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$\mathbb{R}^3$
We consider the chemotaxis-Navier–Stokes system with generalised fluid dissipation in 
$\mathbb{R}^3$:which models the motion of swimming bacteria in water flows. First, we prove blow-up criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin-type criterion for 
\begin{eqnarray*} \begin{cases} \partial _t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi (c)n \nabla c),\\[5pt] \partial _t c+u \cdot \nabla c=\Delta c-nf(c),\\[5pt] \partial _t u +u \cdot \nabla u+\nabla P=-(\!-\Delta )^\alpha u-n\nabla \phi, \\[5pt] \nabla \cdot u=0, \end{cases} \end{eqnarray*}
$\alpha \gt \frac {3}{4}$ and the Beir
$\tilde {\textrm {a}}$o da Veiga-type criterion for 
$\alpha \gt \frac {1}{2}$. Then, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and bacteria density for 
$\alpha \geq \frac {5}{4}$. Furthermore, in the scenario of 
$\frac {3}{4}\lt \alpha \lt \frac {5}{4}$, we establish uniform regularity estimates and optimal time-decay rates of global solutions if only the 
$L^2$-norm of initial data is small. To our knowledge, this work provides the first result concerning the global existence and large-time behaviour of strong solutions for the chemotaxis-Navier–Stokes equations with possibly large oscillations.
Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted sourceare obtained, in the range of exponents 
\begin{align*} |x|^{-2}\partial _tu=\Delta u+|x|^{\sigma }u^p, \quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{align*}
$p\gt 1$, 
$\sigma \ge -2$. More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as 
$t\to \infty$ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case 
$\sigma =-2$, we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher–KPP equation is derived and employed in order to deduce these properties.
The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular stochastic partial differential equations of the form
\begin{equation*}\partial_t u = \mathfrak{L} u+ F(u, \xi),\end{equation*}
where the differential operator 
$\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space 
$\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group 
$\mathbb{G}$ such that the differential operator 
$\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on 
$\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator 
$\partial_t -\Delta_v - v\cdot \nabla_x$ and heat-type operators associated with sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations
\begin{equation*}\partial_t u = \sum_{i} X^2_i u + u (\xi-c)\end{equation*}
on the compact quotient of an arbitrary Carnot group.
We consider the harmonic map heat flow for maps 
$\mathbb {R}^{2} \to \mathbb {S}^2$. It is known that solutions to the initial value problem exhibit bubbling along a well-chosen sequence of times. We prove that every sequence of times admits a subsequence along which bubbling occurs. This is deduced as a corollary of our main theorem, which shows that the solution approaches the family of multi-bubble configurations in continuous time.
Well-posedness in time-weighted spaces of certain quasilinear (and semilinear) parabolic evolution equations 
$u'=A(u)u+f(u)$ is established. The focus lies on the case of strict inclusions 
$\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the nonlinearities 
$u\mapsto f(u)$ and 
$u\mapsto A(u)$. Based on regularizing effects of parabolic equations it is shown that a semiflow is generated in intermediate spaces. In applications this allows one to derive global existence from weaker a priori estimates. The result is illustrated by examples of chemotaxis systems.
In this paper, we prove the global exstence of weak solutions for a porous medium dynamics of m species moving between two domains separated by a zero-thickness membrane. On this membrane, Kedem–Katchalsky conditions are considered, and the study is characterized by natural structural conditions applied to the nonlinear reactive terms. The global existence is established under the assumption that these reactive terms are bounded in 
$L^1$. This problem has already been analyzed in the linear diffusion case by Ciavolella and Perthame in Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540). The present work constitutes an extension for nonlinear diffusion, particularly of the porous medium type, in the form 
$\partial _t v_i - \Delta v_i^{r_i} = R_i$, for an exponent 
$r_i < 2$. The case 
$r_i \geq 2$ remains an open problem. This paper is an adaptation of the ideas from Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540), with new strategies to overcome the appearance of nonlinearity and degeneracy in the diffusion term.
