Nonlinear Analysis-Real World Applications最新文献

In this article, we study the long-time asymptotic properties of a non-linear and non-local equation of diffusive type which describes the rock–paper–scissors game in an interconnected population. We fully characterize the self-similar solution and then prove that the solution of the initial–boundary value problem converges to the self-similar profile with an algebraic rate.

In this paper, we study the existence of solutions for the following eigenvalue problem: $\left(\mathrm{LP}\right)\left(\begin{array}{ccc}(-\Delta +d_1)(-\Delta +d_2)u+m\left(x\right)u=\lambda a\left(x\right)u& & \text{in}\Omega \\ u⁄\equiv 0,u\ge 0& & \text{in}\Omega \\ \Delta u=u=0& & \text{on}\partial \Omega \end{array}\right)$ where $\Omega \subset R^N$ is a smooth bounded domain, $d_1,d_2\in R$ and $a\left(\cdot \right),m\left(\cdot \right)\in L^{\infty }\left(\Omega \right)$ may have indefinite sign.

This paper deals with a PDE model of two species competing for a single limiting nutrient resource in the unstirred chemostat in which one microbial species is of the variable yield. The introduction of the variable yield makes the conservation law fail. We first investigate the uniqueness of positive steady-state solution and dynamical behavior of the single species model. Then we establish the existence and structure of coexistence solutions of two species system. It turns out that the positive bifurcation branch connects two semi-trivial solution branch. Finally, we analyze the dynamical behavior of two species system, and the result shows that the two species system is uniformly persistent.

In this paper, we study the orbital stability of traveling wave solutions to the Gallay and Mascia (GM) reduction of the Gatenby–Gawlinski model. The heteroclinic solutions provided by Gallay and Mascia represent the propagation of a tumor front into healthy tissue. Orbital stability is crucial to investigating models as it determines which solutions are likely to be observed in practice. Through constructing the unstable manifold to connect fixed states of the GM model and applying a shooting argument, we constructed front solutions. After numerically generating front solutions, we studied stability by constructing the spectrum for various parameters of the GM model. We see no evidence of point eigenvalues in the right half-plane, leaving the essential spectrum as the only possible source of instability. These findings show that Gallay and Mascia’s derived heteroclinic solutions are likely to be observed physically in biological systems and are stable for various tumor growth speeds.

In this work, we pose and solve the time-optimal navigation problem considered on a slippery mountain slope modeled by a Riemannian manifold of an arbitrary dimension, under the action of a cross gravitational wind. The impact of both lateral and longitudinal components of gravitational wind on the time geodesics is discussed. The varying along-gravity effect depends on traction in the presented model, whereas the cross-gravity additive is taken entirely in the equations of motion, for any direction and gravity force. We obtain the conditions for strong convexity and the purely geometric solution to the problem is given by a new Finsler metric, which belongs to the type of general $(\alpha ,\beta )$-metrics. The proposed model enables us to create a direct link between the Zermelo navigation problem and the slope-of-a-mountain problem under the action of a cross gravitational wind. Moreover, the behavior of the Finslerian indicatrices and time-minimizing trajectories in relation to the traction coefficient and gravitational wind force are explained and illustrated by a few examples in dimension two. This also compares the corresponding solutions on the slippery slopes under various cross- and along-gravity effects, including the classical Matsumoto’s slope-of-a-mountain problem and Zermelo’s navigation.

The existence and number of limit cycles of planar piecewise linear systems with two improper nodes are studied. By constructing the Poincaré half maps and the successor function, we prove that such systems have at most one limit cycle, and when the limit cycle exists, it must be hyperbolic. Furthermore, we explicitly give the parameter regions where the limit cycle exists.

This study develops an epidemic model to analyze the dynamics of SARS-CoV-2 and dengue coinfection in a population. The population is divided into sixteen compartments for humans and three for vectors. The model’s validity is ensured by maintaining bounded and non-negative solutions. The Basic Reproduction Number (BRN) is calculated for each sub-model to assess stability at equilibrium points. Sensitivity analysis identifies key parameters influencing the model. The complete coinfection model is analyzed to identify equilibrium points and evaluate stability conditions. The reciprocal influence of SARS-CoV-2 and dengue diseases is examined. An optimal control problem is formulated, incorporating six strategies: COVID-19 protection, mosquito bite prevention, treatment for COVID-19 and dengue, mosquito control, and coinfection treatment. Numerical simulations validate the effectiveness of these control strategies for the coinfection model and its sub-models.

This paper examines a nonlinear Kirchhoff plate equation, where the control acts in bilinear form within the boundary of the mentioned equation. The objective is to construct a distributed control to guide such a system from the initial state to the desired state in the final time, while minimizing a quadratic functional cost defined as the sum of the norm difference between the aforementioned state and a desired equation with an energy term. We show how to approximate the solution of the nonlinear Kirchhoff plate equation to a desired objective, indicating the existence of optimal control in specific cases. and deriving the optimally conditions for a closed convex set. Moreover, it is shown that sufficient conditions ensures the uniqueness of control optimal. Furthermore, we provide a concise numerical methodology that involves the integration of finite element and finite difference discretization methods. The approach incorporates Newton’s linearization method to assess the computational performance of the controlled problem, using the Freefem++ software.

This work considers the Keller–Segel consumption system $\left(\begin{array}{cc}{u}_t=\Delta \left(u{v}^{-\alpha }\right)+au-b{u}^{\gamma },& x\in \Omega ,t>0,\\ v_t=\Delta v-uv,& x\in \Omega ,t>0\end{array}\right)$under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset R^n,n\ge 2$, where the parameters $a>0$, $b>0$, $\gamma \ge 2$ and $\alpha \in (0,1)$, the initial data $u_0\in C^0\left(\bar{\Omega }\right)$, $v_0\in W^{1,\infty }\left(\Omega \right)$, $u_0\ge 0(⁄\equiv 0)$ and $v_0>0$ in $\bar{\Omega }$ with ${\Vert v_0\Vert }_{L^{\infty }\left(\Omega \right)}<exp\left(\frac{ln\left(\frac{1-\alpha }{\alpha }\cdot \frac{8}{n}\right)}{\alpha }\right).$