Nonnegative Classical Solutions Research Articles

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.

Boundedness and stabilization in a three-species chemotaxis-competition system

In this paper, we study the following chemotaxis-competition system of parabolic-parabolic-parabolic-elliptic type{ut=d1Δu−χ1∇⋅(u∇z)+μ1u(1−u−a1v),(x,t)∈Ω×(0,∞),vt=d2Δv−χ2∇⋅(v∇z)+μ2v(1−a2u−v),(x,t)∈Ω×(0,∞),wt=d3Δw−χ3∇⋅(w∇z)+μ3w(1−b1u−b2v−w),(x,t)∈Ω×(0,∞),0=d4Δz−αz+β1u+β2v+β3w,(x,t)∈Ω×(0,∞), under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂Rn, n≥2, where χi>0,μi>0,βi>0,i=1,2,3 and α,a1,a2,b1,b2,d1,d2,d3,d4>0. Firstly, it is shown that this system possesses a global bounded nonnegative classical solution for any nonnegative initial data (u0,v0,w0)∈(C0(Ω‾))3 provided that χi are small enough or μi are large enough. Moreover, this paper proceeds to establish asymptotic stabilization of global bounded solutions to the above system as follows:• When a1,a2∈(0,1), the global bounded classical solution (u,v,w,z) converges to (u⁎,v⁎,w⁎,β1u⁎+β2v⁎+β3w⁎α) if b1(1−a1)+b2(1−a2)<1−a1a2 or (u⁎,v⁎,0,β1u⁎+β2v⁎α) if b1(1−a1)+b2(1−a2)≥1−a1a2 in L∞-norm as t→∞, where u⁎=1−a11−a1a2,v⁎=1−a21−a1a2,w⁎=1−a1a2−b1(1−a1)−b2(1−a2)1−a1a2.• When a1≥1>a2>0, the global bounded classical solution (u,v,w,z) converges to (0,1,1−b2,β2+β3(1−b2)α) if b2∈(0,1) or (0,1,0,β2α) if b2≥1 in L∞-norm as t→∞.

Boundedness Through Nonlocal Dampening Effects in a Fully Parabolic Chemotaxis Model with Sub and Superquadratic Growth

This work deals with a chemotaxis model where an external source involving a sub and superquadratic growth effect contrasted by nonlocal dampening reaction influences the motion of a cell density attracted by a chemical signal. We study the mechanism of the two densities once their initial configurations are fixed in bounded impenetrable regions; in the specific, we establish that no gathering effect for the cells can appear in time provided that the dampening effect is strong enough. Mathematically, we are concerned with this problem ut=Δu-χ∇·(u∇v)+auα-buα∫ΩuβinΩ×(0,Tmax),τvt=Δv-v+uinΩ×(0,Tmax),uν=vν=0on∂Ω×(0,Tmax),u(x,0)=u0(x)≥0,v(x,0)=v0(x)≥0,x∈Ω¯,◊\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t=\\Delta u-\\chi \ abla \\cdot (u\ abla v)+au^\\alpha -bu^\\alpha \\int _\\Omega u^\\beta &{}\ extrm{in}\\ \\Omega \ imes (0, T_{max}),\\\\ \ au v_t=\\Delta v-v+u &{}\ extrm{in}\\ \\Omega \ imes (0, T_{max}),\\\\ u_\ u =v_\ u =0 &{}\ extrm{on}\\ \\partial \\Omega \ imes (0, T_{max}),\\\\ u(x, 0)=u_0(x)\\ge 0, v(x,0)=v_0(x)\\ge 0, &{}x \\in {\\bar{\\Omega }}, \\end{array}\\right. } \\quad {\\Diamond } \\end{aligned}$$\\end{document}for τ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au =1$$\\end{document}, n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\in {\\mathbb {N}}$$\\end{document}, χ,a,b>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi ,a,b>0$$\\end{document} and α,β≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha , \\beta \\ge 1$$\\end{document}. Herein u stands for the population density, v for the chemical signal and Tmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{max}$$\\end{document} for the maximal time of existence of any nonnegative classical solution (u, v) to system (◊\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Diamond $$\\end{document}). We prove that despite any large-mass initial data u0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_0$$\\end{document}, whenever(The subquadratic case) 1≤α<2andβ>n+42-α,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le \\alpha <2 \\quad \ ext {and} \\quad \\beta >\\frac{n+4}{2}-\\alpha ,$$\\end{document}(The superquadratic case) β>n2and2≤α<1+2βn,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta >\\frac{n}{2} \\quad \ ext {and} \\quad 2\\le \\alpha < 1+ \\frac{2\\beta }{n},$$\\end{document} actually Tmax=∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{max}=\\infty $$\\end{document} and u and v are uniformly bounded. This paper is in line with the result in Bian et al. (Nonlinear Anal 176:178–191, 2018), where the same conclusion is established for the simplified parabolic-elliptic version of model (◊\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Diamond $$\\end{document}), corresponding to τ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au =0$$\\end{document}; more exactly, this work extends the study to the fully parabolic case Bian et al. (Nonlinear Anal 176:178–191, 2018).

Liouville type theorems for the generalized higher order Choquard-Pekar equation

We study the generalized higher order Choquard-Pekar equation(⁎)(−Δ)mu=(∫RNΨ(|x−y|)up(y)dy)uqin RN, where N,m≥1 are integers, p,q>0. Under suitable assumption on Ψ, when N>2m the Liouville theorems in the ranges(i)(p≥1 or p+q≥2) and ∫|y|>1|y|−p(N−2m)Ψ(|y|)dy=∞; or(ii)p+q≥2 and limsupr→∞r2N−(N−2m)(p+q)Ψ(r)>0 for nonnegative classical solutions of (⁎) were got in previous studies. By using ‘‘Bootstrap’’ iteration procedure, we extend the above results by establishing the Liouville theorems for nonnegative classical solutions of (⁎) with N>2m in the ranges p≥1 and p+q<NN−2m. Moreover, we also consider the higher order Choquard-Pekar equation, i.e., the problem (⁎) with Ψ(r)=r−σ and 0<σ<N. By using the method of scaling sphere we also establish the Liouville theorems for nonnegative classical solutions with N>2m in the ranges 1≤p≤2N−σN−2m, 0<q≤N+2m−σN−2m, and p+q<3N+2m−2σN−2m, which also extend the recent studies on higher order Choquard-Pekar equation. Extensions to the integral representation of (⁎) with more general assumption p>0 are also included.

Classical Solutions for the Generalized Kawahara–KdV System

In this article, we investigate the generalized Kawahara–KdV system. A new topological approach is applied to prove the existence of at least one classical solution and at least two non-negative classical solutions. The arguments are based upon recent theoretical results.

Classical solutions for the generalized Kadomtsev–Petviashvili I equations

PurposeThis paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.Design/methodology/approachThis paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.FindingsThis paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.Originality/valueThis article is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Existence of solutions for impulsive wave equations

&lt;abstract&gt;&lt;p&gt;We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. To prove our main results we give a suitable integral representation of the solutions of the considered problem. Then, we construct two operators so that any fixed point of their sum is a solution.&lt;/p&gt;&lt;/abstract&gt;

Dynamics analysis on a chronic epidemic model with diffusion

In this paper, we propose a reaction-diffusion model of chronic infectious diseases with a special nonlinear incidence rate describing the mechanism of some chronic infections diseases. We first prove that the model has a unique nonnegative global classical solution, and admits a connected global attractor. Further, the threshold dynamics such as the uniform persistence of the model solution and the global stability of the steady-state solution are studied. The theory is verified by numerical simulation. Our results show that chronic infectious diseases have a strong ability to survive, and it is difficult to eradicate chronic infectious diseases completely.

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.

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.

Optimal Liouville-type theorems for polyharmonic elliptic gradient systems and applications

In this paper, we investigate the following polyharmonic systems (d≥1) in RN,{(−Δ)du1=f1(u1,u2),(−Δ)du2=f2(u1,u2). By using the method of Rellich-Pohozaev type identities, Sobolev and interpolation inequalities on SN−1 and feedback and measure arguments, we prove Liouville-type theorems for nonnegative classical solutions under an optimal Sobolev growth condition on f1 and f2. Furthermore, we prove priori bounds, decay and singularity estimates of solutions. This system can be applied to noncooperative Schrödinger system with polyharmonic operator which arises from nonlinear optics and Bose-Einstein condensates.

Multiple Nonnegative Solutions for a Class of Fourth-Order BVPs Via a New Topological Approach

In this paper, we study a class of fourth-order boundary value problems with integral boundary conditions. The nonlinearity may have time-singularity and change sign. Moreover, it satisfies general polynomial growth conditions. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions. An example of application illustrates the existence result.

Classical solutions for 1-dimensional and 2-dimensional Boussinesq equations

In this article we investigate the IVPs for 1-dimensional and 2-dimensional Boussinesq equations. A new topological approach is applied to prove the existence of at least one classical solution and at least two nonnegative classical solutions for the considered IVPs. The arguments are based upon recent theoretical results.

Blow-up phenomenon for a semilinear pseudo-parabolic equation involving variable source

We consider a semilinear pseudo-parabolic equation with nonlinearities of variable exponent subject to Dirichlet boundary conditions We prove that the nonnegative classical solutions blow-up in finite time with arbitrary positive initial energy and suitable large initial values. Also, employing a differential inequality technique, we obtain an upper bound for blow-up time if and the initial value satisfies some conditions.

Multiple nonnegative solutions for a class IVPs for second order ODEs

We study a class of initial value problems for second order ODEs. The interesting points of our results are that the nonlinearity depends on the solution and its derivative and may change sign. Moreover, it satisfies general polynomial growth conditions. 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.

Classification of Nonnegative Solutions to Static Schrödinger--Hartree--Maxwell Type Equations

In this paper, we are mainly concerned with the physically interesting static Schr\"{o}dinger-Hartree-Maxwell type equations \begin{equation*} (-\Delta)^{s}u(x)=\left(\frac{1}{|x|^{\sigma}}\ast |u|^{p}\right)u^{q}(x) \,\,\,\,\,\,\,\,\,\,\,\, \text{in} \,\,\, \mathbb{R}^{n} \end{equation*} involving higher-order or higher-order fractional Laplacians, where $n\geq1$, $0<s:=m+\frac{\alpha}{2}<\frac{n}{2}$, $m\geq0$ is an integer, $0<\alpha\leq2$, $0<\sigma<n$, $0<p\leq\frac{2n-\sigma}{n-2s}$ and $0<q\leq\frac{n+2s-\sigma}{n-2s}$. We first prove the super poly-harmonic properties of nonnegative classical solutions to the above PDEs, then show the equivalence between the PDEs and the following integral equations \begin{equation*} u(x)=\int_{\mathbb{R}^n}\frac{R_{2s,n}}{|x-y|^{n-2s}}\left(\int_{\mathbb{R}^{n}}\frac{1}{|y-z|^{\sigma}}u^p(z)dz\right)u^{q}(y)dy. \end{equation*} Finally, we classify all nonnegative solutions to the integral equations via the method of moving spheres in integral form. As a consequence, we obtain the classification results of nonnegative classical solutions for the PDEs. Our results completely improved the classification results in \cite{CD,DFQ,DL,DQ,Liu}. In critical and super-critical order cases (i.e., $\frac{n}{2}\leq s:=m+\frac{\alpha}{2}<+\infty$), we also derive Liouville type theorem.

Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space

In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire positive solutions to the simplest models of polyharmonic equations with power-type nonlinearityΔmu=±uαin Rn with n⩾1, m⩾1, and α∈R. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants n, m and α. Remarkably, we are able to provide necessary and sufficient conditions on the exponent α to guarantee the existence of such solutions in Rn. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive everywhere.

Existence of solutions for a class of IBVP for nonlinear hyperbolic equations

We study a class of initial boundary value problems of hyperbolic type. A new topological approach is applied to prove the existence of non-negative classical solutions. The arguments are based upon a recent theoretical result.

Pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments

The current paper is concerned with pointwise persistence in full chemotaxis models with local as well as nonlocal time and space dependent logistic source in bounded domains. We first prove the global existence and boundedness of nonnegative classical solutions under some conditions on the coefficients in the models. Next, under the same conditions on the coefficients, we show that pointwise persistence occurs, that is, any globally defined positive solution is bounded below by a positive constant independent of its initial condition when the time is large enough. It should be pointed out that in [21], the authors established the persistence of mass for globally defined positive solutions, which indicates that any extinction phenomenon, if occurring at all, necessarily must be spatially local in nature, whereas the population as a whole always persists. The pointwise persistence proved in the current paper implies that not only the population as a whole persists, but also it persists at any location eventually. It also implies the existence of strictly positive entire solutions.

Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions

<p style="text-indent:20px;">Uniform-in-time bounds of nonnegative classical solutions to reaction-diffusion systems in all space dimension are proved. The systems are assumed to dissipate the total mass and to have locally Lipschitz nonlinearities of at most (slightly super-) quadratic growth. This pushes forward the recent advances concerning global existence of reaction-diffusion systems dissipating mass in which a uniform-in-time bound has been known only in space dimension one or two. As an application, skew-symmetric Lotka-Volterra systems are shown to have unique classical solutions which are uniformly bounded in time in all dimensions with relatively compact trajectories in <inline-formula><tex-math id="M1">\begin{document}$ C(\overline{\Omega})^m $\end{document}</tex-math></inline-formula>.