Nonnegative Solutions Research Articles

Stochastic dynamics of the fractal-fractional Ebola epidemic model combining a fear and environmental spreading mechanism

<abstract><p>Recent Ebola virus disease infections have been limited to human-to-human contact as well as the intricate linkages between the habitat, people and socioeconomic variables. The mechanisms of infection propagation can also occur as a consequence of variations in individual actions brought on by dread. This work studies the evolution of the Ebola virus disease by combining fear and environmental spread using a compartmental framework considering stochastic manipulation and a newly defined non-local fractal-fractional (F-F) derivative depending on the generalized Mittag-Leffler kernel. To determine the incidence of infection and person-to-person dissemination, we developed a fear-dependent interaction rate function. We begin by outlining several fundamental characteristics of the system, such as its fundamental reproducing value and equilibrium. Moreover, we examine the existence-uniqueness of non-negative solutions for the given randomized process. The ergodicity and stationary distribution of the infection are then demonstrated, along with the basic criteria for its eradication. Additionally, it has been studied how the suggested framework behaves under the F-F complexities of the Atangana-Baleanu derivative of fractional-order $ \rho $ and fractal-dimension $ \tau $. The developed scheme has also undergone phenomenological research in addition to the combination of nonlinear characterization by using the fixed point concept. The projected findings are demonstrated through numerical simulations. This research is anticipated to substantially increase the scientific underpinnings for understanding the patterns of infectious illnesses across the globe.</p></abstract>

A Nonexistence Result for Choquard-Type Hamiltonian System

In this article, we establish a nonexistence result of nontrivial non-negative solutions for the following Choquard-type Hamiltonian system by the Pohožaev identity , when , , , , , and , where and denotes the convolution in .

Mathematical analysis of neurological disorder under fractional order derivative

<abstract><p>Multiple sclerosis (MS) is a common neurological disorder that affects the central nervous system (CNS) and can cause lesions that spread over space and time. Our study proposes a mathematical model that illustrates the progression of the disease and its likelihood of recurrence. We use Caputo fractional-order (FO) derivative operators to represent non-negative solutions and to establish a steady-state point and basic reproductive number. We also employ functional analysis to prove the existence of unique solutions and use the Ulam-Hyres (UH) notion to demonstrate the stability of the solution for the proposed model. Furthermore, we conduct numerical simulations using an Euler-type numerical technique to validate our theoretical results. Our findings are presented through graphs that depict various behaviors of the model for different parameter values.</p></abstract>

BLOW-UP PROBLEMS FOR FUJITA-TYPE PARABOLIC SYSTEM INVOLVING TIME-DEPENDENT COEFFICIENTS ON GRAPHS

In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053].

Harnack inequality for polyharmonic equations

Some new types of mean value formulas for the polyharmonic functions were established. Based on the formulas, the Harnack inequality for the nonnegative solutions to the polyharmonic equations was proved.

Multiple nonsemitrivial solutions for quasilinear elliptic systems via associated eigenvalue problems

In this paper, the existence, nonexistence and multiplicity of nonnegative solutions for quasilinear elliptic systems of resonant type involving $ (p,q) $-Laplacian are investigated. By using the generalized variant of the Nehari manifold and fibering method, we prove a suitable manifold with associated map for the systems and exploit the relation between them to study the existence and multiplicity of nonnegative solutions and obtain bifurcation results. Furthermore, we also apply the approach to treat a class of $ (p,q) $-Laplacian elliptic systems with concave-convex nonlinearities.

A priori bounds and existence of non-negative solutions to a quasi-linear Schrödinger equation involving p-Laplacian

A priori bounds and existence of non-negative solutions to a quasi-linear Schrödinger equation involving p-Laplacian

SOR-based alternately linearized implicit iteration method for nonsymmetric algebraic Riccati equations

<abstract><p>In this paper, we propose a class of successive over relaxation-based alternately linearized implicit iteration method for computing the minimal nonnegative solution of nonsymmetric algebraic Riccati equations. Under certain conditions, we prove the convergence of the iterative method. Finally, numerical examples are given to show the iterative method is efficient.</p></abstract>

On a Hamiltonian elliptic system with concave and convex nonlinearities

In this work we study a Hamiltonian elliptic system of equations with Dirichlet boundary condition and with non-linearities that are concave near the origin and are convex and superlinear at infinity. The concavity of the non-linearities depends on non-negative parameters $ \lambda $ and $ \mu $ and we provide regions for the pairs $ (\lambda, \mu) $ guaranteeing existence and non-existence of non-negative solutions. This work is inspired by the seminal work for the single equation done by Ambrosetti, Brezis and Cerami in [1].

Note on a parabolic problem with multi-coupled nonlinearity and space-time weighted functions

This paper deals with two semilinear parabolic equations involving multi-coupled nonlinearity, where the weighted functions depend on space and time variables. We determine the exponent regions where the nonnegative nontrivial solutions blow up for any initial data. Moreover, some exponent regions on the global solutions are given with some conditions on the initial data and the ten exponents.

Global asymptotic stability of constant equilibrium point in attraction-repulsion chemotaxis model with logistic source term

This paper deals with nonnegative solutions of the Neumann initial-boundary value problem for an attraction-repulsion chemotaxis model with logistic source term of Equation 1 in bounded convex domains \(\Omega\subset\mathbb{R}^{n},~ n\geq1\), with smooth boundary. It is shown that if the ratio \(\frac{\mu}{\chi \alpha-\xi \gamma}\) is sufficiently large, then the unique nontrivial spatially homogeneous equilibrium given by \((u_{1},u_{2},u_{3})=(1,~\frac{\alpha}{\beta},~\frac{\gamma}{\eta})\) is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data \((u_{10},u_{20},u_{30})\) such that \(u_{10}\not\equiv0\), the above problem possesses uniquely determined global classical solution \((u_{1},u_{2},u_{3})\) with \((u_{1},u_{2},u_{3})|_{t=0}=(u_{10},u_{20},u_{30})\) which satisfies $$\|u_{1}(\cdot,t)-1\|_{L^{\infty}(\Omega)}\rightarrow{0},~~ \|u_{2}(\cdot,t)-\frac{\alpha}{\beta}\|_{L^{\infty}(\Omega)}\rightarrow{0},~~ \|u_{3}(\cdot,t)-\frac{\gamma}{\eta}\|_{L^{\infty}(\Omega)}\rightarrow{0}.$$ \(\mathrm{as}~t\rightarrow{\infty}\).

A New Mathematical Model of COVID-19 with Quarantine and Vaccination

A mathematical model revealing the transmission mechanism of COVID-19 is produced and theoretically examined, which has helped us address the disease dynamics and treatment measures, such as vaccination for susceptible patients. The mathematical model containing the whole population was partitioned into six different compartments, represented by the SVEIQR model. Important properties of the model, such as the nonnegativity of solutions and their boundedness, are established. Furthermore, we calculated the basic reproduction number, which is an important parameter in infection models. The disease-free equilibrium solution of the model was determined to be locally and globally asymptotically stable. When the basic reproduction number R0 is less than one, the disease-free equilibrium point is locally asymptotically stable. To discover the approximative solution to the model, a general numerical approach based on the Haar collocation technique was developed. Using some real data, the sensitivity analysis of R0 was shown. We simulated the approximate results for various values of the quarantine and vaccination populations using Matlab to show the transmission dynamics of the Coronavirus-19 disease through graphs. The validation of the results by the Simulink software and numerical methods shows that our model and adopted methodology are appropriate and accurate and could be used for further predictions for COVID-19.

Two nonnegative solutions for two-dimensional nonlinear wave equations

We study a class of initial value problems for two-dimensional nonlinear wave equations. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions. The arguments are based upon a recent theoretical result.

Complex dynamic analysis of a reaction-diffusion network information propagation model with non-smooth control

Rumors do the social’s perception seal, leaving people untouched with the true thing. Since the harmfulness of rumors is well known, in order to cut off the net of rumors and make the society run in order, it is necessary to have a deeper understanding of the spread of rumors. In this paper, the non-smooth reaction–diffusion system with considering the encouraging effect of secondary propagation of Internet platform on rumor propagation is used to study the dynamic system of rumor propagation. Firstly, the existence of the non-negative solution is proved by using the theory of upper and lower solutions for mixed monotone system. Secondly, the value of the basic reproduction number is calculated and the existence of positive equilibrium are discussed. Thirdly, the backward bifurcation and the saddle-node bifurcation are investigated. Fourthly, the stability of rumor propagation equilibrium and Hopf bifurcation are analyzed theoretically. Finally, some numerical simulations which proves the validity of the above theory are provided.

Classification of nonnegative solutions to Schrödinger equation with logarithmic nonlinearity

Classification of nonnegative solutions to Schrödinger equation with logarithmic nonlinearity

CLASSICAL SOLUTIONS FOR A BVP FOR A CLASS IMPULSIVE FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we investigate a BVP for a class impulsive fractional partial differential equations. We propose a new topological approach to prove the existence of at least one classical solution and at least two nonnegative classical solutions. The arguments are based upon recent theoretical results.

Linear parabolic operators of Monge-Ampère type I: Nondivergence-form degenerate/singular PDEs

A Harnack inequality for nonnegative strong solutions to inhomogeneous, linear, nondivergence-form parabolic PDEs of Monge-Ampère type is proved. Such parabolic PDEs are shaped by convex functions having certain geometric properties and they include instances of Grushin-type parabolic PDEs as well as other PDEs whose degeneracies/singularities are quantified by the geometric properties of the shaping convex functions.

Non-autonomous chemostat models with non-monotonic growth

In this paper, we investigate four non-autonomous chemostat models with non-monotonic consumption function, where wall growth and nutrient recycling are also taken into account. In each case, we prove the existence and uniqueness of non-negative global solution that generates a non-autonomous dynamical system. In addition, we also prove the existence of a unique (global) pullback attractor whose internal structure provides detailed information about the long-time behaviour of the state variables, for instance, conditions to ensure the extinction and the persistence of the species. We also display numerical simulations to illustrate the theoretical results.

A positivity-preserving RD-FV scheme for diffusion problems on triangular meshes

In this paper, we propose a positivity-preserving residual distribution based finite volume scheme (RD-FV) for steady diffusion problems on unstructured triangular meshes. In this scheme, we use the RD method to obtain the auxiliary values of vertex, then adopt the positivity-preserving finite volume scheme to update the cell-centered values for diffusion problems, in which a nonlinear two-point flux approximation is utilized to obtain nonnegative solution. Thus a positivity-preserving conservative property for the RD-FV scheme is guaranteed. Several numerical experiments show the optimal convergence rates for the numerical solution, that is, approximate second-order accuracy for the solution and first-order accuracy for the flux, and demonstrate the positivity-preserving property of our new scheme.

Exhaustive existence and non-existence results for Hardy–Hénon equations in {{,mathrm{{textbf{R}}},}}^n

This paper concerns solutions to the Hardy–Hénon equation -Δu=|x|σup\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\Delta u = |x|^\\sigma u^p \\end{aligned}$$\\end{document}in {{,mathrm{{textbf{R}}},}}^n with n ge 1 and arbitrary p, sigma in {{,mathrm{{textbf{R}}},}}. This equation was proposed by Hénon in 1973 as a model to study rotating stellar systems in astrophysics. Although there have been many works devoting to the study of the above equation, at least one of the following three assumptions p>1,sigma ge -2, and n ge 3 is often assumed. The aim of this paper is to investigate the equation in other cases of these parameters, leading to a complete picture of the existence/non-existence results for non-trivial, non-negative solutions in the full generality of the parameters. In addition to the existence/non-existence results, the uniqueness of solutions is also discussed.