Journal of Dynamics and Differential Equations最新文献

Let (xi ) be a real analytic vector field with an elementary isolated singularity at (0in mathbb {R}^3) and eigenvalues (pm bi,c) with (b,cin mathbb {R}) and (bne 0). We prove that all cycles of (xi ) in a sufficiently small neighborhood of 0, if they exist, are contained in the union of finitely many subanalytic invariant surfaces, each one entirely composed of a continuum of cycles. In particular, we solve Dulac’s problem for such vector fields, i.e., finiteness of limit cycles.

In the N-body problem, it is classical that there are conserved quantities of center of mass, linear momentum, angular momentum and energy. The level sets (mathfrak {M}(c,h)) of these conserved quantities are parameterized by the angular momentum c and the energy h, and are known as the integral manifolds. A long-standing goal has been to identify the bifurcation values, especially the bifurcation values of energy for fixed non-zero angular momentum, and to describe the integral manifolds at the regular values. Alain Albouy identified two categories of singular values of energy: those corresponding to bifurcations at relative equilibria; and those corresponding to “bifurcations at infinity”, and demonstrated that these are the only possible bifurcation values. This work completes the identification of bifurcations for the four-body problem with equal masses, confirming that, in this setting, Albouy’s necessary conditions for bifurcation are also sufficient conditions: bifurcations of the integral manifolds occur at all of the singular values of energy. A recent study examined the bifurcations at infinity; this work evaluates the four bifurcations at relative equilibria. To establish that the topology of the integral manifolds changes at each of these values, and to describe the manifolds at the regular values of energy, the homology groups of the integral manifolds are computed for the five energy regions on either side of the singular values. The homology group calculations establish that all four energy levels are indeed bifurcation values, and allows some of the global properties of the integral manifolds to be explored.

In this paper, we prove the reducibility for some linear quasi-periodic Hamiltonian derivative wave and half-wave equations under the Brjuno–Rüssmann non-resonance conditions. This is an extension of previous results of reducibility on Hamiltonian PDEs that required stronger (Diophantine) non-resonance conditions.

We clarify the geometric structure of non-negative traveling waves for the spatial one-dimensional degenerate parabolic equation (U_{t}=U^{p}(U_{xx}+mu U)-delta U). This equation has a nonlinear term with a parameter (p>0) and the cases (0<p<1) and (p>1) have been investigated in the author’s previous studies. It has been pointed out that the classifications of the traveling waves for these two cases are not the same and thus a bifurcation phenomenon occurs at (p=1). However, the classification of the case (p=1) remains open since the conventional approaches do not work for this case, which have prevented us to understand how the traveling waves bifurcate. The difficulty for the case (p=1) is that the corresponding ordinary differential equation through the Poincaré compactification has the non-hyperbolic equilibrium at infinity and we need to estimate the asymptotic behaviors of the trajectories near it. In this paper, we solve this problem by using a new asymptotic approach, which is completely different from the asymptotic analysis performed in the previous studies, and clarify the structure of the traveling waves in the case of (p=1). We then discuss the rich structure of traveling waves of the equation from a geometric point of view.

This paper is concerned with traveling fronts of spatially periodic reaction–diffusion equations with combustion nonlinearity in (mathbb {R}^N). It is known that for any given propagation direction (ein mathbb {S}^{N-1}), the equation admits a pulsating front connecting two equilibria 0 and 1. In this paper we firstly give exact asymptotic behaviors of the pulsating front and its derivatives at infinity, and establish uniform decay estimates of the pulsating fronts at infinity on the propagation direction (ein mathbb {S}^{N-1}). Following the uniform estimates, we then show continuous Fréchet differentiability of the pulsating fronts with respect to the propagation direction. Lastly, using the differentiability, we establish the existence, uniqueness and stability of curved fronts with V-shape in (mathbb {R}^2) by constructing suitable super- and subsolutions.

We deal with a four-component reaction–diffusion system with mass conservation in a bounded domain with the Neumann boundary condition. This system serves as a model describing the segregation pattern which emerges during the maintenance phase of asymmetric cell devision. By utilizing the mass conservation, the stationary problem of the system is reduced to a two-component elliptic system with nonlocal terms, formulated as the Euler–Lagrange equation of an energy functional. We first establish the spectral comparison theorem, relating the stability/instability of equilibrium solutions to the four-component system to that of the two-component system. This comparison follows from examining the eigenvalue problems of the linearized operators around equilibrium solutions. Subsequently, with an appropriate scaling, we prove a (Gamma )-convergence of the energy functional. Furthermore, in a cylindrical domain, we prove the existence of equilibrium solutions with monotone profile representing a segregation pattern. This is achieved by applying the gradient flow and the comparison principle to the reduced two-component system.

The viscous regularization of an ill-posed diffusion equation with bistable nonlinearity predicts a hysteretic behavior of dynamical phase transitions but a complete mathematical understanding of the intricate multiscale evolution is still missing. We shed light on the fine structure of propagating phase boundaries by carefully examining traveling wave solutions in a special case. Assuming a trilinear constitutive relation we characterize all waves that possess a monotone profile and connect the two phases by a single interface of positive width. We further study the two sharp-interface regimes related to either vanishing viscosity or the bilinear limit.

We consider the parabolic-difference system ( Big ({dot{u}}(t,x), v(t, x)Big )=Big (D, u_{xx}(t, x)hspace{-0.06cm}-hspace{-0.06cm}f(u(t, x))+Hv(t-h, cdot )(x), ,, g(u(t, x))+B v(t-h, cdot )(x)Big )), ( t>0, xin {{mathbb {R}}},) which appears in a model for hematopoietic cells population. We prove the global stability of semi-wavefronts ((phi _c, varphi _c)) for this system. More precisely, for an initial history ((u_0, v_0)) we study the convergence to zero of the associated perturbation (P(t)=(u(t)-phi _c, v(t)-varphi _c)), as (trightarrow +infty ), in a suitable Banach space Y; we prove that if the initial perturbation satisfies (P_0in C([-h, 0], Y)), then (P(t)rightarrow 0) in two cases: (i) (v_0=varphi _c), for all (hge 0) or (ii) (v_0not equiv varphi _c) for all (hle h_*) and some (h_*=h_*(B)). This result is obtained by analyzing an abstract integral equation with infinite delay. Also, our main result allow us to obtain a result about the uniqueness of these semi-wavefronts.

We are concerned with the well-posedness of the nonlinear wave system, which is a first-order hyperbolic system, in the vicinity of a right-angled spatial corner. The problem can be expressed as an initial boundary value problem (IBVP) involving a second-order hyperbolic equation in a spatial domain with a corner. The main difficulty in establishing the local well-posedness of the problem arises from the lack of smoothness in the spatial domain due to the presence of the corner point. Additionally, the Neumann-type boundary conditions on both edges of the corner angle do not satisfy the linear stability condition, posing challenges in obtaining higher-order a priori estimates for the boundary terms in the analysis. To address the corner singularity, modified extension methods will be employed in this paper. Furthermore, new techniques will be developed to control the boundary terms, leveraging the observation that the boundary operators are co-normal.

We are interested in reaction–diffusion systems, with a conservation law, exhibiting a Hopf bifurcation at the spatial wave number ( k = 0 ). With the help of a multiple scaling perturbation ansatz a Ginzburg–Landau equation coupled to a scalar conservation law can be derived as an amplitude system for the approximate description of the dynamics of the original reaction–diffusion system near the first instability. We use the amplitude system to show the global existence of all solutions starting in a small neighborhood of the weakly unstable ground state for original systems posed on a large spatial interval with periodic boundary conditions.