In this paper, we prove existence results of a one-dimensional periodic solution to equations with the fractional Laplacian of order $s\in (1/2,1)$, singular nonlinearity and gradient term under various situations, including nonlocal contra-part of classical Lienard vector equations, as well other nonlocal versions of classical results know only in the context of second-order ODE. Our proofs are based on degree theory and Perron's method, so before that we need to establish a variety of priori estimates under different assumptions on the nonlinearities appearing in the equations. Besides, we obtain also multiplicity results in a regime where a priori bounds are lost and bifurcation from infinity occurs.

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.

This paper is concerned with the periodic (in time) solutions to an one-dimensional semilinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By combining variational methods with saddle point reduction technique, we obtain the existence of at least three periodic solutions whenever the period is a rational multiple of the length of the spatial interval. Our method is based on a delicate analysis for the asymptotic character of the spectrum of the wave operator with x-dependent coefficients, and the spectral properties play an essential role in the proof.

We study the existence and multiplicity of periodic weak solutions for a non-local equation involving an odd subcritical nonlinearity which is asymptotically linear at infinity. We investigate such problem by applying the pseudo-index theory developed by Bartolo, Benci and Fortunato [11] after transforming the problem to a degenerate elliptic problem in a half-cylinder with a Neumann boundary condition, via a Caffarelli-Silvestre type extension in periodic setting. The periodic nonlocal case, considered here, presents, respect to the cases studied in the literature, some new additional difficulties and a careful analysis of the fractional spaces involved is necessary.

We investigate a system of singular–degenerate parabolic equations with non-local terms, which can be regarded as a spatially heterogeneous competition model of Lotka–Volterra type. Applying the Leray–Schauder fixed-point theorem, we establish the existence of coexistence periodic solutions to the problem, which, together with the existing literature, gives a complete picture for such a system for all parameters.

We prove the linear and nonlinear instability of periodic traveling wave solutions for a generalized version of the symmetric regularized long wave (SRLW) equation. Using analytic and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so the linear instability of periodic profiles is obtained. An application of this approach is made to obtain the linear/nonlinear instability of cnoidal wave solutions for the modified SRLW (mSRLW) equation. We also prove the stability of dnoidal wave solutions associated to the equation just mentioned.

This paper is concerned with the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts for time-periodic reaction–diffusion equations with bistable nonlinearity in ℝm with m ≥ 2. It should be mentioned that the existence and stability of two-dimensional time-periodic V-shaped travelling fronts and three-dimensional time-periodic pyramidal travelling fronts have been studied previously. In this paper we consider two cases: the first is that the wave speed of a one-dimensional travelling front is positive and the second is that the one-dimensional wave speed is zero. For both cases we establish the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts. In particular, for the first case we furthermore show the asymptotic behaviours of level sets of the cylindrically symmetric travelling fronts.

Analytical study of the regularization of the Boussinesq system is performed in frequency space using Fourier theory. Existence and uniqueness of weak solutions with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray–Hopf solution of the Boussinesq system are established as the regularizing parameter $\alpha$ vanishes. The proofs are done in the frequency space and use energy methods, the Arselà-Ascoli compactness theorem and a Friedrichs-like approximation scheme.

Let Ω⊂ℝN be a smooth bounded domain and let f⁄≡0 be a possibly discontinuous and unbounded function. We give a necessary and sufficient condition on f for the existence of positive solutions for all λ>0 of Dirichlet periodic parabolic problems of the form Lu=h(x,t,u)+λf(x,t), where h is a nonnegative Carathéodory function that is sublinear at infinity. When this condition is not fulfilled, under some additional assumptions on h we characterize the set of λs for which the aforementioned problem possesses some positive solution. All results remain true for the corresponding elliptic problems.

Global convergence is established in this paper for monotone and subhomogeneous discrete dynamical systems on product Banach spaces. This result is then used to obtain the asymptotic periodicity of solutions to a class of periodic and cooperative reaction-diffusion systems.

We show that almost all perturbations P — λ, λ € C, of an arbitrary constant coefficient partial differential operator P are globally hypoelliptic on the torus. We also give a characterization of the values λ € C for which the operator is globally hypoelliptic; in particular, we show that the addition of a term of order zero may destroy the property of global hypoellipticity of operators of principal type, contrary to that happens with the usual (local) hypoellipticity.