The Keller–Segel system on bounded convex domains in critical spaces

Consider the classical Keller–Segel system on a bounded convex domain $$\varOmega \subset {\mathbb {R}}^3$$ Ω ⊂ R 3 . In contrast to previous works it is not assumed that the boundary of $$\varOmega $$ Ω is smooth. It is shown that this system admits a local, strong solution for initial data in critical spaces which extends to a global one provided the data are small enough in this critical norm. Furthermore, it is shown that this system admits for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution.

Time periodic solution to a two-species chemotaxis-Stokes system with $ p $-Laplacian diffusion

<p style='text-indent:20px;'>In this paper, we consider a two-species chemotaxis-Stokes system with <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian diffusion in two-dimensional smooth bounded domains. It is proved that the existence of time periodic solution for any <inline-formula><tex-math id="M3">\begin{document}$ \frac{15}{7}\leq p&lt;3 $\end{document}</tex-math></inline-formula> and any large periodic source <inline-formula><tex-math id="M4">\begin{document}$ g_1(x,t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ g_2(x,t) $\end{document}</tex-math></inline-formula>.</p>

Asymptotic solution of the inverse problem for restoring the modular type source in Burgers’ equation with modular advection

For a singularly perturbed Burgers’ type equation with modular advection that has a time-periodic solution with an internal transition layer, asymptotic analysis is applied to solve the inverse problem for restoring the function of the source using known information about the observed solution of a direct problem at a given time interval (period).

Hopf bifurcation from spike solutions for the weak coupling Gierer–Meinhardt system

The Hopf bifurcation from spike solutions for the classical Gierer–Meinhardt system in a onedimensional interval is considered. The existence of time-periodic solution near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of the limit cycle are determined, and it is shown that the limit cycle is unstable.