Search

We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the C ∞ [ 0 , 1 ] steady-state solution by the method of upper and lower solution, and the uniqueness in both W 1 , 2 (0 , 1) and L ∞ (0 , 1) by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in C ([ 0 , 1 ] × [ 0 , + ∞)) ∩ C 2 , 1 ([ 0 , 1 ] × (0 , + ∞)) , which, for any continuous initial value, converges to the steady-state solution in C [ 0 , 1 ] (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species. [ABSTRACT FROM AUTHOR]

In this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the L 1 norm of potentials is known. We establish the minimization characterization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. The conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough. [ABSTRACT FROM AUTHOR]

In this paper, we continue to study a spatially heterogeneous predator–prey system where the interaction is governed by a Holling type II functional response, which has been studied in Du and Shi (2007) [14] . We further study the asymptotic profile of positive solutions and give a complete understanding of coexistence region. Moreover, a good understanding of the number, stability and asymptotic behavior of positive solutions is gained for large m . Finally, we further compare the difference of steady-state solutions between m > 0 and m = 0 . It turns out that the spatial heterogeneity of the environment and the Holling type II functional response play a very important role in this model. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if the solution of the planar N + 1 -body problem has N bodies rotating at the vertices of a regular N -gon then the N bodies have equal mass and the remaining body must be at the center. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of ground state solution for an indefinite variational problem of the type ( P ) { − Δ u + ( V ( x ) − W ( x ) ) u = f ( x , u ) in R N , u ∈ H 1 ( R N ) , where N ≥ 2 , V , W : R N → R and f : R N × R → R are continuous functions verifying some technical conditions and f possesses a critical growth. Here, we will consider the case where the problem is asymptotically periodic, that is, V is Z N -periodic, W goes to 0 at infinity and f is asymptotically periodic. [ABSTRACT FROM AUTHOR]

In this paper, we consider the radial singular solution of Δ 2 u = − 1 u p , in R N with u > 0 , p > 0 . By using some elementary ordinary differential equation arguments, we prove the above equation admits no radial singular solution for N = 3 , p ≥ 3 . In addition, the exact asymptotic behaviors of the radial singular solution as r → 0 is established for p = 1 , N ≥ 4 , which have a significant difference with the explicit singular solution. As a product, the singularity of any radial singular solution is of type II if p = 1 , N ≥ 4 . [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with an integrable two-component peakon system, which was proposed by Xia, Qiao and Zhou. We present a precise blow-up scenario and a new blow-up result for strong solutions to the system. Moreover, we prove that the strong solutions of the system maintain corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively. [ABSTRACT FROM AUTHOR]

In this paper, the formation of singularities for the nonlocal Whitham-type equations is studied. It is shown that if the lowest slope of flows can be controlled by its highest value with the bounded Whitham-type integral kernel initially, then the finite-time blow-up will occur in the form of wave-breaking. This refined wave-breaking result is established by analyzing the monotonicity and continuity properties of a new system of the Riccati-type differential inequalities involving the extremal slopes of flows. Our theory is illustrated via the Whitham equation, Camassa–Holm equation, Degasperis–Procesi equation, and their μ -versions as well as hyperelastic rod equation. [ABSTRACT FROM AUTHOR]

We point out a few mistakes in our earlier paper [2] and propose some corrections. Our main results remain valid provided some more restrictive assumptions. [ABSTRACT FROM AUTHOR]

In this paper we consider analytic planar differential systems having a first integral of the form H(x,y)=A(x)+B(x)y+C(x)y2 and an integrating factor κ(x) not depending on y. Our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results. Theorem A gives a characterization of isochronicity, a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone. Theorem B is intended for being applied in combination with Theorem A in an algebraic setting that we shall specify. Finally, Theorem C is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers. Four different applications are given to illustrate these results. [ABSTRACT FROM AUTHOR]