Neumann Initial Boundary Value Problem Research Articles

A Riemann–Hilbert approach to the existence results for the Benjamin–Ono equation on a half-line

The main problem addressed in this paper is to study the local existence in time of solutions to the non-homogeneous Neumann initial boundary value problem for the Benjamin–Ono equation on a half-line. In this result, we observe the influence of the boundary data on the behavior of solutions. In order to obtain the characterization of the solution it is essential to use the theory concerning the Riemann–Hilbert problem. We prove local existence in time of the solutions.

Regularizing effect in a Keller–Segel system with density-dependent sensitivity for Lp-initial data of cell density

The fully parabolic Keller–Segel system{ut=Δu−χ∇⋅(u(u+1)α−1∇v)inΩ×(0,∞),vt=Δv−v+uinΩ×(0,∞) is considered in a bounded domain Ω⊂RN (N∈N) with smooth boundary, where χ,α∈R. It is shown that an associated Neumann initial-boundary value problem admits a global classical solution for initial data with low regularity, in particular, for Lp-initial data (p>max⁡{1,α,NN+1(α+1)}) of u when α<2N and when α=2N and the initial mass is small. This extends a well-known result on global solvability for reasonably regular initial data.

Boundedness in a chemotaxis-May–Nowak model with exposed state

This paper is concerned with the Neumann initial–boundary value problem of the chemotaxis-May–Nowak model with exposed state for virus dynamics. It is proved that the problem admits a unique global classical solution which is uniformly bounded for all sufficiently smooth initial data in smoothly bounded domains Ω⊂Rn, n≤3.

Smoothness effects of a quadratic damping term of mixed type on a chemotaxis-type system modeling propagation of urban crime

This paper focuses on the two-dimensional Neumann initial–boundary value problem of a chemotaxis-type system with a mixed-type quadratic damping term constituting of the product of two unknown functions. It is shown that such quadratic damping term seems sufficient to exclude the possibility of blowup in infinite time. Precisely, the first result indicates that for all reasonably regular initial data and any chemotatic sensitivity, the solution of the initial–boundary value problem is global in time within a suitable generalized framework. Meanwhile, the second result demonstrates that such generalized solution enjoys the eventual boundedness and regularity properties, i.e., it becomes bounded and smooth after some waiting time. Finally, a statement on the asymptotic stability of certain steady states is derived as a by-product.

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 $$\|u_{1}(\cdot,t)-1\|_{L^{\infty}(\Omega)}\rightarrow{0},~~ \|u_{2}(\cdot,t)-\frac{\alpha}{\beta}\|_{L^{\infty}(\Omega)}\rightarrow{0},~~ \|u_{3}(\cdot,t)-\frac{\gamma}{\eta}\|_{L^{\infty}(\Omega)}\rightarrow{0}.$$ \(\mathrm{as}~t\rightarrow{\infty}\).

On a chemotaxis-type Solow-Swan model for economic growth with capital-induced labor migration

We study the Neumann initial-boundary value problem of the parabolic chemotaxis system(⋆){ut=Δu−χ∇⋅(u∇v)+μu(1−u),vt=Δv−v+κu1−αvα with parameters χ,κ>0, μ≥0 and α∈(0,1), in a bounded domain Ω⊂Rn with n≥1 and smooth boundary. The system (⋆) was originally proposed in [15] as a spatial Solow-Swan model for economic growth with the capital-induced labor migration. Our result asserts that for any suitably regular initial data, whenever either μ>0 or 1−α<2n, the corresponding Neumann initial-boundary value problem possesses a global classical solution, which is uniformly bounded.

Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

&lt;p style='text-indent:20px;'&gt;This paper is concerned with a predator-prey model with stage structure for the predator, with a cross-diffusion term modeling the effect that mature predators move toward the direction of gradient of prey. It is first shown that the corresponding Neumann initial-boundary value problem in an &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-dimensional bounded smooth domain possesses a unique global classical solution which is uniformly-in-time bounded for the weak cross-diffusion. It is further shown that, in the presence of cross-diffusion, the model admits threshold-type dynamics in terms of the cross-diffusion coefficient; that is, the homogenous steady state keeps stability for weak attractive prey-taxis, while the stationary patterns will occur for strong attractive prey-taxis. This implies that such cross diffusion does contribute to the rich dynamics of predator-prey model with stage structure for predators.&lt;/p&gt;

A note on construction of nonnegative initial data inducing unbounded solutions to some two-dimensional Keller–Segel systems

&lt;abstract&gt;&lt;p&gt;It was shown that unbounded solutions of the Neumann initial-boundary value problem to the two-dimensional Keller–Segel system can be induced by initial data having large negative energy if the total mass $ \Lambda \in (4\pi, \infty)\setminus 4\pi \cdot \mathbb{N} $ and an example of such an initial datum was given for some transformed system and its associated energy in Horstmann–Wang (2001). In this work, we provide an alternative construction of nonnegative nonradially symmetric initial data enforcing unbounded solutions to the original Keller–Segel model.&lt;/p&gt;&lt;/abstract&gt;

The role of local kinetics in a three-component chemotaxis model for Alopecia Areata

This manuscript considers the homogeneous Neumann initial-boundary value problem for a three-component chemotaxis system describing the spatio-temporal Alopecia Areata dynamics. The model addresses the complex interactions between CD4+ T cells, CD8+ T cells and interferon-gamma (IFN-γ): Both types of immune cells secrete the chemical that diffuses and degrades; conversely, the T cells are activated by the chemical but decay due to density-dependent death. In addition to random motion, the T cells bias their movement toward the concentration gradient of IFN-γ. To compare with previous chemotaxis models, a distinctive feature of this system is that CD8+ T cells additionally proliferate in a nonlinear manner with the help of CD4+ T cells. Given suitably regular initial data, it is shown that either small logistic damping in the two-dimensional case or strong logistic damping in the three-dimensional setting can prevent any blow-up of classical solutions to the problem. Moreover, we find a specific parameter regime for which the unique coexistence equilibrium is globally convergent.

Approaching optimality in blow-up results for Keller\u2013Segel systems with logistic-type dampening

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system in smooth bounded domains Omega subset {mathbb {R}}^n, n ge 1, are known to be global in time if lambda ge 0, mu > 0 and kappa > 2. In the present work, we show that the exponent kappa = 2 is actually critical in the four- and higher dimensional setting. More precisely, if n≥4,κ∈(1,2)andμ>0orn≥5,κ=2andμ∈0,n-4n,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\qquad n&\\ge 4,&\\quad \\kappa \\in (1, 2) \\quad&\\text {and} \\quad \\mu > 0 \\\\ \\text {or}\\qquad n&\\ge 5,&\\quad \\kappa = 2 \\quad&\\text {and} \\quad \\mu \\in \\left( 0, \\frac{n-4}{n}\\right) , \\end{aligned}$$\\end{document}for balls Omega subset {mathbb {R}}^n and parameters lambda ge 0, m_0 > 0, we construct a nonnegative initial datum u_0 in C^0({{overline{Omega }}}) with int nolimits _Omega u_0 = m_0 for which the corresponding solution (u, v) of (star ) blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for kappa in (1, frac{3}{2}) (and lambda ge 0, mu > 0). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function w(s, t) = int nolimits _0^{root n of {s}} rho ^{n-1} u(rho , t) ,mathrm {d}rho fulfills the estimate w_s le frac{w}{s}. Using this information, we then obtain finite-time blow-up of u by showing that for suitably chosen initial data, s_0 and gamma , the function phi (t) = int nolimits _0^{s_0} s^{-gamma } (s_0 - s) w(s, t) cannot exist globally.

Immediate regularization of measure-type population densities in a two-dimensional chemotaxis system with signal consumption

This paper deals with the Neumann initial-boundary value problem for a classical chemotaxis system with signal consumption in a disk. In contrast to previous studies which have established a comprehensive theory of global classical solutions for suitably regular nonnegative initial data, the focus in the present work is on the question to which extent initially prescribed singularities can be regularized despite the presence of the nonlinear cross-diffusive interaction. The main result in this paper asserts that at least in the framework of radial solutions immediate regularization occurs under an essentially optimal condition on the initial distribution of the population density. More precisely, it will turn out that for any radially symmetric initial data belonging to the space of regular signed Borel measures for the population density and to L2 for the signal density, there exists a classical solution to the Neumann initial-boundary value problem, which is smooth and approaches the given initial data in an appropriate trace sense.

Global existence and boundedness of a chemotaxis model with indirect production and general kinetic function

This paper is devoted to the chemotaxis model with indirect production and general kinetic function $$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u),\qquad&x\in \Omega ,\,t>0,\\&v_t=\Delta v-v+w,\qquad&x\in \Omega ,\,t>0,\\&\tau w_t+\lambda w=g(u),\qquad&x\in \Omega ,\,t>0, \end{aligned} \right. \end{aligned}$$ in a bounded domain $$\Omega \subset \mathbb {R}^n(n\le 3)$$ with smooth boundary $$\partial \Omega $$ , where $$\chi , \tau , \lambda $$ are given positive parameters, f and g are known functions. We find several explicit conditions involving the kinetic function f, g, the parameters $$\chi $$ , $$\lambda $$ , and the initial data $$\Vert u_0\Vert _{L^1(\Omega )}$$ to ensure the global-in-time existence and uniform boundedness for the corresponding 2D/3D Neumann initial-boundary value problem. Particularly, when $$f\equiv 0$$ , and g is a linear function, the global bounded classical solutions to the corresponding 2D Neumann initial-boundary value problem with arbitrarily large initial data and chemotactic sensitivity are established. Our results partially extend the results of Hu and Tao (Math Models Methods Appl Sci 26:2111–2128, 2016), Tao and Winkler (J Eur Math Soc 19:3641–3678, 2017), etc.

On boundedness, blow-up and convergence in a two-species and two-stimuli chemotaxis system with/without loop

In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We detect the product of two species masses as a feature to determine boundedness, gradient estimate, blow-up and exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one species mass to be suitably large, is sufficient to guarantee global boundedness, higher order gradient estimates and $$W^{j,\infty }(j\ge 1)$$ -exponential convergence with rates of convergence to constant equilibria; and then, in a special case, we detect a straight line of masses on which blow-up occurs for large product of masses. Our findings provide new understandings about the underlying model, and thus, improve and extend greatly the existing knowledge relevant to this model.

Global existence of solutions for a p‐Laplacian equation with nonlocal Fisher–KPP type reaction terms

This work is concerned with the Neumann initial boundary value problem and Cauchy problem of a parabolic p‐Laplacian equation with nonlocal Fisher–KPP type reaction terms. We establish a uniform boundedness and global existence of solutions to the equation by applying the method of multipliers and modified Moser's iteration technique for some ranges of parameters. The ranges of parameters have similar structure to that of the classical critical Fujita exponent.

Uniform Boundedness and Global Existence of Solutions to a Quasilinear Diffusion Equation with Nonlocal Fisher-KPP Type Reaction Term

This paper deals with the Cauchy problem and Neumann initial boundary value problem for a quasilinear diffusion equation with nonlocal Fisher-KPP type reaction terms. We establish the uniform boundedness and global existence of solutions to the problems by using multipliers technique and modified Moser's iteration argument for some ranges of parameters. Moreover, the ranges of parameters have similar structure to that of the classical critical Fujita exponent.

Global bounded solutions to a Keller–Segel system with singular sensitivity

This paper deals with the Neumann initial–boundary value problem for the Keller–Segel chemotaxis system with singular sensitivity ut=d1Δu−χ∇⋅(uv∇v),vt=d2Δv−v+uin a smooth bounded domain Ω⊂Rn (n≥1), where d1>0, d2>0 and χ∈R. When d1=d2+χ, for all initial data 0≤u0∈C0(Ω̄) and 0<v0∈W1,∞(Ω), we prove that the problem possesses a unique global classical solution which is uniformly bounded in Ω×(0,∞).

A critical virus production rate for blow-up suppression in a haptotaxis model for oncolytic virotherapy

In a two-dimensional smoothly bounded domain, we consider the oncolytic virotherapy model given by ut=Δu−∇⋅(u∇v)−ρuz,vt=−(u+w)v,wt=DwΔw−w+uz,zt=DzΔz−z−uz+βw,with parameters Dw>0,Dz>0,β>0 and ρ≥0.According to previous findings, a corresponding Neumann initial–boundary value problem possesses a globally defined classical solution whenever the prescribed initial data are suitably regular and satisfy appropriate positivity assumptions. In the present study it is shown that if β<1, then any such solution is uniformly bounded. This complements a recent result which in the case ρ=0 has asserted the existence of unbounded solutions whenever β>1.

Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms

This paper is concerned with a general asymptotic stabilization of arbitrary global positive bounded solutions for the Lotka Volterra reaction diffusion systems, with an additional chemotactic influence and constant coefficients. We consider the dynamics of a mathematical model involving two biological species, both of which move according to random diffusion and are attracted/ repulsed by chemical stimulus produced by the other. The biological species present the ability to orientate their movement towards the concentration of the chemical secreted by the other species. The nonlinear system consists of two parabolic equations with Lotka-Volterra-type kinetic terms coupled with chemotactic cross-diffusion, along with two elliptic equations describing the behavior of the chemicals. We prove that the solution to the corresponding Neumann initial boundary value problem is global and bounded for regular and positive initial data. Moreover, for different ranges of parameters, we show that any positive and bounded solution converges to a spatially constant homogeneous state.

Global Bounded Solutions for the Keller-Segel Chemotaxis System with Singular Sensitivity

In this paper, we consider the Neumann initial-boundary value problem for the Keller-Segel chemotaxis system with singular sensitivity (0.1) is considered in a bounded domain with smooth boundary, Ω ⊂ Rn (n ≥ 1), where d1 > 0, d2 > 0 with parameter χ ∈ R. When d1 = d2 + χ, satisfying for all initial data 0 ≤ n0 ∈ C0 and 0 v0∈ W1,∞ (Ω), we prove that the problem possesses a unique global classical solution which is uniformly bounded in Ω × (0, ∞).

Neumann inhomogeneous initial-boundary value problem for the 2D nonlinear Schrödinger equation

This paper is the first attempt to give a rigorous mathematical study of Neumann initial boundary value problems for the multidimensional dispersive evolution equations considering as example famous nonlinear Schrodinger equation. We consider the inhomogeneous initial-boundary value problem for the nonlinear Schrodinger equation, formulated on upper right-quarter plane with initial data $$u({\mathbf {x}},t)\left| _{t=0}\right. =u_{0}({\mathbf {x}})$$ and Neumann boundary data $$u_{x_{1}}\left| _{\partial _{1}{\mathbf {D}}}\right. =h_{1}(x_{2},t),u_{x_{2}}\left| _{\partial _{2}{\mathbf {D}}}\right. =h_{2}(x_{1},t)$$ given in a suitable weighted Lebesgue spaces. We are interested in the study of the influence of the Neumann boundary data on the asymptotic behavior of solutions for large time. We show that problem admits global solutions whose long-time behavior essentially depends on the boundary data. To get a nonlinear theory for the multidimensional model. we propose general method based on Riemann–Hilbert approach and theory Cauchy type integral equations. The advantage of this method is that it can also be applied to non-integrable equations with general inhomogeneous boundary data.