In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals $0$ at a given constant state. We show that the system admits a unique, global strong solution for small initial data in the maximal $L_p$-$L_q$ regularity class. Consequently, we also prove the decay estimates of the solutions to the nonlinear problem. To obtain the global well-posedness for the critical case, we show $L_p$-$L_q$ decay properties of solutions to the linearized equations under an additional assumption for low frequencies.

This paper is concerned with nonplanar traveling fronts for a class of nonlocal dispersal equations with bistable nonlinearity. A comparison principle is introduced and employed to establish the existence of traveling fronts with pyramidal shape in $\mathbb R^3$. Although the speed of the pyramidal traveling fronts is not unique, their global average speed is unique. This is possibly the first time that the nonplanar traveling waves for the nonlocal dispersal equations has been studied.

For Besov spaces $B^s_{p,r}(\mathbb R)$ with $s > \max\{ 2 + \frac1p , \frac52\} $, $p \in (1,\infty]$ and $r \in [1 , \infty)$, it is proved that the data-to-solution map for the FORQ equation is not uniformly continuous from $B^s_{p,r}(\mathbb R)$ to $C([0,T]; B^s_{p,r}(\mathbb R))$. The proof of non-uniform dependence is based on approximate solutions, the Littlewood-Paley decomposition and estimates on the linear transport equation.

This work deals with a parabolic chemotaxis model with nonlinear diffusion and nonlocal reaction source. The problem is formulated on the whole space and, depending on a specific interplay between the coefficients associated to such diffusion and reaction, we establish that all given solutions are uniformly bounded in time. To be precise, we study these attractive (sign ``$+$'') and repulsive (sign ``$-$'') following models, formally described by the Cauchy problems \begin{equation}\label{problem_abstract} \tag{$\Diamond$} \begin{cases} \displaystyle \rho_t=\Delta \rho^m \pm \nabla \cdot \Big(\rho \nabla \Big(\frac{|x|^{2-n}}{2-n}*\rho\Big)\Big) \\[10pt] \displaystyle \hskip 50pt +a\rho^\eta-b\rho^\alpha \int_{\mathbb R^n} \rho^\beta dx & x\in \mathbb R^n, t\in (0,T_{\max}), \\ \rho(x,0)=\rho_{0}(x) & x\in \mathbb R^n, \end{cases} \end{equation} for $n\geq 3$, $m,a,b,\alpha,\eta > 0$ and $\beta\geq 1$. By denoting with $T_{\max}$ the maximum time of existence of any nonnegative weak solution $\rho$ to problems \eqref{problem_abstract}, we prove that despite any large-mass initial data $\rho_0$, for any $\eta > 0$ and arbitrarily small diffusive parameter $m > 0$, whenever $\alpha+\beta$ surpasses some computable expression depending on $m, \eta$ and $n$, $T_{\max}=\infty$ and $\rho$ is uniformly bounded. On the one hand, this paper is in line with claims established for $a=b=0$, where the same conclusion holds true in, respectively: \begin{align*} \text{$\bullet$} \ & \text{the repulsive scenario, under the assumption $m > 0$}\hskip 300pt \\ & \text{(adaptation of the case $m > 1-\frac{2}{n}$, in Carrillo and Wang [17]);} \\ \text{$\bullet$} \ & \text{the attraction scenario, under the assumption $\frac{2n}{n+2} < m < 2-\frac{2}{n}$} \\ & \text{and for small initial data (Chen and Wang in [18]).} \end{align*} On the other hand, for the attractive case with $a=b=m=1$ and $\eta=\alpha$, this investigation also extends a result derived by Bian, Chen and Latos in [3].