In this article, we study the scalar one-dimensional nonlocal quasilinear problem of the form$$u_t=a(\Vert u_x\Vert^2)u_{xx}+\nu f(u) , $$with Dirichlet boundary conditions on the interval$[0,\pi]$, where$a: \mathbb{R}^+\to [m,M]\subset (0,+\infty)$ and$f: \mathbb{R}\to \mathbb{R}$ are continuous functions that satisfy suitable additional conditions. We give a complete characterization of the bifurcations and hyperbolicity for the corresponding equilibria. With respect to bifurcation, the existing result requires that the function$a(\cdot)$ be non-decreasing and shows that bifurcations are pitchfork supercritical bifurcations from zero. We extend these results to the case of a general smooth nonlocal diffusion function$a(\cdot)$ and show that bifurcations may be pitchfork or saddle-node, both subcritical or supercritical. Concerning hyperbolicity, we specifying necessary and sufficient conditions for its occurrence. We also explore some examples to exhibit the variety of possibilities, depending on the choice of the function$a(\cdot)$, that may occur as the parameter$\nu$ varies.

We lay down the preliminary work to apply the Functional Analytic Approach to quasi-periodic boundary value problems for the Helmholtz equation. This consists in introducing a quasi-periodic fundamental solution and the related layer potentials, showing how they are used to construct the solutions of quasi-periodic boundary value problems, and how they behave when we perform a singular perturbation of the domain. To show an application, we study a nonlinear quasi-periodic Robin problem in a domain with a set of holes that shrink to points.

We prove a maximal regularity result for abstract linear evolution autonomous equations with a fractional time derivative in the sense of Caputo. We employ it to show theorem of existence and uniqueness of local solutions for fully nonlinear equations and a theorem of existence of a stable manifold which is analogous to well known results in the case of a derivative of order one. We conclude with some examples and applications to mixed Cauchy-Dirichlet and Cauchy-Neumann problems.

In this paper, we consider the following Kirchhoff type equation\begin{equation*} \begin{cases}-\displaystyle\Big ( a+b\int_{\mathbb{R}^3}|\nabla u|^2 \Big ) \Delta u-\lambda u=f(u), & \textrm{in}\,\,\mathbb{R}^3,\\[3mm]u\in H^1(\mathbb{R}^3),\end{cases}\end{equation*}with an $L^2$ constraint $\int_{\mathbb{R}^3}|u|^2dx=m$, where $a,b,m > 0$, $\lambda\in\mathbb{R}$ will arise as a Lagrange multiplier and the nonlinearity $f$ is merely continuous and satisfies general mass supercritical conditions. Both in the Sobolev subcritical and critical cases, we establish the existence of ground states to this problem and derive some basic behavior of the ground state energy $E_m$ when $m > 0$ varies. Our results generalize and improve the ones in [L. Jeanjean, S.-S. Lu, Calc. Var. 59, 174 (2020)] and other related literature.