Nonlinear Analysis-Real World Applications最新文献

In this work, we are concerned with the mathematical modeling and analysis of Ebola virus disease dynamics. Firstly, we design and analyze a nonlinear ordinary differential equations model integrating both direct and indirect transmission pathways with density-independent treatment and a composite nonlinear incidence function. We begin the analysis by proving the global existence of a unique positive and bounded solution. Then we compute the basic reproduction number on which relies the global dynamics of the model. We precisely show the existence of a unique disease-free equilibrium and that of a unique endemic equilibrium, and prove their global stability under appropriate assumptions on the basic reproduction number. Moreover, we perform the global sensitivity analysis of the basic reproduction number to assess the variability in the model predictions. We find that the forecasts closely agree with the 2014 outbreaks of the disease in Liberia and Sierra Leone. Secondly, we enrich this first model by extending it to a partially degenerate reaction–diffusion system via the inclusion of Fickian diffusion for susceptible and non-hospitalized infectious individuals in order to understand the dynamics of the disease transmission in a spatially homogeneous environment. We prove the global stability of the disease-free equilibrium and the uniform persistence when the basic reproduction number lies below and above one, respectively. Finally, numerical simulations are performed to illustrate some theoretical results obtained.

This paper deals with the global boundedness and stability of classical solutions to an important alarm-taxis ecosystem that is significant in understanding the behaviors of prey and predators. Specifically, it studies the case where prey attracts the secondary predators when threatened by the primary predators. The secondary consumers pursue the signal generated by the interaction between the prey and direct consumers. However, obtaining the necessary gradient estimates for global existence seems difficult in the critical case due to the strong coupled structure. Therefore, a new approach is developed to estimate the gradient of prey and primary predators, which takes advantage of slightly higher damping power. Subsequently, the boundedness of classical solutions in two-dimension with Neumann boundary conditions can be established by energy estimates and semigroup theory. Moreover, by constructing Lyapunov functional, it is proved that the coexistence homogeneous steady states are asymptotically stable, and the convergence rate is exponential under certain assumptions on the system coefficients.

Based on some new elementary estimates for the space–time derivatives of the heat kernel, we use a bootstrapping approach to establish quantitative estimates on the optimal decay rates for the $L^q\left(R^d\right)$ ($1\le q\le \infty$, $d\in N$) norm of the space–time derivatives of solutions to the (modified) Patlak-Keller–Segel equations with initial data in $L^1\left(R^d\right)$, which implies the joint space–time analyticity of solutions. When the $L^1\left(R^d\right)$ norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the radius of space–time analyticity in time. As a byproduct, the space analyticity is obtained for any initial data in $L^1\left(R^d\right)$. The decay estimates and space–time analyticity are also established for solutions bounded in both space and time variables. The results can be extended to a more general class of equations.

The asymptotics of weak solutions to the Boussinesq equations with no-slip boundary and moderately ill-prepared data is investigated in rotation-dominant limit regime as the Rossby number, the Froude number and the vertical viscosity tend to zero simultaneously. The new ingredient of this paper is to give a first proof of the three scale singular limit coupled with Ekman boundary layer by introducing an asymptotic profile to the original system.

This paper is concerned with the existence and regularity of global attractor $A$ for a Kirchhoff wave equation with strong damping and memory in $H$ and $H^1$, respectively. In order to obtain the existence of $A$, we mainly use the energy method in the priori estimations, and then verify the asymptotic compactness of the semigroup by the method of contraction function. Finally, by decomposing the weak solutions into two parts and some elaborate calculations, we prove the regularity of $A$.

We study the asymptotic behaviour of the solutions to Navier–Stokes unforced equations under Navier boundary conditions in a wide class of merely Lipschitz domains of physical interest. The paper draws its main motivation from celebrated results by Foias and Saut (1984) under Dirichlet conditions; here the choice of the boundary conditions requires carefully considering the geometry of the domain $\Omega$, due to the possible lack of the Poincaré inequality in presence of symmetries. In non-axially symmetric domains we show the validity of the Foias–Saut result about the limit at infinity of the Dirichlet quotient, in axially symmetric domains we provide two invariants of the flow which completely characterize the motion and we prove that the Foias–Saut result holds for initial data belonging to one of the invariants.