Nonnegative Solutions Research Articles

On Liouville-type theorems for 𝑘-Hessian equations with gradient terms

In this paper, we investigate several Liouville-type theorems related to k k -Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to S k [ D 2 u ] ≥ h ( u , | D u | ) S_k[D^2u]\ge h(u,|Du|) in R n \mathbb {R}^n . The results depend on some qualified growth conditions of h h at infinity. A Liouville-type result to subsolutions of a prototype equation S k [ D 2 u ] = f ( u ) + g ( u ) ϖ ( | D u | ) S_k[D^2u]=f(u)+g(u)\varpi (|Du|) is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to S k [ D 2 u ] = f ( u ) + g ( u ) | D u | q S_k[D^2u]=f(u)+g(u)|Du|^q for 0 ≤ q > k + 1 0\le q>k+1 is also given.

Uniqueness and stability of nonnegative solutions for a class of nonpositone problems in a ball

Uniqueness and stability of nonnegative solutions for a class of nonpositone problems in a ball

Nonnegative solutions of a coupled k-Hessian system involving different fractional Laplacians

Nonnegative solutions of a coupled k-Hessian system involving different fractional Laplacians

Dengue Transmission Dynamics: A Fractional-Order Approach with Compartmental Modeling

This work presents a quantitative analysis of the transmission dynamics of dengue using the Caputo–Fabrizio fractional-order derivative. It presents an extensive framework for modeling a dengue epidemic, including the various stages of infection and encompassing a wide range of transmission pathways. The proposed model is subjected to a rigorous qualitative study, including the determination of a non-negative solution, the assessment of the basic reproduction number, and an evaluation of local stability. Numerical solutions are obtained using the Newton method. The fractional-order operator, developed using the Caputo–Fabrizio approach, provides a refined perspective on the transmission dynamics of dengue. This study contributes to a deeper understanding of the disease’s transmission mechanisms, considering both fractional-order dynamics and diverse transmission routes, thus offering insights for enhanced disease management and control.

On a fractional (p,q)-Laplacian equation with critical nonlinearities

In this paper, we consider the following fractional ( p , q ) -Laplacian equations with critical Hardy-Sobolev exponents { ( − Δ ) p s 1 u + ( − Δ ) q s 2 u = λ | u | r − 2 u + μ | u | p s 1 ∗ ( α ) − 2 u | x | α in Ω , u = 0 in R N ∖Ω , where 0 < s 2 < s 1 < 1 < q < r ≤ p < N s 1 , 0 $ ]]> λ , μ > 0 are two parameters, 0 ≤ α < p s 1 and p s 1 ∗ ( α ) = p ( N − α ) N − p s 1 is the fractional Hardy-Sobolev critical exponent, Ω ⊂ R N is an open bounded domain with smooth boundary. By using variational methods, we show that the problem has a nontrivial nonnegative weak solution.

Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N

Abstract In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.

Optimal regularity and fine asymptotics for the porous medium equation in bounded domains

Abstract We prove the optimal global regularity of nonnegative solutions to the porous medium equation in smooth bounded domains with the zero Dirichlet boundary condition after certain waiting time T * {T^{*}} . More precisely, we show that solutions are C 2 , α ⁢ ( Ω ¯ ) {C^{2,\alpha}(\overline{\Omega})} in space, with α = 1 m {\alpha=\frac{1}{m}} , and C ∞ {C^{\infty}} in time (uniformly in x ∈ Ω ¯ {x\in\overline{\Omega}} ), for t &gt; T * {t&gt;T^{*}} . Furthermore, this allows us to refine the asymptotics of solutions for large times, improving the best known results so far in two ways: we establish a faster rate of convergence O ⁢ ( t - 1 - γ ) {O(t^{-1-\gamma})} , and we prove that the convergence holds in the C 1 , α ⁢ ( Ω ¯ ) {C^{1,\alpha}(\overline{\Omega})} topology.

A Co-infection Model for Monkeypox and HIV/AIDS: Sensitivity and Bifurcation Analyses

Monkeypox can make people very sick. The skin becomes infected with bacteria, thus causing severe skin damage. This can lead to corneal infection with loss of vision, pneumonia, difficulty swallowing, diarrhoea and vomiting leading to harsh malnutrition or dehydration, several organs inflammation or death. HIV-AIDS is a life-threatening and chronic condition. In HIV-infected individuals whose immune systems have been compromised, monkeypox mortality alone may be much higher. The co-infection of monkeypox and HIV/AIDS infections has been studied from a mathematical perspective by constructing a 13-compartment deterministic model. Basic mathematical analyses were performed on the co-infection model and the sub-models. The disease equilibrium points, the non-negativity of solutions, the basic reproduction numbers, the invariant region and the stability patterns. When the basic reproduction number is less than unity, the disease-free equilibrium points of each sub-model are globally asymptomatically stable. Certain calculations were done using the maple 18 programming language. The sensitivity analysis reveals that the parameters of the basic reproduction of the monkeypox sub-model with positive sensitivity indices are the probability of catching the monkeypox virus, the rate of effective contact, the compartment Im ’s coefficient of infection and the monkeypox vaccine’s waning rate, while the parameters of the basic reproduction of the HIV/AIDS sub-model with positive sensitivity indices are the probability of catching HIV virus, the rate effective contact, the compartment Ih ’s coefficient of infection and the compartment Ah ’s coefficient of infection. Via the centre manifold theorem, the bifurcation analysis reveals a forward bifurcation pattern for the monkeypox sub-model and the HIV/AIDS sub-model, and under a certain condition, a critical value of the monkeypox basic reproduction exists such that an effective management and possible elimination of the monkeypox infection would require that the monkeypox basic reproduction number should be kept below unity and above the critical value.

Finding an NARE whose minimal nonnegative solution represents first passage quantities in the two-dimensional Brownian motion

Finding an NARE whose minimal nonnegative solution represents first passage quantities in the two-dimensional Brownian motion

Estimation of Parameters for the Mathematical Model of the Spread of Hepatitis B in Burkina Faso Using Grey Wolf Optimizer

In this paper, we developed a mathematical model of differential susceptibility, taking into account vaccination and treatment, to simulate the transmission of the hepatitis B virus in the population of Burkina Faso. The existence and uniqueness of non-negative solutions are proved. The model has a globally asymptotically stable disease free equilibrium when the basic reproduction number R0 &lt; 1 and an endemic equilibrium when R0 &gt; 1. We estimated the parameters of the model based on hepatitis B cases from 1997 to 2020 by using a Grey Wolf Optimizer Algorithm (GWO). The results demonstrated the efficacy of the GWO algorithm in estimating the model parameters. A sensitivity analysis was carried out to determine the determining factors in the spread of hepatitis B in Burkina Faso. The estimated parameters were used to simulate the spread of hepatitis B in Burkina Faso from 1997 to 2020.

Nonnegative weak solutions for a mathematical model of atherosclerosis in the early stage

Nonnegative weak solutions for a mathematical model of atherosclerosis in the early stage

On a singular mathematical model for brain lactate kinetics

In this paper, we consider a singular nonlinear differential system that characterizes the intricate dynamics of brain lactate kinetics between cells and capillaries, as described by System (1.1) below. We begin by establishing the existence and uniqueness of nonnegative solutions for our system through the application of Schauder's fixed‐point theorem. Subsequently, we explore the behavior of these solutions as the viscosity term approaches zero, shedding light on the system's dynamic evolution in such scenarios. To provide empirical validation for our theoretical findings, we offer a series of numerical simulations. These simulations not only confirm the results we have obtained but also reinforce prior research, underscoring the model's efficiency in capturing the complexities of the brain's lactate kinetics. Our work contributes not only to the theoretical underpinning of this field but also to its practical implications, making it a valuable resource for both researchers and practitioners seeking to comprehend and manipulate these vital biological processes.

On an SIS epidemic model with power‐like nonlinear incidence and with/without cross‐diffusion

AbstractWe study global existence, boundedness, and convergence of nonnegative classical solutions to a Neumann initial‐boundary value problem for the following possibly cross‐diffusive SIS (susceptible–infected–susceptible) epidemic model with power‐like infection mechanism generalizing the standard mass action incidence: in a bounded smooth domain . The infection force of the form with is a natural extension of the classical mass action type , and the cross‐diffusive term with describes the effect that susceptible individuals tend to move away from higher density of infected populations. Global existence and boundedness of classical solutions are established in certain parameter ranges, and threshold/nonthreshold long‐time behaviors of global bounded solutions are also detected. Our findings significantly improve and extend previous related studies.

Effect of unfavorable regions on the spreading solution in a diffusion equation

We consider a diffusion equation on the real line with growth rates (reaction terms) being negative in a bounded unfavorable region and bistable on two sides. The equation can be used to model a species living in a habitat with a polluted or hunting zone but still tries to survive or even spread to the whole space. Using the zero number argument, we first show the general convergence to stationary solution for any nonnegative global solutions, and then we prove a spreading–transition–vanishing trichotomy result for the asymptotic behavior of global solutions. The key point is to find a quite special stationary solution to distinguish the spreading and transition solutions. Finally, we construct precise upper and lower solutions to estimate the asymptotic speed for the spreading solution.

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

In this paper, we first propose an eco-epidemiological prey–predator model with infectious diseases in prey. Then study the ODE model, and diffusive model with the homogeneous Neumann and Dirichlet boundary conditions, respectively. For the ODE model and the diffusive model with the homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative equilibrium states (nonnegative constant equilibrium solutions). The results show that these two problems has no periodic solutions, and the diffusive model with the homogeneous Neumann boundary conditions has no yet Turing patterns. For the diffusive model with the homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of positive equilibrium solutions, and prove that the positive equilibrium solution is unique when it exists. Then we study the global asymptotic stabilities of trivial and semi-trivial nonnegative equilibrium solutions.

On a mathematical model for cancer invasion with repellent pH-taxis and nonlocal intraspecific interaction

Starting from a mesoscopic description of cell migration and intraspecific interactions, we obtain by upscaling an effective reaction–diffusion–taxis equation for the cell population density involving spatial nonlocalities in the source term and biasing its motility and growth behavior according to environmental acidity. We prove global existence, uniqueness, and boundedness of a nonnegative solution to a simplified version of the coupled system describing cell and acidity dynamics. A 1D study of pattern formation is performed. Numerical simulations illustrate the qualitative behavior of solutions.

Existence and uniqueness of solutions for Leontief's Input–Output Model, graph theory and sensitivity analysis

We provide a complete study of existence and uniqueness (uniqueness up to multiples in the case d = 0 ) of non-negative and non-trivial solutions x for the linear system ( I − A ) x = d with A ≥ 0 , d ≥ 0 (which, in particular, applies to Leontief's Input–Output Model). This study is done in terms of the block triangular form of the matrix A and is related to a directed graph associated to A . In particular, this study of existence and uniqueness up to multiples of solutions provides a framework that allows us to rigorously perform a sensitivity analysis of the normalized solutions of the linear system ( I − A ) x = d .

Elliptic equations with critical and supercritical growth at the boundary

In this study, we consider two classes of elliptic problems with nonlinear boundary conditions of concave–convex type. In the first problem, we obtain two nonzero and nonnegative solutions when the nonlinear term exhibits critical growth. In the second problem, we obtain infinitely many solutions (with no prescribed sign) by assuming that the nonlinearity is even and subcritical near the origin but with no growth condition at infinity.

Asymptotic behaviour for a parabolic p-Laplacian equation with an advection force

The rescaling method is presented to allow us establishing the interface functions and the nonnegative local solutions to the evolution of nonlinear degenerate parabolic p-Laplacian process with conservation laws that are posed in one-dimensional space. This equation has a self-similar solution which represents the main feature of this work. The Cauchy problem (CP) for this equation with specific restrictions in the range of parameters and the negative advection coefficient is considered. In this work, there are several regions to discuss the qualitative analysis for the local weak solutions and the asymptotic interfaces in the irregular domains. The dominating of the advection force over the p-Laplacian type diffusion will appear clearly in these regions. Moreover, the solutions of the CP for the degenerate parabolic p-Laplacian advection equations are asymptotically equal to the solutions of advection equations under some restrictions.

Mathematical Modelling and Numerical Simulation of Hepatitis B Viral Infection: The Case of Burkina Faso

Hepatitis is a viral infection that can cause inflammation of the liver and lead to severe liver damage and even death. The study of hepatitis in Burkina Faso is crucial for several reasons. Indeed, understanding the epidemiology of hepatitis in Burkina Faso can help develop effective prevention and control strategies, and its study can contribute to a better understanding of the global burden of the disease and the development of effective interventions in other parts of the world. To this aim, a new differential susceptibility and infectivity mathematical model of Hepatitis B transmission was developed in order to simulate the potential spread of the Hepatitis B virus in the population of Burkina Faso. Once the mathematical model is presented, the existence and uniqueness of non-negative solutions are proved. The model has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R01 and a globally asymptotically stable endemic equilibrium when R0 &gt; 1. The global asymptotic stability of the disease-free equilibrium has been studied using the Castillo Chavez method [5]. The Lyapunov function and the LaSalle invariance principle are used to prove the global asymptotic stability of the endemic equilibrium [26, 4, 18]. To simulate the proposed model, a Matlab numerical code has been developed. Numerical simulations are performed using data of Burkina Faso. The obtained numerical results confirm analytical results as well as the evolution of hepatitis B in Burkina Faso.