We investigate the Brusselator system with diffusion and Dirichlet boundary conditions on one dimensional space interval. Our proof demonstrates that, for certain parameter values, a periodic orbit exists. This proof is computer-assisted and rooted in the rigorous integration of partial differential equations. Additionally, we present the evidence of the occurrence of period-doubling bifurcation.

This article is concerned with the existence and multiplicity of positive weak solutions for the following fractional Kirchhoff-Choquard problem: \begin{align*} & M ( \|u\|^2 ) (-\Delta)^s u = \lambda f(x)|u|^{q-2}u + \Big ( \int _{\Omega} \frac{|u(y)|^{2^{*}_{\mu ,s}}}{|x-y|^ \mu}\, dy \Big ) |u|^{2^{*}_{\mu ,s}-2}u \;\text{in} \; \Omega,\\& u > 0\quad \text{in} \; \Omega, \,\,u = 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega, \end{align*}where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$, here $M$ models Kirchhoff-type coefficient of the form $M(t) = a + bt^{\theta-1}$, where $a, b > 0$ are given constants.$(-\Delta)^s$ is fractional Laplace operator, $\lambda > 0$ is a real parameter. We explore using the variational methods, the existence of solution for ${q} \in (1,2^*_s)$ and $\theta \geq 1$.Here, $2^*_s = \frac{2N}{N-2s}$ and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.

In this article, we obtain higher Hölder regularity results for weak solutions to nonlocal problems driven by the fractional double phase operator\begin{align*}\mathcal L u(x):= & 2 \; {\rm P.V.} \int_{\mathbb R^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps_1}}dy \nonumber \\ & +2 \; {\rm P.V.} \int_{\mathbb R^N} a(x,y) \frac{|u(x)-u(y)|^{q-2}(u(x)-u(y))} {|x-y|^{N+qs_2}}dy,\end{align*}where $1 < p \leq q < \infty$, $0 < s_2, s_1 < 1$ and the modulating coefficient $a(\cdot,\cdot)$ is a non-negative bounded function. Specifically, we prove higher space-time Hölder continuity result for weak solutions of time depending nonlocal double phase problems for a particular subclass of the modulating coefficients. Using suitable approximation arguments, we further establish higher (global) Hölder continuity results for weak solutions to the stationary problems involving the operator $\mathcal L$ with modulating coefficients that are locally continuous.

In this paper, we focus on the study of Holder regularity in time and space variables, to solutions of a parabolic Hamilton-Jacobi problem, where the standard time derivative is replaced by a variable-order fractional derivative of Caputo type.