Nonnegative Initial Conditions Research Articles

On maximum and comparison principles for parabolic problems with the p-Laplacian

We investigate strong and weak versions of maximum and comparison principles for a class of quasilinear parabolic equations with the $p$-Laplacian $$ \partial_t u - \Delta_p u = \lambda |u|^{p-2} u + f(x,t) $$ under zero boundary and nonnegative initial conditions on a bounded cylindrical domain $\Omega \times (0, T)$, $\lambda \in \mathbb{R}$, and $f \in L^\infty(\Omega \times (0, T))$. Several related counterexamples are given.

The global behavior of a quadratic difference equation

In this paper we will present the Julia set and the global behavior of a quadratic second order difference equation of type xn+1 = axnxn-1 + ax2n-1 + bxn-1 where a > 0 and 0 ? b < 1 with non-negative initial conditions.

Global Dynamics of Rational Difference Equations xn+1=xn+xn-1/q+ynyn-1 and yn+1=yn+yn-1/p+xnxn-1

Global dynamics of a system of nonlinear difference equations was investigated, which had five kinds of equilibria including isolated points and a continuum of nonhyperbolic equilibria along the coordinate axes. The local stability of these equilibria was analyzed which led to nine regions in the parameters space. The solution of the system converged to the equilibria or the boundary point (+∞,0) or (0,+∞) in each region depending on nonnegative initial conditions. These results completely described the behavior of the system.

Large time behavior of solution to an attraction–repulsion chemotaxis system with logistic source in three dimensions

This paper studies the attraction–repulsion chemotaxis system with logistic source ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u), vt=Δv−α1v+β1u, wt=Δw−α2w+β2u in a smooth bounded convex domain Ω⊂R3, subject to nonnegative initial data and homogeneous Neumann boundary conditions, where χ, ξ, αi and βi(i=1,2) are positive parameters and the logistic source function f fulfills f(s)=s−μsγ+1,s≥0,μ>0andγ≥1. It is shown that this system possesses a unique global bounded classical solution under the conditions αi≥12 and μ≥max⁡{(412χβ1+9ξβ2)γ,(9χβ1+412ξβ2)γ}. Furthermore, whenever u0≢0 and for any γ∈N, the solution of the system approaches to the steady state ((1μ)1γ,(1μ)1γβ1α1,(1μ)1γβ2α2) in the norm of L∞(Ω) as t→∞.

Predation with indirect effects in fluctuating environments

We investigate the long-term dynamics for a predation model of Plankton community with indirect effects, under fluctuating environments. A random version and a stochastic version with multiplicative noise of the model are discussed and compared. We prove that the solutions to both versions are nonnegative and bounded given any nonnegative positive initial conditions. We also prove that both the random system and the stochastic system possess a unique random attractor under the same set of assumptions, by using the classical theory of random dynamical systems. In addition, we provide conditions under which coexistence of species exists for the random system.

On the (strict) positivity of solutions of the stochastic heat equation

We give a new proof of the fact that the solutions of the stochastic heat equation, started with nonnegative initial conditions, are strictly positive at positive times. The proof uses concentration of measure arguments for discrete directed polymers in Gaussian environments, originated in M. Talagrand’s work on spin glasses and brought to directed polymers by Ph. Carmona and Y. Hu. We also get slightly improved bounds on the lower tail of the solutions of the stochastic heat equation started with a delta initial condition.

Global Period-Doubling Bifurcation of Quadratic Fractional Second Order Difference Equation

We investigate the local stability and the global asymptotic stability of the difference equationxn+1=αxn2+βxnxn-1+γxn-1/Axn2+Bxnxn-1+Cxn-1,n=0,1,…with nonnegative parameters and initial conditions such thatAxn2+Bxnxn-1+Cxn-1>0, for alln≥0. We obtain the local stability of the equilibrium for all values of parameters and give some global asymptotic stability results for some values of the parameters. We also obtain global dynamics in the special case, whereβ=B=0, in which case we show that such equation exhibits a global period doubling bifurcation.

Global dynamics of quadratic second order difference equation in the first quadrant

We investigate the global behavior of a quadratic second order difference equationxn+1=Axn2+Bxnxn-1+Cxn-12+Dxn+Exn-1+F,n=0,1,…with non-negative parameters and initial conditions. We find the global behavior for all ranges of parameters and determine the basins of attraction of all equilibrium points.

Basins of attraction of equilibrium and boundary points of second-order difference equations

We investigate the global behaviour of the difference equation of the formwith non-negative parameters and initial conditions such that . We give a precise description of the basins of attraction of different equilibrium points, and show that the boundaries of the basins of attractions of different locally asymptotically stable equilibrium points or non-hyperbolic equilibrium points are in fact the global stable manifolds of neighbouring saddle or non-hyperbolic equilibrium points. Different types of bifurcations when one or more parameters are 0 are explained.

On a fully discrete finite-difference approximation of a nonlinear diffusion–reaction model in microbial ecology

In this work, we introduce a numerical method to approximate the solutions of a multidimensional parabolic partial differential equation with nonlinear diffusion and reaction, subject to nonnegative initial data and homogeneous boundary conditions of the Neumann type. The equation considered is a model for both the growth of biological films and the propagation of mutant genes which are advantageous to a population. The initial-boundary-value problem under investigation is fully discretized temporally and spatially following a finite-difference methodology which results in a simple, linear, implicit scheme that is consistent with respect to the continuous problem. The method is a two-step technique that preserves the positivity and the boundedness of initial profiles. We provide some simulations on the growth of microbial colonies, and comparisons versus a standard approach.

Open problems and conjectures on rational systems in three dimensions

We present some open problems and conjectures on Ratio- nal Systems in three dimensions, or higher, with nonnegative parameters and with nonnegative initial conditions such that the denominators are always positive. We also employ the method of Full Limiting Sequences to confirm an outstanding conjecture on k th -order rational difference equations.

On the boundedness character of rational systems in the plane

We study first-order systems of two rational difference equations, In particular, we assume non-negative parameters and non-negative initial conditions. We develop several approaches which allow us to resolve some conjectures regarding the boundedness of first-order systems of two rational difference equations.

On the dynamics of the recursive sequence [formula omitted

In this paper, we investigate the global behavior of the difference equation x n + 1 = α x n − 1 β + γ ∑ k = 1 t x n − 2 k ∏ k = 1 t x n − 2 k , n = 0 , 1 , … where β is a positive parameter and α , γ are non-negative parameters, with non-negative initial conditions.

On the recursive sequence [formula omitted

In this paper, we investigate the global behavior and boundedness of the difference equation x n + 1 = a + bx n A + Bx n - 1 k , n = 0 , 1 , … , with positive coefficients and non-negative initial conditions.

On the dynamics of the recursive sequencexn+1=xn−1β+γxn−22xn−4+γxn−2xn−42

In this paper, we investigate the global behavior of the difference equation x n + 1 = x n − 1 β + γ x n − 2 2 x n − 4 + γ x n − 2 x n − 4 2 , n = 0 , 1 , … with positive parameters and non-negative initial conditions.

Some boundedness results for systems of two rational difference equations

Abstract We study k th order systems of two rational difference equations $$ x_n = \frac{{\alpha + \sum\nolimits_{i = 1}^k {\beta _i x_{n - 1} + } \sum\nolimits_{i = 1}^k {\gamma _i y_{n - 1} } }} {{A + \sum\nolimits_{j = 1}^k {B_j x_{n - j} + } \sum\nolimits_{j = 1}^k {C_j y_{n - j} } }}, y_n = \frac{{p + \sum\nolimits_{i = 1}^k {\delta _i x_{n - i} + } \sum\nolimits_{i = 1}^k {\varepsilon _i y_{n - i} } }} {{q + \sum\nolimits_{j = 1}^k {D_j x_{n - j} + } \sum\nolimits_{j = 1}^k {E_j y_{n - j} } }} n \in \mathbb{N} $$. In particular, we assume non-negative parameters and non-negative initial conditions, such that the denominators are nonzero. We develop several approaches which allow us to extend well known boundedness results on the k th order rational difference equation to the setting of systems in certain cases.

BOUNDEDNESS OF SOLUTIONS OF A HAPTOTAXIS MODEL

In this paper we prove the existence of global solutions of the haptotaxis model of cancer invasion for arbitrary non-negative initial conditions. Uniform boundedness of the solutions is shown using the method of bounded invariant rectangles applied to the reformulated system of reaction-diffusion equations in divergence form with a diagonal diffusion matrix. Moreover, the analysis of the model shows how the structure of kinetics of the model is related to the growth properties of the solutions and how this growth depends on the ratio of the sensitivity function (describing the size of haptotaxis) and the diffusion coefficient. One of the implications of our analysis is that in the haptotaxis model with a logistic growth term, cell density may exceed the carrying capacity, which is impossible in the classical logistic equation and its reaction-diffusion extension.

On the rational recursive sequence

Abstract In this paper we consider the difference equation (E) with positive parameters and nonnegative initial conditions. We use the explicit formula for the solutions of equation (E) in investigating their behavior.

Stochastic Cahn–Hilliard equation with singular nonlinearity and reflection

We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space–time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche and Zambotti, we use a method based on infinite dimensional equations, approximation by regular equations and convergence of the approximated semigroup. We obtain existence and uniqueness of a solution for nonnegative initial conditions, results on the invariant measures, and on the reflection measures.

On the characterization of rational difference equations

We explore the implications of monotonic character for difference equations of order greater than one. Several techniques are developed culminating in the complete characterization of the behaviour of solutions to the k th order rational difference equation in the case As is customary we assume non-negative parameters and non-negative initial conditions.