Nonlinear Analysis-Real World Applications最新文献

We establish the necessary conditions for the existence and multiplicity of periodic and subharmonic solutions to a second-order nonlinear ordinary differential equation (ODE). This ODE describes the motion of a bead on a rotating circular hoop subjected to a constant angular velocity $\omega$ and a $T$-periodic forcing. Our approach involves estimating bounds for the angular velocity and period using upper and lower solution methods.

In this paper, we study the Cauchy problem of the three-dimensional compressible micropolar equations in the absence of heat-conductivity. By leveraging Fourier theory and employing a refined energy method, we establish the global well-posedness of the equations for small initial data within Besov spaces. As a byproduct, we also derive the optimal time decay of solutions if the low frequency of initial data belonging to ${\dot{B}}_{2,\infty }^{-{\sigma }_1}\left(R^3\right)$.

The main focus of this article is to investigate the behavior of two strongly competitive species in a spatially heterogeneous environment using a Lotka–Volterra-type reaction–advection–diffusion model. The model assumes that one species diffuses at a constant rate, while the other species moves toward a more favorable environment through a combination of constant diffusion and directional movement. The study finds that no stable coexistence can be guaranteed when both species disperse randomly. In contrast, stable coexistence between the two species is possible when one of the species exhibits advection–diffusion. The study also reveals the existence of unstable coexistence imposed by bistability in a strongly competitive system, regardless of the diffusion type. The results are obtained by analyzing the stability of semitrivial solutions. The study concludes that the species moving toward a better environment has a competitive advantage, allowing them to survive even when their population density is initially low. Finally, the study identifies the unique globally asymptotically stable coexistence steady states of the system at high advection rates, particularly for relatively moderate interspecific competition parameters in species with directional movement. These findings underscore the crucial role of directed movement and interspecific competition coefficients in shaping the dynamics and coexistence of strongly competing species.

In this article, we analyze the phase synchronization and frequency synchronization for the bidirectionally coupled Kuramoto model under the effect of inertia. Unlike the classical Kuramoto model equipped with all-to-all coupled interaction, in the setting of this model, each oscillator ${\theta }_i$ only interacts directly with ${\theta }_{i+1}$ and ${\theta }_{i-1}$. The bidirectional interaction is a typical setting of the concatenation in power systems. Additionally, it is necessary to impose the effect of inertia in the Kuramoto model in the applications such as power systems and Josephson junction array. In this article, we first present a theory of the global convergence for frequency synchronization for the identical case. For the non-identical case, we prove that the second-order bidirectionally coupled Kuramoto model exhibits a frequency synchronization if the coupling strength is large, inertia is small, and all oscillators are initially confined to a sector. We emphasize that the arc length of this sector possesses a positive lower bound which is independent of the number of oscillators. If, in addition, all natural frequencies are identical, we further show that the phase synchronization emerges. Moreover, we demonstrate the numerical simulations to support the main results. On the other hand, we observe that the model equipped with large inertia can exhibit the synchronization. Exploring the synchronization theory for large inertia case is left as the future work.

In this paper we consider a non-linear Robin problem driven by the Orlicz $g$-Laplacian operator. Using variational technique combined with a suitable truncation and Morse theory (critical groups), we prove two multiplicity theorems with sign information for all the solutions. In the first theorem, we establish the existence of at least two non-trivial solutions with fixed sign. In the second, we prove the existence of at least three non-trivial solutions with sign information (one positive, one negative, and the other change sign) and order. The result of the nodal solution is new for the non-linear $g$-Laplacian problems with the Robin boundary condition.

The existence of backward bifurcation indicates an obstacle to disease eradication even when the basic reproduction number falls below unity. Bifurcation analysis allows us to identify causes for backward bifurcation, thereby helping to design a strategy to avoid such phenomena for disease eradication. In this study, we perform an in-depth bifurcation analysis of a malaria model incorporating cross-border mobility between two countries to explore mobility’s role in backward bifurcation. Our analysis reveals that cross-border mobility can be a primary driving force for backward bifurcation in malaria dynamics. This novel result with cross-border mobility bringing backward bifurcation advances the traditional idea of disease-induced death being the primary driver of backward bifurcation. Using the malaria case in Nepal with cross-border mobility between Nepal–India, we validated analytical results by numerical simulations. Our model predicts that the disease-free equilibrium exists only if cross-border mobility or infection abroad are absent and malaria eradication is possible in Nepal. Otherwise, there is the coexistence of three endemic equilibria with a lower and higher stable epidemic level. Results on the bifurcation of our model may be helpful to control dynamics in order to maintain the malaria epidemic at a low level if it cannot be eradicated due to the entry of cases through cross-border mobility.

This work is concerned with a competition system with nonlinear coupled reaction terms. By using Schauder’s fixed point theorem, we first prove the existence of a traveling wave solution connecting two uniform stationary states that do not satisfy the competitive ordering. Then some asymptotic spreading properties of the two species are obtained, and on this basis, we derive the multiplicity of asymptotic spreading speed of the considered system. Finally, numerical simulations corroborate the existence of traveling wave solutions satisfying different asymptotic conditions, which are theoretically established by the current paper and the reference.

We study a two-stage structured population model for which the juveniles diffuse purely by random walk while the adults exhibit long range dispersal. Questions on the persistence or extinction of the species are examined. It is shown that the population eventually dies out if the principal spectrum point ${\lambda }_p$ of the linearized system at the trivial solution is nonpositive. However, the species persists if ${\lambda }_p>0$. Moreover, at least one positive steady state exists when ${\lambda }_p>0$. The uniqueness and global stability of the positive steady-state solution is obtained under some special cases. We also establish a sup/inf characterization of ${\lambda }_p$.

This paper focuses on the existence and nonexistence of a time periodic traveling wave solution of a time-periodic reaction–diffusion SEIR epidemic model. The main feature of the model is the possible deficiency of the classical comparison principle such that many known results do not directly work. If the basic reproduction number of the model, denoted by $R_0$, is larger than one, there exists a minimal wave speed $c^{\ast }>0$ satisfying for each $c>c^{\ast }$, the system admits a nontrivial time periodic traveling wave solution with wave speed $c$ and for $c<c^{\ast }$, there exists no nontrivial time periodic traveling waves such that the system; if $R_0<1$, the system admits no nontrivial time periodic traveling waves.

This work mainly focuses on spatial decay properties of solutions to the Zakharov–Kuznetsov equation. For the two- and three-dimensional cases, it was established that if the initial condition $u_0$ verifies ${〈\sigma \cdot x〉}^r{u}_0\in L^2\left(\left(\sigma \cdot x\ge \kappa \right)\right),$ for some $r\in N$, $\kappa \in R$, being $\sigma$ be a suitable non-null vector in the Euclidean space, then the corresponding solution $u\left(t\right)$ generated from this initial condition verifies ${〈\sigma \cdot x〉}^{r}u\left(t\right)\in L^2\left(\left(\sigma \cdot x>\kappa -\nu t\right)\right)$, for any $\nu >0$. Additionally, depending on the magnitude of the weight $r$, it was also deduced some localized gain of regularity. In this regard, we first extend such results to arbitrary dimensions, decay power $r>0$ not necessarily an integer, and we give a detailed description of the gain of regularity propagated by solutions. The deduction of our results depends on a new class of pseudo-differential operators, which is useful for quantifying decay and smoothness properties on a fractional scale. Secondly, we show that if the initial data $u_0$ has a decay of exponential type on a particular half space, that is, $e^{b\sigma \cdot x}{u}_0\in L^2\left(\left(\sigma \cdot x\ge \kappa \right)\right),$ then the corresponding solution satisfies