Traveling wave solutions for a two-species competitive Keller–Segel chemotaxis system

In this paper, the traveling wave problem for a two-species competition reaction–diffusion–chemotaxis Lotka–Volterra system is investigated. Upper and lower solutions method and fixed point theory are employed to show the existence of traveling wave solutions connecting the coexistence constant steady state with zero state for all large enough wave speed $c$, and conversely, when $c$ is small, we prove there is no traveling wave solution.     

A remark on the existence of entire positive solutions for a class of semilinear elliptic systems

Under the simple conditions on f and g, we show that entire positive radial solutions exist for the semilinear elliptic system Δu=p(|x|)f(v), Δv=q(|x|)g(u), x∈RN, N?3, where the functions are continuous.     

Existence and qualitative features of entire solutions for delayed reaction diffusion system: the monostable case

The paper is concerned with the existence and qualitative features of entire solutions for delayed reaction diffusion monostable systems. Here the entire solutions mean solutions defined on the $ (x,t)\in\mathbb{R}^{N+1} $. We first establish the comparison principles, construct appropriate upper and lower solutions and some upper estimates for the systems with quasimonotone nonlinearities. Then, some new types of entire solutions are constructed by mixing any finite number of traveling wave fronts with different speeds $ c\geq c_* $ and propagation directions and a spatially independent solution, where $c_*>0$ is the critical wave speed. Furthermore, various qualitative properties of entire solutions are investigated. In particularly, the relationship between the entire solution, the traveling wave fronts and a spatially independent solution are considered, respectively. At last, for the nonquasimonotone nonlinearity case, some new types of entire solutions are also investigated by introducing two auxiliary quasimonotone controlled systems and establishing some comparison theorems for Cauchy problems of the three systems.     

Spreading speeds and traveling waves for abstract monostable evolution systems

This paper is devoted to the development of the theory of spreading speeds and traveling waves for abstract monostable evolution systems with spatial structure. Under appropriate assumptions, we show that the spreading speeds coincide with the minimal wave speeds for monotone traveling waves in the positive and negative directions. Then we use this theory to study the spatial dynamics of a parabolic equation in a periodic cylinder with the Dirichlet boundary condition, a reaction-diffusion model with a quiescent stage, a porous medium equation in a tube, and a lattice system in a periodic habitat.     

Existence and uniqueness of entire solutions for a reaction-diffusion equation

We consider entire solutions of u_t=u_xx-f(u), i.e. solutions that exist for all (x,t)∈R², where f(0)=f(1)=0<f^′(0). In particular, we are interested in the entire solutions which behave as two opposite wave fronts of positive speed(s) approaching each other from both sides of the x-axis and then annihilating in a finite time. In the case f^′(1)>0, we show that such entire solution exists and is unique up to space-time translations. In the case f^′(1)<0, we derive two families of such entire solutions. In the first family, one cannot be any space-time translation of the other. Yet all entire solutions in the second family only differ by a space-time translation.     

On existence of periodic solutions for gradient systems with applications

In this paper we prove the existence of periodic solutions for gradient systems in finite and infinite dimensional spaces. The techniques of the proofs are based on the application of a global inverse functions theorem, the Schäefer fixed point theorem and the Faedou–Galerkin method. We apply our results in order to solve nonlinear reaction–diffusion equations with Dirichlet and Neumann boundary conditions.     

Pulsating fronts and front-like entire solutions for a reaction–advection–diffusion competition model in a periodic habitat

This paper is devoted to the study of pulsating fronts and pulsating front-like entire solutions for a reaction–advection–diffusion model of two competing species in a periodic habitat. Under certain assumptions, the competition system admits a leftward and a rightward pulsating fronts in the bistable case. In this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts. Some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the x-axis, which turn out to be unique and Liapunov stable 2-dimensional manifolds of solutions, furthermore, the leftward and rightward pulsating fronts are on the boundary of these 2-dimensional manifolds. The others behave as the two pulsating fronts propagating from one side of the x-axis, the faster one then invades the slower one as $t\to +\infty$. These kinds of pulsating front-like entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat.     

Counting and ordering periodic stationary solutions of lattice Nagumo equations

We study the rich structure of periodic stationary solutions of Nagumo reaction diffusion equation on lattices. By exploring the relationship with Nagumo equations on cyclic graphs we are able to divide these periodic solutions into equivalence classes that can be partially ordered and counted. In order to accomplish this, we use combinatorial concepts such as necklaces, bracelets and Lyndon words.     

Generalized traveling-wave solutions of nonlinear reaction–diffusion equations with delay and variable coefficients

The paper presents a number of new exact solutions to nonlinear reaction–diffusion equations with delay of the form $c\left(x\right)u_t={\left[a\left(x\right)u_x\right]}_x+b\left(x\right)F(u,w),w=u(x,t-\tau ),$where $\tau >0$ is the delay time, and $F(u,w)$ is an arbitrary function of two arguments. Solutions are sought in the form of a generalized traveling-wave, $u=U\left(z\right)$ with $z=t+\theta \left(x\right)$. It is shown that one of the two functional coefficients $a\left(x\right)$ and $b\left(x\right)$ of the equation considered can be specified arbitrarily. Examples of delay reaction–diffusion equations and their solutions are given. New exact solutions of few other nonlinear delay PDEs are also obtained.     

Computational analysis of glycolytic reaction in open spatial reactor

We consider the computational analysis of processes within the spatially-distributed model simulating the glycolytic reaction taking place in the one-side fed open chemical reactor. The main point of the simulation is the decomposition of the reaction–diffusion system into unidirectional reaction in a bulk supplied by feedback terms stated as boundary conditions on the lower boundary of the reactor, i.e. the unique plane where an exchange with an outer medium is possible within the real experimental situation. Analysis of the curvature of the reagents distribution curves proves kinematic character of the observed lateral waves corresponding to the picture of experimentally observed glycolytic traveling waves. At the same time, their origin relates to diffusion of the reagents in a vertical cross-section of the reactor. Study of the solutions for the concerned reaction–diffusion model in the case of stochastically different diffusion coefficients reveals the Turing structures.     

A remark on uniqueness of large solutions for elliptic systems of competitive type

We prove that the semilinear system Δu=a(x)upvq, Δv=b(x)urvs in a smooth bounded domain Ω⊂RN has a unique positive solution with the boundary condition u=v=+∞ on ∂Ω, provided that p,s>1, q,r>0 and (p−1)(s−1)−qr>0. The main novelty is imposing a growth on the possibly singular weights a(x), b(x) near ∂Ω, rather than requiring them to have a precise asymptotic behavior.     

Viscosity solutions of a class of degenerate quasilinear parabolic equations with Dirichlet boundary condition

This paper is concerned with viscosity solutions for a class of degenerate quasilinear parabolic equations in a bounded domain with homogeneous Dirichlet boundary condition. The equation under consideration arises from a number of practical model problems including reaction–diffusion processes in a porous medium. The degeneracy of the problem appears on the boundary and possibly in the interior of the domain. The goal of this paper is to establish some comparison properties between viscosity upper and lower solutions and to show the existence of a continuous viscosity solution between them. An application of the above results is given to a porous-medium type of reaction–diffusion model which demonstrates some distinctive properties of the solution when compared with the corresponding semilinear problem.     

New periodic solutions for a class of singular Hamiltonian systems

In this paper, we apply a variant of the famous Mountain Pass Lemmas of Ambrosetti-Rabinowitz and Ambrosetti-Coti Zelati with (PSC)c type condition of Palais-Smale-Cerami to study the existence of new periodic solutions with a prescribed energy for symmetrical singular second order Hamiltonian conservative systems with weak force type potentials.     

Finding all solutions of nonlinearly constrained systems of equations

A new approach is proposed for finding all-feasible solutions for certain classes of nonlinearly constrained systems of equations. By introducing slack variables, the initial problem is transformed into a global optimization problem (P) whose multiple global minimum solutions with a zero objective value (if any) correspond to all solutions of the initial constrained system of equalities. All-globally optimal points of (P) are then localized within a set of arbitrarily small disjoint rectangles. This is based on a branch and bound type global optimization algorithm which attains finite-convergence to each of the multiple global minima of (P) through the successive refinement of a convex relaxation of the feasible region and the subsequent solution of a series of nonlinear convex optimization problems. Based on the form of the participating functions, a number of techniques for constructing this convex relaxation are proposed. By taking advantage of the properties of products of univariate functions, customized convex lower bounding functions are introduced for a large number of expressions that are or can be transformed into products of univariate functions. Alternative convex relaxation procedures involve either the difference of two convex functions employed in BB [23] or the exponential variable transformation based underestimators employed for generalized geometric programming problems [24]. The proposed approach is illustrated with several test problems. For some of these problems additional solutions are identified that existing methods failed to locate.