Homogeneous Neumann Boundary Conditions Research Articles

Probabilistic analysis of a class of 2D-random heat equations via densities

We give new probabilistic results for a class of random two-dimensional homogeneous heat equations with mixed homogeneous Dirichlet and Neumann boundary conditions and an arbitrary initial condition on a rectangular domain. The diffusion coefficient is assumed to be an arbitrary second-order random variable, while the initial condition is a stochastic process admitting a Karhunen-Loève expansion. We then construct pointwise convergent approximations for the main moments and the density of the solution. The theoretical results are numerically illustrated.

Isogeometric multi-resolution full waveform inversion based on the finite cell method

Full waveform inversion (FWI) is an iterative identification process that serves to minimize the misfit of model-based simulated and experimentally measured wave field data. Its goal is to identify a field of parameters for a given physical object. For many years, FWI is very successful in seismic imaging to deduce velocity models of the earth or of local geophysical exploration areas. FWI has also been successfully applied in various other fields, including non-destructive testing (NDT) and biomedical imaging. The inverse optimization process of FWI relies on forward and backward solutions of the (elastic or acoustic) wave equation, as well as on efficient computations of adequate optimization directions. Many approaches employ (low order) finite element or finite difference methods, using parameterized material fields whose resolution is chosen in relation to the elements or nodes of the discretized wave field. In our previous paper (Bürchner et al., 2023), we investigated the potential of using the finite cell method (FCM) as the wave field solver. The FCM offers the advantage that highly complex geometric models can be incorporated easily. Furthermore, we demonstrated that the identification of the model’s density outperforms that of the velocity — particularly in cases where unknown voids characterized by homogeneous Neumann boundary conditions need to be detected. The paper at hand extends this previous study in the following aspects: The isogeometric finite cell analysis (IGA-FCM) – a combination of isogeometric analysis (IGA) and FCM – is applied as the wave field solver, with the advantage that the polynomial degree and subsequently also the sampling frequency of the wave field can be increased quite easily. Since the inversion efficiency strongly depends on the accuracy of the forward and backward wave field solutions and of the gradients of the functional, consistent, and lumped mass matrix discretization are compared. The resolution of the grid describing the unknown material density – thus allowing to identify voids in a physical object – is then decoupled from the knot span grid. Finally, we propose an adaptive multi-resolution algorithm that locally refines the material grid using an image processing-based refinement indicator. The developed inversion framework allows fast and memory-efficient wave simulations and object identification. While we study the general behavior of the proposed approach using 2D benchmark problems, a final 3D problem shows that it can also be used to identify void regions in geometrically complex spatial structures.

Behavior in time of solutions of a Keller–Segel system with flux limitation and source term

In this paper we consider radially symmetric solutions of the following parabolic–elliptic cross-diffusion system ut=Δu-∇·(uf(|∇v|2)∇v)+g(u),0=Δv-m(t)+u,∫Ωvdx=0,u(x,0)=u0(x),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\Delta u - \ abla \\cdot (u f(|\ abla v|^2 )\ abla v) + g(u), &{} \\\\ 0= \\Delta v -m(t)+ u, \\quad \\int _{\\Omega }v \\,dx=0, &{} \\\\ u(x,0)= u_0(x), &{} \\end{array}\\right. } \\end{aligned}$$\\end{document}in Omega times (0,infty ), with Omega a ball in {mathbb {R}}^N, Nge 3, under homogeneous Neumann boundary conditions, where g(u)= lambda u - mu u^k, lambda>0, mu >0, and k >1, f(|nabla v|^2 )= k_f(1+ |nabla v|^2)^{-alpha }, alpha >0, which describes gradient-dependent limitation of cross diffusion fluxes. The function m(t) is the time dependent spatial mean of u(x, t) i.e. m(t):= frac{1}{|Omega |} int _{Omega } u(x,t) ,dx. Under smallness conditions on alpha and k, we prove that the solution u(x, t) blows up in L^{infty }-norm at finite time T_{max} and for some p>1 it blows up also in L^p-norm. In addition a lower bound of blow-up time is derived. Finally, under largeness conditions on alpha or k, we prove that the solution is global and bounded in time.

Spatiotemporal dynamics in a toxin-producing predator–prey model with threshold harvesting

In this paper, we propose a toxin-producing predator–prey model with threshold harvesting and study spatiotemporal dynamics of the model under the homogeneous Neumann boundary conditions. At first, the persistence property of solutions to the system is investigated. Then the explicit requirements for the existence of nonconstant steady state solutions are derived by studying the relevant characteristic equation. These steady states occur from related constant steady states via steady state bifurcation. Throughout the analysis of the amplitude equations of Turing pattern by the multiple scale method, pattern formation can be found. Finally, we display umericalsimulations to verify the theoretical outcomes.

Stability and Hopf bifurcation of a modified Leslie–Gower predator–prey model with Smith growth rate and B–D functional response

This paper is concerned with a modified Leslie–Gower predator–prey diffusive dynamics system with Smith growth subject to homogeneous Neumann boundary condition, in which the functional response is Beddington–DeAngelis (Denote it by B–D) functional response term. Firstly, by applying the theory of stability and the Hopf bifurcation, we discuss the local stability and the existence of the Hopf bifurcation at the positive constant equilibrium solution of the ODE model, which the model undergoes the Hopf bifurcation when bifurcation parameter δ crosses the bifurcation critical value b0. Moreover, stability of the bifurcation periodic solution is analyzed. Secondly, the Turing instability and the direction of Hopf bifurcation of the corresponding to PDE system are investigated by using Normal form theory and Centre manifold theory. Finally, we study the numerical simulations of this system to illustrate the theoretical analysis.

Bifurcations of a diffusive predator–prey model with prey‐stage structure and prey‐taxis

This paper is concerned with a diffusive predator–prey model with prey‐taxis and prey‐stage structure under the homogeneous Neumann boundary condition. The stability of the unique positive constant equilibrium of the predator–prey model is derived. Hopf bifurcation and steady‐state bifurcation are also concluded.

Critical mass for Keller–Segel systems with supercritical nonlinear sensitivity

This paper is concerned with the following radially symmetric Keller–Segel systems with nonlinear sensitivity [Formula: see text] and [Formula: see text], posed on [Formula: see text] [Formula: see text] and subjected to homogeneous Neumann boundary conditions. It is well-known that [Formula: see text] is the critical exponent of the systems in the sense that all solutions exist globally if [Formula: see text] and there exist finite-time blowup solutions if [Formula: see text]. Here we consider the supercritical case [Formula: see text] and show a critical mass phenomenon. Precisely, we prove that there exists a critical mass [Formula: see text] such that (1) for arbitrary nonincreasing nonnegative initial data [Formula: see text] with [Formula: see text] and [Formula: see text], the corresponding solution blows up in finite time if [Formula: see text], and if [Formula: see text] we can only prove that the solution blows up in finite time or infinite time; (2) for some nonincreasing nonnegative initial data with [Formula: see text], the corresponding solutions are globally bounded. Our results extend that of Winkler’s paper [M. Winkler, How unstable is spatial homogeneity in Keller–Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic–elliptic cases, Math. Ann. 373 (2019) 1237–1282], where he proved similar results for the system with [Formula: see text].

Quantum-information theory of magnetic field influence on circular dots with different boundary conditions

Influence of the transverse uniform magnetic field B on position (subscript ρ) and momentum (γ) Shannon quantum-information entropies S ρ,γ Fisher informations I ρ,γ and informational energies O ρ,γ is studied theoretically for the 2D circular quantum dots (QDs) whose circumference supports homogeneous either Dirichlet or Neumann boundary condition (BC). Comparative analysis reveals similarities and differences of the influence on the properties of the structure of the surface interaction with the magnetic field. A conspicuous distinction between the two spectra are crossings at the increasing induction of the Neumann energies with the same radial quantum number n and adjacent non-positive angular indices m. At the growing B, either system undergoes Landau condensation when its characteristics turn into their uniform field counterparts. It is shown that for the Dirichlet system this transformation takes place at the smaller magnetic intensities; e.g. the Dirichlet sum on its approach from above to a fundamental limit is at any B smaller than the corresponding Neumann quantity what physically means that the former geometry provides more total information about the position and motion of the particle. It is pointed out that the widely accepted disequilibrium uncertainty relation , with d being a dimensionality of the system, is violated by the Neumann QD in the magnetic field. Comparison with electrostatic harmonic confinement is performed; in particular, contrary to the hard-wall QDs, for this configuration the sums S ρ + S γ and the products I ρ I γ and O ρ O γ do not depend on the field. Physical interpretation is based on the different roles of the two BCs and their interplay with the field: Dirichlet (Neumann) surface is a repulsive (attractive) interface.

Global boundedness in a Keller-Segel system with flux limitation and logistic source

We consider the parabolic chemotaxis system{ut=Δu−∇⋅(uf(|∇v|2)∇v)+ru−μuγ,x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, in a smooth bounded domain Ω⊂Rn(n≥1) with the homogeneous Neumann boundary conditions, where r,μ>0, γ>1 and the function f satisfiesf(ξ)=(1+ξ)−α2,for all ξ≥0, with α>0. In the case n≤2, it is shown that the corresponding initial value problem possesses a global bounded classical solution for any α,μ>0. In the case n≥3, if γ=2 and α=n−22n, there exists μ0>0 such that for any μ≥μ0, a global bounded classical solution exists.

Well‐posedness and blow‐up of the fractional Keller–Segel model on domains

AbstractThis work deals with well‐posedness and blow‐up in the setting of Lebesgue and Besov spaces to the time‐fractional Keller–Segel model for chemotaxis under homogeneous Neumann boundary conditions in a smooth domain of . The KS model consists in a coupled system of partial differential equations. In particular, we also treat the unique continuation of the solution and the persistence of continuous dependence on the initial data for the continued solution.

Boundedness and weak stabilization in a degenerate chemotaxis model arising from tumor invasion

This paper is concerned with the degenerate system{ut=∇⋅(f(u,w)∇u−g(u)∇v),x∈Ω,t>0,vt=Δv+wz,x∈Ω,t>0,wt=−wz,x∈Ω,t>0,zt=Δz−z+u,x∈Ω,t>0 in a bounded domain Ω⊂RN (N≥2) under the no-flux boundary condition for u and the homogeneous Neumann boundary condition for v,z with non-negative initial data u0,v0,w0,z0. Here, the diffusivity f and the sensitivity g are assumed to fulfill f(u,w)≥um−1 (m>1), 0≤g(u)≤uα(α∈R). It is shown that if α+1<m+4N(N≥2) or α+1=m+4N(N≥3) with small mass of u0, then the system possesses a global bounded weak solution which converges to the constant equilibrium in the weak⁎ topology in L∞(Ω) as t→∞.

Non-existence of patterns for a class of weighted degenerate operators

A classical result by Casten-Holland and Matano asserts that constants are the only positive and stable solutions to semilinear elliptic PDEs subject to homogeneous Neumann boundary condition in bounded convex domains. In other terms, this result asserts that stable patterns do not exist in convex domains.In this paper we consider a weighted version of the Laplace operator, where the weight may be singular or degenerate at the origin, and prove the nonexistence of patterns, extending the results by Casten-Holland and Matano to general weak solutions (not necessarily stable) and under a suitable assumption on the nonlinearity and the domain.Our results exhibit some intriguing behavior of the problem according to the weight and the geometry of the domain. Indeed, our main results follow from a geometric assumption on the second fundamental form of the boundary in terms of a parameter which describes the degeneracy of the operator. As a consequence, we provide some examples and show that nonexistence of patterns may occur also for non convex domains whenever the weight is degenerate.

Dynamics analysis of a spatiotemporal SI model

In this article, we investigate a susceptible-infected (SI) model with the saturated treatment, the non-monotonic incidence rate, the logistic growth, and the homogeneous Neumann boundary conditions. The global existence and the uniform boundedness of the parabolic system are performed. After that, we investigate the global stability of the disease free equilibrium (DFE) and the endemic equilibrium (EE), respectively. In the end, we give a priori estimates, some propositions of the nonconstant steady states to the elliptic system. Meanwhile, we find that the diffusion rates of the susceptible and the infected population can affect the nonexistence of the nonconstant steady states. An interesting finding is DFE and basic reproduction number do not exist when the intrinsic growth rate of the susceptible class less than the rate of susceptible individuals being vaccinated.

Analysis and simulation of a delayed HIV model with reaction–diffusion and sliding control

Selection of an appropriate therapy threshold to restrain the virus load is still challenging on structured treatment interruptions (STIs) for HIV. In this paper, we ponder that how the sliding control and multistability to regulate the treatment effect through comprehensive dynamics of a virus-immune model. Firstly, based on piecewise therapy, we propose a delayed reaction–diffusion virus-immune model under the homogeneous Neumann boundary condition. Secondly, the existence and stabilities of five kinds of equilibria as well as the direction and stability of spatial Hopf bifurcation at regular equilibrium are investigated. Thirdly, the sliding domain and the boundary node bifurcations are addressed by theoretical analysis. Finally, we appraise the effects of therapy threshold, sliding domain and multistability on HIV therapy by simulations, and further seek out the appropriate therapy threshold for infected patients with given physiological parameters and present the corresponding principles. Our explorations will provide evidence for HIV and other disease therapies.

Global existence for a class of non-equilibrium reaction–diffusion systems with flux limitation

For a class of general reaction–diffusion systems with flux limitation in one dimension, when homogeneous Neumann boundary conditions are considered, we will prove global upper and lower boundness estimates for the positive classical solutions and boundness for their first-order spatial derivatives. In particular, the non-equilibrium radiation flux-limited diffusion coupled with material heat conduction model is included as a special case of the general reaction–diffusion systems. The key point of our method is to at first prove that the second-order spatial derivative of the solution of the second diffusion equation can be linearly controlled by the first-order spatial derivative of the solution of the flux-limited diffusion equation. Then, with this control, we can apply the Bernstein method to the flux-limited diffusion equation, while a stratifying time technique is introduced to get a complete first order spatial derivative estimate. With these global prior estimates, the global existence and uniqueness of positive classical solutions for the general flux-limited diffusion systems are proved.

Boundedness, stability and pattern formation for a predator-prey model with Sigmoid functional response and prey-taxis

This article concerns the structure of the nonconstant steady states for a predator-prey model of Leslie-Gower type with Sigmoid functional and prey-taxis subject to the homogeneous Neumann boundary condition. The existence of bounded classical global solutions is discussed in bounded domains with arbitrary spatial dimension and any prey-taxis sensitivity coefficient. The local stability of the homogeneous steady state is analyzed to show that the prey-taxis sensitivity coefficient destabilizes the stability of the homogeneous steady state when prey defends. Then we study the existence and stability of the nonconstant positive steady state of the system over 1D domain by applying the bifurcation theory and present properties of local branches such as pitchfork and turning direction. Moreover, we discuss global bifurcation, homogeneous steady state solutions, nonconstant steady states solutions, spatio-temporal periodic solutions and spatio-temporal irregular solutions which demonstrate the coexistence and spatial distribution of prey and predator species. Finally, we perform numerical simulations to illustrate and support our theoretical analysis.

Critical mass phenomena in higher dimensional quasilinear Keller–Segel systems with indirect signal production

In this paper, we deal with quasilinear Keller–Segel systems with indirect signal production, complemented with homogeneous Neumann boundary conditions and suitable initial conditions, where is a bounded smooth domain, and We show that in the case , there exists such that if either or , then the solution exists globally and remains bounded, and that in the case , if either or , then there exist radially symmetric initial data such that and the solution blows up in finite or infinite time, where the blow‐up time is infinite if . In particular, if , there is a critical mass phenomenon in the sense that is a finite positive number.

Turing‐Turing and Turing‐Hopf bifurcations in a general diffusive Brusselator model

AbstractA general reaction‐diffusion Brusselator model subject to homogeneous Neumann boundary conditions is investigated in this paper. First, the stability of the unique positive equilibrium is studied, and we identify the existence of the Hopf bifurcation. Then, occurrence conditions of the Turing instability, the Turing‐Turing, and the Turing‐Hopf bifurcations are established. To explore the spatiotemporal solutions resulting from the bifurcation, the amplitude equations of the Turing‐Turing and the Turing‐Hopf bifurcations are established via the method of multiple time scale. It is found that the model admits the nonconstant steady state, the mixed nonconstant steady state, the spatially homogeneous periodic solution, and the spatially nonhomogeneous periodic solution. As such, the nonhomogeneous spatiotemporal solutions appear in the model. In the end, numerical simulations verify the validity of the theoretical results.

Steady state solutions for the Gierer-Meinhardt system in the whole space

We are concerned with the study of positive solutions to the Gierer-Meinhardt system{−Δu+λu=upvq+ρ(x)in RN,N≥3,−Δv+μv=umvsin RN, which satisfy u(x),v(x)→0 as |x|→∞. In the above system p,q,m,s>0, λ,μ≥0 and ρ∈C(RN), ρ≥0. It is a known fact that posed in a smooth and bounded domain of RN, the above system subject to homogeneous Neumann boundary conditions has positive solutions if p>1 and σ=mq(p−1)(s+1)>1. In the present work we emphasize a different phenomenon: we see that for λ,μ>0 large, positive solutions with exponential decay exist if 0<σ≤1. Further, for λ=μ=0 we derive various existence and nonexistence results and underline the role of the critical exponents p=NN−2 and p=N+2N−2.

Resonant solutions for elliptic systems with Neumann boundary conditions

We consider a sublinear perturbation of an elliptic eigenvalue system with homogeneous Neumann boundary conditions. For oscillatory nonlinearities and using bifurcation from infinity, we prove the existence of an unbounded sequence of turning points and an unbounded sequence of resonant solutions.&#x0D; See also https://ejde.math.txstate.edu/special/02/d1/abstr.html