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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]

Abstract There is the paper by H. Tietze published in 1905 on differential transcendence of solutions of difference Riccati equations. In this paper, we clarify the essence of Tietze's treatment and make it purely algebraic. As an application, the q -Airy equation is studied. [ABSTRACT FROM AUTHOR]

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity { u t = Δ u − ∇ ⋅ ( χ ( v ) u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R n ( n ≥ 3 ) is a bounded domain with smooth boundary, ξ , μ > 0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4] , Tao [10] , Tao and Winkler [11] ); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system. [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]

Abstract In this paper, we show the existence of the first nontrivial family of analytic stationary patch solutions of the SQG and gSQG equations. This answers an open problem in F. de la Hoz et al. (2016) [13]. [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 introduce a new concept of completely linear degeneracy for quasilinear hyperbolic systems in several space variables, and then get an interesting property for multidimensional hyperbolic conservation laws satisfying our new definition. For applications, we give some examples arising from mathematics and physics at last. [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 consider the following nonlinear coupled elliptic system ( A ε ) { − ε 2 Δ u + P ( x ) u = μ 1 u 3 + β u v 2 in R N , − ε 2 Δ v + Q ( x ) v = μ 2 v 3 + β u 2 v in R N , u > 0 , v > 0 in R N , u → 0 , v → 0 as | x | → + ∞ , where ε > 0 is a parameter, μ 1 , μ 2 > 0 and β > 0 are constants, and P ( x ) and Q ( x ) are two nonnegative, smooth functions with different nondegenerate critical points and separated zero sets. Due to the Lyapunov–Schmidt reduction method and the Maximum Principle, we show that when β is less than a small positive number, there exists an ε 0 > 0 such that for any 0 < ε < ε 0 , ( A ε ) has a segregated vector solution where ( u ε , v ε ) and u ε is trapped in a neighborhood of the nondegenerate critical points of P ( x ) as well as the zero sets of P ( x ) , and v ε is trapped in a neighborhood of the nondegenerate critical points of Q ( x ) as well as the zero sets of Q ( x ) . Moreover, the amplitudes of u ε (res v ε ) around the nondegenerate critical points and the zero sets of P ( x ) (res Q ( x ) ) are of a different order compared with ε . To the best of our knowledge, these multi-scale solutions to the system have not been obtained previously. [ABSTRACT FROM AUTHOR]

This paper presents the major achievements of the 20th century regarding Karamata functions and the theory of differential equations, made mostly by V. Marić, M. Tomić, E. Omey, J.L. Geluk. The connection between these notions was first noticed by V.G. Avakumović (1910–1990). Slowly and regularly varying functions were introduced by J. Karamata (1902–1967). A group of mathematicians from the Karamata School of classical mathematical analysis were pioneers in research on these functions and their role in the theory of differential equations. Special attentions is given to the study of the Thomas–Fermi, Emden–Fowler and Friedmann equations, as well as the classical second order linear differential equations. [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, we study the asymptotic behavior of solutions of a class of non-autonomous delay differential equations. These equations have important practical applications and generalize those on which Bernfeld and Haddock conjectured that each solution of the equations tends to a constant. It is shown that every solution of the equations is bounded and tends to a constant as t → + ∞ , which improves and extends some existing ones. [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]

This paper considers a semilinear integro-differential equation of Volterra type which interpolates semilinear heat and wave equations. Global existence of solutions is showed in spaces of Besov type based in Morrey spaces, namely Besov–Morrey spaces. Our initial data is larger than the previous works and our results provide a maximal existence class for semilinear interpolated heat-wave equation. Some symmetries, self-similarity and asymptotic behavior of solutions are also investigated in the framework of Besov–Morrey spaces. [ABSTRACT FROM AUTHOR]

In this paper, we first establish the existence and uniqueness of L p ( p > 1 ) solutions for multidimensional backward stochastic differential equations (BSDEs) under a weak monotonicity condition together with a general growth condition in y for the generator g . Then, we overview several conditions related closely to the weak monotonicity condition and compare them in an effective way. Finally, we put forward and prove a stability theorem and a comparison theorem of L p ( p > 1 ) solutions for this kind of BSDEs. [ABSTRACT FROM AUTHOR]

It is well-known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations. In the present work, a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem. The new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove existence of optimal design for the new model. This work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional (see [14] ). [ABSTRACT FROM AUTHOR]

The Bohl exponents, similarly as the Lyapunov exponents, are one of the most important numerical characteristics of dynamical systems used in control theory. By contrast with properties of the Lyapunov characteristics, properties of the Bohl exponents are much less investigated. In this paper we present some new properties of the upper Bohl exponents of the diagonal discrete time-varying linear system. [ABSTRACT FROM AUTHOR]

Tachiya investigated a class of infinite products of rational functions arithmetically and established that their values at certain algebraic points are algebraic numbers if and only if the infinite products are rational functions. In this paper we prove further arithmetical results for the values of these infinite products both qualitatively and quantitatively, which can be carried out by studying these infinite products as formal power series carefully. [ABSTRACT FROM AUTHOR]

Let f ( n ) be a multiplicative function satisfying | f ( n ) | ≤ 1 , q ( ≤ N 2 ) be a prime number and a be an integer with ( a , q ) = 1 , χ be a non-principal Dirichlet character modulo q . In this paper, we shall prove that ∑ n ≤ N f ( n ) χ ( n + a ) ≪ N q 1 4 log ⁡ log ⁡ ( 6 N ) + q 1 4 N 1 2 log ⁡ ( 6 N ) + N log ⁡ log ⁡ ( 6 N ) and that ∑ n ≤ N f ( n ) χ ( n + a 1 ) ⋯ χ ( n + a t ) ≪ N q 1 4 log ⁡ log ⁡ ( 6 N ) + q 1 4 N 1 2 log ⁡ ( 6 N ) + N log ⁡ log ⁡ ( 6 N ) , where t ≥ 2 , a 1 , … , a t are pairwise distinct integers modulo q . [ABSTRACT FROM AUTHOR]

The paper is concerned with orders of the Kolmogorov, Gelfand and linear widths of function classes on sets with tree-like structure. As examples we consider weighted Sobolev classes on a John domain and some function classes on trees. [ABSTRACT FROM AUTHOR]

In this paper we prove two global existence results for Hughes' model for pedestrian flows under assumptions that ensure that the traces of the solutions along the turning curve are zero for all positive times. [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]

• New analysis of the ligand-induced dimerisation process for VEGFR is presented. • A key idea in the mathematical model and analysis is kinetic binding cooperativity. • The nonlinear model is in contrast with the corresponding linear model for GPCRs. • Numerical and asymptotic analysis reveal insights into the underlying LID dynamics. • Our simulations show excellent fit to recent experimental data for VEGFR. The vascular endothelial growth factor (VEGF) receptor (VEGFR) system plays a role in cancer and many other diseases. It is widely accepted that VEGFR receptors dimerise in response to VEGF binding. However, analysis of these mechanisms and their implications for drug development still requires further exploration. In this paper, we present a mathematical model representing the binding of VEGF to VEGFR and the subsequent ligand-induced dimerisation. A key factor in this work is the qualitative and quantitative effect of binding cooperativity, which describes the effect that the binding of a ligand to a receptor has on the binding of that ligand to a second receptor, and the dimerisation of these receptors. We analyse the ordinary differential equation system at equilibrium, giving analytical solutions for the total amount of ligand bound. For time-course dynamics, we use numerical methods to explore possible behaviours under various parameter regimes, while perturbation analysis is used to understand the intricacies of these behaviours. Our simulation results show an excellent fit to experimental data, towards validating the model. [ABSTRACT FROM AUTHOR]