Existence Of Time-periodic Solutions Research Articles

Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when α∈(0,1). We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.

Chemotaxis driven formation of time periodic pattern and steady state pattern

This paper is concerned with a chemotaxis-fluid model with realistic boundary conditions matching the experiments of Kessler et al. Different from the initial and boundary value problem, the existence of time periodic solution is proved under no any smallness assumptions on chemotactic coefficient. Subsequently, we investigate the reason of pattern formation. It is generally known that there is no pattern formation for the homogeneous boundary value problem. Therefore, the pattern formation is partly caused by non-homogeneous boundary conditions, but chemotaxis is also an indispensable condition. In fact, it is proved that the density of bacteria will tend to be evenly distributed when the chemotactic intensity tends to 0. At last, the idea of finding time periodic pattern is also applied to prove the steady-state pattern formation of chemotaxis-Navier-Stokes model.

Time periodic strong solutions to the Keller-Segel system coupled to Navier-Stokes equation

We consider the time periodic solutions to the Keller-Segel (KS) system coupled to Navier-Stokes equation in R N ( N ≥ 4 ) . Based on the theory of real interpolation as Meyer and Yamazaki used, we first consider the existence of time periodic solution in B C ( R , L s , ∞ ( R N ) ) when S ( n ) = n − 1 2 . Next for the case S ( n ) = 0 , we study the mild solution and point out it can become strong solution in B C ( R , L s , ∞ ( R N ) ) if the external force g satisfies a natural condition which comes from the strong solvability of the inhomogeneous Stokes equations.

On time periodic solutions to the conformal cubic wave equation on the Einstein cylinder

We consider the conformal wave equation on the Einstein cylinder with a defocusing cubic non-linearity. Motivated by a method developed by Rostworowski and Maliborski on the existence of time periodic solutions to the spherically symmetric Einstein–Klein–Gordon system, we study perturbations around the zero solution as a formal series expansion and assume that the perturbations bifurcate from one mode. In the center of this work stands a rigorous proof on how one can choose the initial data to cancel out all secular terms in the resonant system. Interestingly, our analysis reveals that the only possible choice for the existence of time periodic solutions bifurcating from the first mode is when the error terms in the expansion are all proportional to this dominant one mode. Finally, we use techniques from ordinary differential equations and establish the existence of time periodic solutions for the initial data proportional to the first mode of the linearized operator.

Time periodic solutions for the non-isentropic compressible quantum hydrodynamic equations with viscosity in R^3

This article concerns the existence and uniqueness of a time periodic solution for the non-isentropic quantum hydrodynamic equations with viscosity. By applying the Leray-Schauder theory, subtle energy estimates and a limiting method, we obtain the existence of time periodic solutions under some smallness assumptions on the time periodic external force in \(\mathbb{R}^3\). The uniqueness can be proved by similar energy estimates. In particular, the quantum effects and the energy equation are taken into account in this paper which play a significant role in the uniform (in the domain R and the positive constant \(\epsilon\)) estimates, especially in the selection of the norm.
 For more information see https://ejde.math.txstate.edu/Volumes/2020/103/abstr.html

Kuznetsov–Ma breather-like solutions in the Salerno model

The Salerno model is a discrete variant of the celebrated nonlinear Schrödinger (NLS) equation interpolating between the discrete NLS (DNLS) equation and completely integrable Ablowitz–Ladik (AL) model by appropriately tuning the relevant homotopy parameter. Although the AL model possesses an explicit time-periodic solution known as the Kuznetsov–Ma (KM) breather, the existence of time-periodic solutions away from the integrable limit has not been studied as of yet. It is thus the purpose of this work to shed light on the existence and stability of time-periodic solutions of the Salerno model. In particular, we vary the homotopy parameter of the model by employing a pseudo-arclength continuation algorithm where time-periodic solutions are identified via fixed-point iterations. We show that the solutions transform into time-periodic patterns featuring small, yet non-decaying far-field oscillations. Remarkably, our numerical results support the existence of previously unknown time-periodic solutions even at the integrable case whose stability is explored by using Floquet theory. A continuation of these patterns towards the DNLS limit is also discussed.

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 isunstable.

The Existence and Stability of Periodic Solutions with a Boundary Layer in a Two-Dimensional Reaction-Diffusion Problem in the Case of Singularly Perturbed Boundary Conditions of the Second Kind

The existence of time-periodic solutions of the boundary-layer type to a two-dimensional reaction–diffusion problem with a small-parameter coefficient of a parabolic operator is proved in the case of singularly perturbed boundary conditions of the second kind. An asymptotic approximation with respect to the small parameter is constructed for these solutions. The set of boundary conditions for which these solutions exist is studied and the local uniqueness and asymptotic Lyapunov stability are established for them. It is shown that, unlike the analogous Dirichlet problem, for which such a solution is unique, there can be several solutions of this kind for the problem under consideration, each of which has its domains of stability and local uniqueness. To prove these facts, results based on the asymptotic principle of differential inequalities are used. Keywords: , , , , , , .

Periodic solutions of an age-structured epidemic model with periodic infection rate

In this paper we consider an age-structured epidemic model of the susceptible-exposed-infectious-recovered (SEIR) type. To characterize the seasonality of some infectious diseases such as measles, it is assumed that the infection rate is time periodic. After establishing the well-posedness of the initial-boundary value problem, we study existence of time periodic solutions of the model by using a fixed point theorem. Some numerical simulations are presented to illustrate the obtained results.

On time-periodic solutions of some nonlinear parabolic equations with nonmonotone multivalued terms

In this paper we study the existence of time periodic solutions to a class of nonlinear parabolic equations with multivalued nonlinear terms subject to the homogeneous Dirichlet boundary condition. We give two types of existence results: one for large periodic solutions with any large data, and the other for small periodic solutions with small data. Both concern the case where the nonlinear terms contain either a upper semicontinuous multivalued term or a lower semicontinuous multivalued term. Some applications of our results are also given.

On “Hard Stars” in General Relativity

We study spherically symmetric solutions to the Einstein-Euler equations which model an idealized relativistic neutron star surrounded by vacuum. These are barotropic fluids with a free boundary, governed by an equation of state which sets the speed of sound equal to the speed of light. We demonstrate the existence of a 1-parameter family of static solutions, or ''hard stars,'' and describe their stability properties: First, we show that small stars are a local minimum of the mass energy functional under variations which preserve the total number of particles. In particular, we prove that the second variation of the mass energy functional controls the ''mass aspect function.'' Second, we derive the linearisation of the Euler-Einstein system around small stars in ''comoving coordinates,'' and prove a uniform boundedness statement for an energy, which is exactly at the level of the variational argument. Finally, we exhibit the existence of time periodic solutions to the linearised system, which shows that energy boundedness is optimal for this problem.

Stable and Unstable Periodic Spiky Solutions for the Gray–Scott System and the Schnakenberg System

The Hopf bifurcations for the classical Gray–Scott system and the Schnakenberg system in an one-dimensional interval are considered. For each system, the existence of time-periodic solutions near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of limit cycles are determined and it is shown that the Hopf bifurcation is supercritical for the Schnakenberg system, and hence the bifurcating periodic solutions are linearly stable. For the Gray–Scott system, there is a critical feeding rate, when the feeding rate of the system is greater than this critical feeding rate, the Hopf bifurcation is supercritical which implies the bifurcating periodic solutions are linearly stable, while when the feeding rate is smaller than this critical feeding rate, the Hopf bifurcation is subcritical, implying the bifurcating periodic solutions are linearly unstable.

EXISTENCE OF TIME PERIODIC SOLUTIONS FOR THE 3-D VISCOUS PRIMITIVE EQUATIONS OF LARGE-SCALE DRY ATMOSPHERE

In this paper, we consider the existence of time periodic solutions of the 3-D viscous primitive equations of large-scale dry atmosphere. We used the Galerkin method. Firstly, by Leray-Schauder fixed point theorem, we prove the existence of approximate solutions of the primitive equations, then we show the convergence of the approximate solutions, and we also get the uniqueness to the primitive equations.

On the motion of the 3D compressible micropolar fluids with time periodic external forces

This paper analyzes the time periodic solutions to the three-dimensional equations modeling the compressible micropolar fluids under some smallness and structure conditions on time periodic forces. By adopting a regularized approximation scheme and using the topological degree theory, we prove the existence of time periodic solutions in a bounded domain. Furthermore, via a limiting process, the existence result can be obtained in the whole space R3.

A note on the existence of time periodic solution of a superlinear heat equation

In this paper, we consider the existence of positive periodic solutions for a superlinear heat equation with and being the Sobolev critical exponent, subject to homogeneous Dirichlet boundary condition. We relax the technical assumption on a(t) and convex assumption on the domain , which generalizes the previous results.

Existence of Time Periodic Solution to Some Double-Diffusive Convection System in the Whole Space Domain

This paper is concerned with the existence of time periodic solutions to some system which describes double-diffusive convection phenomena in the whole space $${\mathbb {R}} ^N $$ with $$N = 3$$ and 4. In previous results for periodic problems of parabolic type equations with non-monotone perturbation terms, it seems that either of the smallness of given data or the boundedness of space domain is essential. In spite of the presence of non-monotone terms, the solvability of our problem in the whole space is shown for large external forces via the convergence of solutions to approximate equations in bounded domains.

Time periodic strong solutions to the incompressible Navier–Stokes equations with external forces of non-divergence form

We discuss the time periodic problem to the incompressible Navier–Stokes equations on the whole space Rn, n≥3, with the external forces of non-divergence form. Firstly, we consider the existence of time periodic solutions in BC(R;Ln,∞(Rn)) assuming the smallness of external forces in BC(R;L1(R3)) and BC(R;Ln3,∞(Rn)) in the case n≥4. Next, we show that the mild solution above becomes a strong solution in the topology of Ln,∞(Rn) with a natural condition of the external force, derived from the strong solvability of the inhomogeneous Stokes equations in Ln,∞(Rn). For this aim, we re-construct a strong solvability of an abstract evolution equation where the associated semigroup is not strongly continuous at t=0.

On time periodic solutions, asymptotic stability and bifurcations of Navier-Stokes equations

In this article, we investigate the time periodic solutions for two-dimensional Navier-Stokes equations with nontrivial time periodic force terms. Under the time periodic assumption of the force term, the existence of time periodic solutions for two-dimensional Navier-Stokes equations has received extensive attention from many authors. With the smallness assumption of the time periodic force, we show that there exists only one time periodic solution and this time periodic solution is globally asymptotically stable in the $$H^1$$H1 sense. Without smallness assumption of the force term, there is no stability analysis theory addressed. It is expected that when the amplitude of the force term is increasing, the time periodic solution is no longer asymptotically stable. In the last part of the article, we use numerical experiments to study the bifurcation of the time periodic solutions when the amplitude of the force is increasing. Extrapolating to the heating of the earth by the sun, the bifurcation diagram hints that when the earth receives a relatively small amount of solar energy regularly, the time periodic fluid patterns are asymptotically stable; while/when the earth receives too much solar energy even though in a time periodic way, the time periodic pattern of the fluid motions will lose its stability.

Time periodic solutions for a sixth order nonlinear parabolic equation

Abstract In this paper, we study the time periodic solution of a sixth order nonlinear parabolic equation, which arises in oil-water-surfactant mixtures based on Leray-Schauder’s fixed point theorem, we prove the existence of time-periodic solutions.

Time-periodic solutions of the compressible Navier–Stokes equations in $${\mathbb{R}^{4}}$$ R 4

This paper deals with the existence of time-periodic solutions to the compressible Navier–Stokes equations effected by general form external force in \({\mathbb{R}^{N}}\) with \({N = 4}\). Using a fixed point method, we establish the existence and uniqueness of time-periodic solutions. This paper extends Ma, UKai, Yang’s result [5], in which, the existence is obtained when the space dimension \({N \ge 5}\).