Nonautonomous Equations Research Articles

On the mass concentration of normalized ground state solutions for non-autonomous Kirchhoff equations

On the mass concentration of normalized ground state solutions for non-autonomous Kirchhoff equations

Equilibrium points and locations in the photogravitational circular Robe’s restricted three-body problem with variable masses

The restricted three-body problem (R3BP) defines the dynamics of an infinitesimal mass moving in the gravitational neighborhood of two primaries, which move in circular orbits around their center of mass on account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries. In this paper, we examine equilibrium points and their locations in the photogravitational circular Robe’s restricted three-body problem (R3BP) with variable masses. The motion of the primaries and variation in masses of the primaries are governed by the Gylden-Mestschersky problem (GMP) and the unified Mestschersky law (UML), respectively, while the second primary is assumed to be a radiation emitter. The non-autonomous equations of the governing dynamical system are deduced and transformed using the Mestschersky transformation (MT), the UML and the particular solutions of the GMP, to a system of the autonomized equations with constant coefficients under the condition that there is no fluid inside the first primary. Next, the equilibrium points (EPs) of the autonomized system are explored using perturbation method and it is seen that axial EP which is defined by the mass parameter and the radiation pressure of the second primary exists. Further, a pair of non-collinear EPs which depends on the mass parameter, radiation pressure of the second primary and a constant of the mass variations of the primaries, is found. The EPs of the non-autonomous system are obtained using the MT and differ from those of the autonomized system by time t . The EPs may be used in different problems of stellar dynamics, and also in other astrophysical applications.

Solutions for non-autonomous fractional integrodifferential equations with delayed force term

Solutions for non-autonomous fractional integrodifferential equations with delayed force term

Dynamic bifurcation of nonautonomous evolution equations under Landesman–Lazer condition with cohomology methods

In this article we study the dynamic bifurcation of nonautonomous evolution equations by using cohomology methods. First, we construct a homotopy equivalence relation between the nonautonomous system and a product flow. Then, we slightly extend some continuation theorems on bifurcations for autonomous equations, and prove some new cohomology consequences on the reduced singular groups. Based on this homotopy equivalence relation and these conclusions, we establish some typical results on the dynamic bifurcation from infinity of the abstract nonautonomous evolution equation. Finally, we consider the parabolic equation ut−Δu=λu+f(x,u)+g(x,t) associated with the Dirichlet boundary condition, where f(x,u) satisfies the appropriate Landesman–Lazer type condition. Some new results on the dynamical behaviors of this equation near resonance of the equation are derived.

Existence and limit behavior of normalized ground state solutions for a class of non-autonomous Kirchhoff equations

Existence and limit behavior of normalized ground state solutions for a class of non-autonomous Kirchhoff equations

Long-time dynamics of a random higher-order Kirchhoff model with variable coefficient rotational inertia

This paper delves into the stochastic asymptotic behavior of a non-autonomous stochastic higher-order Kirchhoff equation with variable coefficient rotational inertia. The equation is solved using the Galerkin method, and a stochastic dynamical system is established on this basis. Uniform estimation demonstrates a family of Dk−absorbing sets in the stochastic dynamical system Φk, and the asymptotic compactness of Φk is proved via decomposition. Finally, the family of Dk−random attractors is acquired for the stochastic dynamical system Φk in Vm+k(Ω)×Vm+kb0(Ω). These results improve and extend those in recent literature (Lv et al., 2021). The findings promote the relevant conclusions of the non-autonomous stochastic higher-order Kirchhoff model and provide a theoretical basis for its subsequent application and research.

Existence, multiplicity and asymptotic behaviour of normalized solutions to non-autonomous fractional HLS lower critical Choquard equation

Existence, multiplicity and asymptotic behaviour of normalized solutions to non-autonomous fractional HLS lower critical Choquard equation

Time-dependent uniform upper semicontinuity of pullback attractors for non-autonomous delay dynamical systems: Theoretical results and applications

In this paper we provide general results on the uniform upper semicontinuity of pullback attractors with respect to the time parameter for non-autonomous delay dynamical systems. Namely, we establish a criteria in terms of the multi-index convergence of solutions for the delay system to the non-delay one, locally pointwise convergence and local controllability of pullback attractors. As an application, we prove the upper semicontinuity of pullback attractors for a non-autonomous delay reaction-diffusion equation to the corresponding nondelay one over any bounded time interval as the delay parameter tends to zero.

Asymptotic behavior of a semilinear non-autonomous wave equation with distributed delay and analytic nonlinearity

This paper is concerned with a semilinear non-autonomous wave equation with distributed delay and analytic nonlinearity. The distributed delay is represented by an integral operator integrating on the delay interval, which considers a segment of the past dynamic information. Firstly we show the global boundedness of solutions and then prove that every globally bounded solution has a relative compact range in the phase space. By the Łojasiewicz–Simon inequality, we prove the convergence to an equilibrium of globally bounded solutions and then show the convergence rate dependence on the Łojasiewicz exponent and the decay conditions on non-autonomous term. The existence and uniqueness of a global strong solution is finally proved. Note that the result of existence and uniqueness of the solution does not need any restrictions on the coefficients between the damping and the delay. The presence of an integral kernel makes distributed delays more difficult to be analyzed compared to its pointwise counterpart.

Existence and exponential stability in pth moment of non-autonomous stochastic integro-differential equations

This study is concerned with the existence and exponential stability of solutions of non-autonomous neutral stochastic integro-differential equations with delay; the linear part of this equation is dependent on time and generates a linear evolution system. The obtained results are applied to some neutral stochastic integro-differential equations. These kinds of equations arise in systems related to couple oscillators in a noisy environment or in viscoelastic materials under random or stochastic influences. Finally, we provide an example to illustrate the results.

Well-posedness of nonautonomous abstract Cauchy problems under dissipativity conditions expressed by metric-like functionals

Under a dissipativity condition expressed by a metric-like functional, the well-posedness of a nonautonomous abstract Cauchy problem is established. The main result not only extends Martin's result on the generation of an evolution operator in a class of uniformly convex Banach spaces but also can be applied to mixed problems for nonautonomous evolution equations of Kirchhoff type with which Kato's quasilinear theory or the theory of quasi-contractive semigroups cannot directly deal. The main result also gives an affirmative answer to Kōmura's conjecture that an evolution operator of linear operators with moving domains can be generated even under a weak stability condition rather than Kato's stability condition, and the result provides an effective means for solving degenerate wave equations.

Stability of random attractors for non-autonomous fractional stochastic p-Laplacian equations driven by nonlinear colored noise

Stability of random attractors for non-autonomous fractional stochastic p-Laplacian equations driven by nonlinear colored noise

Perturbations of non-autonomous second-order abstract Cauchy problems

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.

Non-autonomous fractional nonlocal evolution equations with superlinear growth nonlinearities

We carry out an analysis of the existence of solutions for a class of nonlinear fractional partial differential equations of parabolic type with nonlocal initial conditions. Sufficient conditions for the solvability of the desired problem are presented by transforming it into an abstract non-autonomous fractional evolution equation, and constructing two families of solution operators based on the Mittag-Leffler function, the Mainardi Wright-type function and the analytic semigroup generated by the closed densely defined operator −A(⋅). The discussions are based on the fractional power theory as well as the Banach fixed point theorem in the interpolation space Xpυ (0⩽υ<1, 1<p<∞).

On non-autonomous fractional evolution equations and applications

On non-autonomous fractional evolution equations and applications

Massera type theorems for abstract non-autonomous evolution equations

Massera type theorems for abstract non-autonomous evolution equations

Massera type theorems for abstract non-autonomous evolution equations

Massera type theorems for abstract non-autonomous evolution equations

Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions

Abstract In this article, we investigate the existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions. Using the theory of resolvent operators, some fixed point theorems, and an estimation technique of Kuratowski measure of noncompactness, we first establish some existence results of mild solutions for the proposed equation. Subsequently, we show by applying a newly established lemma that these solutions have regularity property under some conditions. Finally, as a sample of application, the obtained results are applied to a class of non-autonomous nonlocal partial integrodifferential equations.

Adjoints of Sums of M-Accretive Operators and Applications to Non-Autonomous Evolutionary Equations

Adjoints of Sums of M-Accretive Operators and Applications to Non-Autonomous Evolutionary Equations

Asymptotic Behavior of Non-autonomous Random Ginzburg-Landau Equations with Colored Noise on Unbounded Thin Domains

Asymptotic Behavior of Non-autonomous Random Ginzburg-Landau Equations with Colored Noise on Unbounded Thin Domains