Tobias Black

This manuscript is concerned with the one-dimensional system \[ \begin{array}{l} τu_{ttt} + αu_{tt} = b \big(γ(Θ) u_{xt}\big)_x + \big( γ(Θ) u_x\big)_x, \\[1mm] Θ_t = D Θ_{xx} + bγ(Θ) u_{xt}^2, \end{array} \] 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 $τ>0, b>0$ and $α\geq0$ satisfy \begin{align}\tag{$\star$} αb >τ, \end{align} it is shown that for all $Θ_\star>0$ one can find $ν=ν(D,τ,α,b,Θ_\star,γ)>0$ 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 $$\|Θ_0\|_{L^\infty(Ω)}\leq Θ_\star$$ and the derivatives of the initial data are sufficiently small in the sense of satisfying $$\int_Ωu_{0xx}^2 + \int_Ω(u_{0t})_{xx}^2 + \int_Ω(u_{0tt})_x^2 < ν\quad\text{and}\quad \|Θ_{0x}\|_{L^\infty(Ω)} + \|Θ_{0xx}\|_{L^\infty(Ω)} < ν.$$ 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.

This manuscript is concerned with the one-dimensional system $ \begin{eqnarray*} \left\{ \begin{array}{l} \tau u_{ttt} + \alpha u_{tt} = b \big(\gamma(\Theta) u_{xt}\big)_x + \big( \gamma(\Theta) u_x\big)_x, \\ \Theta_t = D \Theta_{xx} + b\gamma(\Theta) u_{xt}^2, \end{array} \right. \end{eqnarray*} $ that arises as a simplified model for heat generation during acoustic wave propagation in one-dimensional Zener type materials in which elastic parameters and viscosities depend on temperature, with $ u $ and $ \Theta $ denoting the scalar displacement variable and the temperature, respectively. Despite the presence of a strong quasilinear coupling, under the assumption that the coefficients $ \tau>0, b>0 $ and $ \alpha\geq 0 $ satisfy $ \begin{eqnarray*} \alpha b >\tau \qquad \qquad (\star) \end{eqnarray*} $ it is shown by means of a self map type argument involving a third-order energy analysis 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 $ \begin{eqnarray*} \displaystyle\int_\Omega u_{0xx}^2 + \displaystyle\int_\Omega (u_{0t})_{xx}^2 + \displaystyle\int_\Omega (u_{0tt})_x^2 < \nu\ \text{and}\ \|\Theta_{0x}\|_{L^\infty(\Omega)} + \|\Theta_{0xx}\|_{L^\infty(\Omega)} < \nu. \end{eqnarray*} $ The constructed solution moreover features an exponential stabilization property in 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.

This paper deals with the Keller--Segel system with signal-dependent sensitivity \begin{align*} &u_t = \Delta u - \chi \nabla \cdot (uS(v)\nabla v), &v_t = \Delta v - v + u, \end{align*} where $\chi>0$ 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 $\chi>0$.

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

In bounded smooth domains $\Omega\subset\mathbb{R}^N$, $N\in\{2,3\}$, considering the chemotaxis--fluid system \[ \begin{cases} \begin{split} & n_t + u\cdot \nabla n &= \Delta n - \chi \nabla \cdot(\frac{n}{c}\nabla c) &\\ & c_t + u\cdot \nabla c &= \Delta c - c + n &\\ & u_t + \kappa (u\cdot \nabla) u &= \Delta u + \nabla P + n\nabla \Phi & \end{split}\end{cases} \] with singular sensitivity, we prove global existence of classical solutions for given $\Phi\in C^2(\bar{\Omega})$, for $\kappa=0$ (Stokes-fluid) if $N=3$ and $\kappa\in\{0,1\}$ (Stokes- or Navier--Stokes fluid) if $N=2$ and under the condition that \[ 0<\chi<\sqrt{\frac{2}{N}}. \]