that for every blow-up point $x$ the quantity $\int_{B_R(x)\cap\Omega} u(\cdot,t )\ln(u(\cdot, t))$ is unbounded as $t\nearrow T$ for all $R > 0$ even though the global mass of $u$ is always conserved. The present manuscript provides some quantitative information on the behavior of such localized $L\log L$ expressions by asserting the existence of $\delta_0=\delta_0(\Omega)>0$ such that any solution to the Neumann problem for ($\star$) blowing up at time $T\in (0,\infty)$ satisfies \[ \limsup_{t\nearrow T} \frac{1}{\ln\frac{T}{T-t}}\int_{B_R(x)\cap\Omega} u(\cdot, t)\ln(u(\cdot, t)) \ge \delta_0 \tag{$\star\star$} \] for all $R > 0$ at each blow-up point $x$. This confirms a certain universality property of the blow-up mechanism seen in the particular examples of radial collapsing solutions constructed in the seminal work [16], especially also beyond the realm of symmetry; apart from that, along with a consequence of ($\star\star$) on the corresponding asymptotics of similarly localized $L^p$ norms of $u$ for $p\in (1,\infty]$, this provides some extension of a known result on non-degeneracy of blow-up points that has concentrated on the choice $p=\infty$ here.

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arXiv2026

Global solutions and large time stabilization in a model for thermoacoustics in a standard linear solid

This manuscript is concerned with the one-dimensional system \[ \begin{array}{l} \tau u_{ttt} + \alpha u_{tt} = b \big(\gamma(\Theta) u_{xt}\big)_x + \big( \gamma(\Theta) u_x\big)_x, \\[1mm] \Theta_t = D \Theta_{xx} + b\gamma(\Theta) u_{xt}^2, \end{array} \] which is connected to the simplified modeling of heat generation in Zener type materials subject to stress from acoustic waves. Under the assumption that the coefficients $\tau>0, b>0$ and $\alpha\geq0$ satisfy \begin{align}\tag{$\star$} \alpha b >\tau, \end{align} it is shown that for all $\Theta_\star>0$ one can find $\nu=\nu(D,\tau,\alpha,b,\Theta_\star,\gamma)>0$ such that an associated Neumann type initial-boundary value problem with Neumann data admits a unique time-global solution in a suitable framework of strong solvability whenever the initial temperature distribution fulfills $$\|\Theta_0\|_{L^\infty(\Omega)}\leq \Theta_\star$$ and the derivatives of the initial data are sufficiently small in the sense of satisfying $$\int_\Omega u_{0xx}^2 + \int_\Omega (u_{0t})_{xx}^2 + \int_\Omega (u_{0tt})_x^2 < \nu\quad\text{and}\quad \|\Theta_{0x}\|_{L^\infty(\Omega)} + \|\Theta_{0xx}\|_{L^\infty(\Omega)} < \nu.$$ The constructed solution moreover features an exponential stabilization property for both components. In particular, the parameter range described by ($\star$) coincides with the full stability regime known for the corresponding Moore--Gibson--Thompson equation despite the fairly strong nonlinear coupling to the temperature variable.

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arXiv2026

Large time stabilization of rough-data solutions in one-dimensional nonlinear thermoelasticity

In an open bounded real interval $\Omega$, the model for one-dimensional thermoelasticity given by \[ u_{tt} = u_{xx} - \big(f(\Theta)\big)_x, \qquad \Theta_t = \Theta_{xx} - f(\Theta) u_{xt}, \] is considered along with homogeneous boundary conditions of Dirichlet type for $u$ and of Neumann type for $\Theta$, under the assumption that $f\in C^1([0,\infty))$ satisfies $f(0)=0$, $f'\in L^\infty((0,\infty))$ and $f'>0$ on $[0,\infty)$. The focus is on initial data which are merely required to be consistent with the fundamental principles of energy conservation and entropy nondecrease, by satisfying \[ u_0\in W_0^{1,2}(\Omega), u_{0t} \in L^2(\Omega), 0 \le \Theta_0\in L^1(\Omega), \Theta_0 \not\equiv 0. \] Despite an apparent lack of favorable compactness properties that have underlain previous related studies on more regular settings, it is shown that corresponding weak solutions stabilize in the sense that \[ \lim_{t\to\infty} \|u(\cdot,t)\|_{L^\infty(\Omega)}=0 \] and \[ {\rm ess} \lim_{\!\!\!\! t\to\infty} \|\Theta(\cdot,t)-\Theta_\infty\|_{L^\infty(\Omega)}=0 \] with some $\Theta_\infty>0$.

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arXiv2026

A simple model for one-dimensional nonlinear thermoelasticity: Well-posedness in rough-data frameworks

In an open bounded interval $\Omega$, the problem \[ u_{tt} = u_{xx} - \big(f(\Theta)\big)_x, \Theta_t = \Theta_{xx} - f(\Theta) u_{xt}, \] is considered under the boundary conditions $u|_{\partial\Omega}=\Theta_x|_{\partial\Omega}=0$, and for $f\in C^2([0,\infty))$ satisfying $f(0)=0$, $f'>0$ on $[0,\infty)$ and $f'\in W^{1,\infty}((0,\infty))$. In the sense of unconditional global existence, uniqueness and continuous dependence, this problem is shown to be well-posed within ranges of initial data merely satisfying \[ u_0\in W_0^{1,2}(\Omega), \quad u_{0t} \in L^2(\Omega) \quad \mbox{and} \quad \Theta_0 \in L^2(\Omega) \mbox{ with $\Theta\ge 0$ a.e.~in $\Omega$,} \] and in classes of solutions fulfilling \[ u\in C^0([0,\infty);W_0^{1,2}(\Omega)), \qquad u_t \in C^0([0,\infty);L^2(\Omega)) \qquad \mbox{and} \qquad \Theta\in C^0([0,\infty);L^2(\Omega)) \cap L^2_{loc}([0,\infty);W^{1,2}(\Omega)). \]

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arXiv2026

Global solvability and stabilization in multi-dimensional small-strain nonlinear thermoviscoelasticity

Despite considerable developments in the literature of the past decades, a standing open problem in the analysis of continuum mechanics appears to consist of determining how far the prototypical model for small-strain thermoviscoelastic evolution in Kelvin-Voigt materials with inertia, as given by \[ u_{tt} = \mu \Delta u_t + (\lambda+\mu)\nabla\nabla\cdot u_t + \hat{\mu} \Delta u + (\hat{\lambda}+\hat{\mu}) \nabla\nabla\cdot u - B\nabla\Theta, \qquad \qquad \kappa \Theta_t = D\Delta\Theta + \mu |\nabla u_t|^2 + (\lambda+\mu) |{\rm div} \, u_t|^2 - B\Theta {\rm div} \, u_t, \qquad \qquad \qquad (\star) \] is globally solvable in multi-dimensional settings and for initial data of arbitrary size. The present manuscript addresses this in the context of an initial value problem in smoothly bounded $n$-dimensional domains with $n\ge 2$, posed under homogeneous boundary conditions of Dirichlet type for the displacement variable $u$, and of Neumann type for the temperature $\Theta$. Within suitably generalized concepts of solvability, global existence of solutions is shown without any size restrictions on the data, and for a system actually more general than ($\star$) by, inter alia, allowing the heat capacity $\kappa$ to depend on $\Theta$. Apart from that, results on large time behavior are derived which particularly assert stabilization of $\Theta$ toward a spatially homogeneous limit. Besides on standard features related to energy conservation and entropy production, in its core parts the analysis relies on an evolution property of certain logarithmic refinements of classical entropy functionals, to the best of our knowledge undiscovered in precedent literature and possibly of independent interest.

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arXiv2026

Hotspot formation driven by temperature-dependent coefficients in one-dimensional thermoviscoelasticity

This manuscript is concerned with a two-component evolution system generalizing the classical model for one-dimensional thermoviscoelastic dynamics in Kelvin-Voigt materials in the presence of temperature-dependent viscosities and elastic stiffnesses. Under suitable assumptions on the growth of these ingredients and on the initial data, the occurrence of finite-time blow-up with respect to the $L^\infty$ norm in the temperature variable is discovered.

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arXiv2026

Large-data global solutions to a quasilinear model for viscuos acoustic wave propagation in a non-isothermal setting

The manuscript considers the model for conversion of mechanical energy into heat during acoustic wave propagation in the presence of temperature-dependent elastic parameters, as given by \[ \left\{ \begin{array}{l} u_{tt} = (\gamma(\Theta) u_{xt})_x + a (\gamma(\Theta) u_x)_x, \\[1mm] \Theta_t = D\Theta_{xx} + \gamma(\Theta) u_{xt}^2. \end{array} \right. \qquad \qquad (\star) \] It is firstly shown that when considered along with no-flux boundary conditions in an open bounded real interval $\Omega$, under the assumption that $\gamma\in C^2([0,\infty))$ is such that $\gamma>0$ and $\gamma'\ge 0$ on $[0,\infty)$ as well as \[ D\cdot (\gamma+D) \cdot \gamma'' + 2\gamma \gamma'^2 \le 0 \qquad \mbox{on } [0,\infty), \] for all suitably regular initial data this problem admits a globally defined classical solution. This complements recent findings in the literature, according to which ($\star$) may admit solutions blowing up in finite time whenever $\gamma$ is positive and nondecreasing on $[0,\infty)$ with $\int_0^\infty \frac{d\xi}{\gamma(\xi)} < \infty$. Apart from that, it is found that if the additional assumption \[ a|\Omega|^2 \le \frac{\pi^2 \gamma(0)}{1+\sqrt{1+\frac{\gamma(0)}{D}}} \] is satisfied, the all these solutions stabilize toward some spatially homogeneous equilibrium in the large time limit.

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arXiv2026

Local strong solutions in a quasilinear Moore-Gibson-Thompson type model for thermoviscoelastic evolution in a standard linear solid

This manuscript is concerned with the evolution system \[ \left\{ \begin{array}{l} u_{ttt} + \alpha u_{tt} = \big(\gamma(\Theta) u_{xt}\big)_x + \big( \widehat{\gamma}(\Theta) u_x\big)_x, \Theta_t = D \Theta_{xx} + \Gamma(\Theta) u_{xt}^2, \end{array} \right. \] which arises as a simplified model for heat generation during acoustic wave propagation in a one-dimensional viscoelastic medium of standard linear solid type. Under the assumptions that $D>0$ and $\alpha\ge 0$, and that $\gamma, \widehat{\gamma}$ and $\Gamma$ are sufficiently smooth with $\gamma>0, \widehat{\gamma}>0$ and $\Gamma\ge 0$ on $[0,\infty)$, for suitably regular initial data a statement on local existence and uniqueness of solutions in an associated Neumann problem is derived in a suitable framework of strong solvability.

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arXiv2026

A dimension-independent critical exponent in a nutrient taxis system

In a ball $\Omega\subset R^n$ with arbitrary $n\ge 1$, the chemotaxis-consumption system \[ \left\{ \begin{array}{l} u_t = \nabla \cdot \big(D(u)\nabla u\big) - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - uv, \end{array} \right. \] is considered under no-flux boundary conditions for $u$, and for prescribed constant positive boundary data for $v$. Under the assumption that $D\in C^3([0,\infty))$ satisfies \[ D(\xi)\ge k_D (\xi+1)^{-\alpha} \qquad \mbox{for all } \xi\ge 0 \qquad \qquad (\star) \] with some $\alpha<1$ and some $k_D>0$, it is shown that for each nonnegative and radially symmetric $u_0\in \bigcup_{q>\max\{2,n\}} W^{1,q}(\Omega)$, a uniquely determined global bounded classical solution exists. This complements a previous result according to which given any positive $D\in C^3([0,\infty))$ fulfilling $D(\xi) \le K_D (\xi+1)^{-\alpha}$ with some $\alpha>1$ and $K_D>0$, one can find nonnegative radial initial data $u_0\in C_0^\infty(\Omega)$ such that no global solution exists.

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arXiv2025

Describing smooth small-data solutions to a quasilinear hyperbolic-parabolic system by $W^{1,p}$ energy analysis

In bounded $n$-dimensonal domains with $n\ge 1$, this manuscript examines an initial-boundary value problem for the system \[ \left\{ \begin{array}{l} u_{tt} = \nabla \cdot (\gamma(\Theta) \nabla u_t) + a \nabla \cdot (\gamma(\Theta) \nabla u) + \nabla\cdot f(\Theta), \Theta_t = D\Delta\Theta + \Gamma(\Theta) |\nabla u_t|^2 + F(\Theta)\cdot \nabla u_t, \end{array} \right. \] which in the case $n=1$ and with $\gamma\equiv \Gamma$ as well as $f\equiv F$ reduces to the classical model for the evolution of strains and temperatures in thermoviscoelasticity. Unlike in previous related studies, the focus here is on situations in which besides $f$ and $F$, also the core ingredients $\gamma$ and $\Gamma$ may depend on the temperature variable $\Theta$. Firstly, a statement on local existence of classical solutions is derived for arbitrary $a>0, D>0$ as well as $0<\gamma\in C^2([0,\infty))$ and $0\le\Gamma\in C^1([0,\infty))$, for functions $f\in C^2([0,\infty);{\mathbb{R}}^n)$ and $F\in C^1([0,\infty);{\mathbb{R}}^n)$ with $F(0)=0$, and for suitably regular initial data of arbitrary size. Secondly, it is seen that for each $p\ge 2$ such that $p>n$ there exists $\delta(p)>0$ with the property that whenever in addition to the above we have \[ \frac{a}{\gamma(0)} \le \delta(p) \qquad \mbox{and} \qquad \frac{|f'(\Theta_\star)| \cdot |F(\Theta_\star)|}{D \cdot \gamma(\Theta_\star)} \le \delta(p), \] for initial data suitably close to the constant level given by $u=0$ and $\Theta=\Theta_\star$, with any fixed $\Theta_\star\ge 0$, these solutions are actually global in time and have the property that $\nabla u_t, \nabla u$ and $\nabla\Theta$ decay exponentially fast in $L^p$. This is achieved by detecting suitable dissipative properties of functionals involving norms of these gradients in $L^p$ spaces.

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arXiv2025

Large-data regular solutions in a one-dimensional thermoviscoelastic evolution problem involving temperature-dependent viscosities

The model \[ \left\{ \begin{array}{l} u_{tt} = \big(γ(Θ) u_{xt}\big)_x + au_{xx} - \big(f(Θ)\big)_x, \\[1mm] Θ_t = Θ_{xx} + γ(Θ) u_{xt}^2 - f(Θ) u_{xt}, \end{array} \right. \] for thermoviscoelastic evolution in one-dimensional Kelvin-Voigt materials is considered. By means of an approach based on maximal Sobolev regularity theory of scalar parabolic equations, it is shown that if $γ_0>0$ is fixed, then there exists $δ=δ(γ_0)>0$ with the property that for suitably regular initial data of arbitrary size an associated initial-boundary value problem posed in an open bounded interval admits a global classical solution whenever $γ\in C^2([0,\infty))$ and $f\in C^2([0,\infty))$ are such that $f(0)=0$ and $|f(ξ)| \le K_f \cdot (ξ+1)^α$ for all $ξ\ge 0$ and some $K_f>0$ and $α<\frac{3}{2}$, and that \[ γ_0 \le γ(ξ) \le γ_0 + δ \qquad \mbox{for all } ξ\ge 0. \] This is supplemented by a statement on global existence of certain strong solutions, particularly continuous in both components, under weaker conditions on the initial data.

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arXiv2025

Large-data solutions in one-dimensional thermoviscoelasticity involving temperature-dependent viscosities

An initial-boundary value problem for \[ \left\{ \begin{array}{ll} u_{tt} = \big(γ(Θ) u_{xt}\big)_x + au_{xx} - \big(f(Θ)\big)_x, \qquad & x\inΩ, \ t>0, \\[1mm] Θ_t = Θ_{xx} + γ(Θ) u_{xt}^2 - f(Θ) u_{xt}, \qquad & x\inΩ, \ t>0, \end{array} \right. \] is considered in an open bounded real interval $Ω$. Under the assumption that $γ\in C^0([0,\infty))$ and $f\in C^0([0,\infty))$ are such that $f(0)=0$, and $k_γ\le γ\le K_γ$ as well as \[ |f(ξ)| \le K_f \cdot (ξ+1)^α \qquad \mbox{for all } ξ\ge 0 \] with some $k_γ>0, K_γ>0, K_f>0$ and $α<\frac{3}{2}$, for all suitably regular initial data of arbitrary size a statement on global existence of a global weak solution is derived.

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arXiv2024

Mathematical modeling and analysis of a tumor invasion problem with angiogenesis and taxis cascade

We propose a mathematical model for tumor invasion supported by angiogenesis and interactions with the surrounding tissue. For the model deduction we employ a multiscale approach starting from lower scales and obtaining by an informal parabolic upscaling a system of reaction-diffusion-taxis equations with a so-called 'taxis cascade', where one species is performing taxis towards a signal whose production/decay is controled by the other, for which it also serves as a tactic cue. We prove global existence and uniqueness of solutions to the obtained PDE-ODE system and perform numerical simulations to illustrate the behavior of solutions.

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arXiv2024

A model for heat generation by acoustic waves in piezoelectric materials: Global large-data solutions

A model for the generation of heat due to mechanical losses during acoustic wave propagation in a solid is considered in a Kelvin-Voigt type framework. In contrast to previous studies on related thermoviscoelastic models, in line with recent experimental findings the present manuscript focuses on situations in which elastic parameters depend on temperature. Despite an apparent loss of mathematically favorable structural properties thereby encountered, in the framework of a suitably generalized concept of solvability a result on global existence of solutions is derived under mild assumptions which, in particular, do not involve any smallness condition on the initial data.

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arXiv2024

A switch in dimension dependence of critical blow-up exponents in a Keller-Segel system involving indirect signal production

In bounded $n$-dimensional domains with $n\ge 3$, this manuscript considers an initial-boundary problem for a quasilinear chemotaxis system with indirect attractant production, as arising, inter alia, in the modeling of effects due to phenotypical heterogeneity in microbial populations. Under the assumption that the rates $D$ and $S$ of diffusion and cross-diffusion are suitably regular functions of the population density, essentially exhibiting asymptotic behavior of the form \[ D(ξ) \simeq ξ^{m-1} \quad \mbox{and} \quad S(ξ) \simeq ξ^σ, \qquad ξ\simeq \infty, \] the identity \[ σ=m-1+\frac{4}{n} \qquad \qquad (n\ge 3), \] is shown to determine a critical line for the occurrence of blow-up. This considerably differs from low-dimensional cases, in which the relation \[ σ=m+\frac{2}{n} \qquad \qquad (n\le 2) \] is known to play a correspondingly pivotal role.

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