Neumann Problem Research Articles

Radial blow-up in quasilinear Keller-Segel systems: approaching the full picture

Abstract The Neumann problem for the Keller-Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = − ∫ Ω u d x , is considered in n-dimensional balls Ω with n ⩾ 2 , with suitably regular and radially symmetric, radially nonincreasing initial data u 0. The functions D and S are only assumed to belong to C 2 ( [ 0 , ∞ ) ) and to satisfy D > 0 and S ⩾ 0 on [ 0 , ∞ ) as well as S ( 0 ) = 0 ; in particular, diffusivities with arbitrarily fast decay are included. In this general context, it is shown that it is merely the asymptotic behavior as ξ → ∞ of the expression I ( ξ ) := S ( ξ ) ξ 2 n D ( ξ ) , ξ > 0 , which decides about the occurrence of blow-up: Namely, it is seen that • if lim ξ → ∞ I ( ξ ) = 0 , then any such solution is global and bounded, that • if lim sup ξ → ∞ I ( ξ ) < ∞ and ∫ Ω u 0 is suitably small, then the corresponding solution is global and bounded, and that • if lim inf ξ → ∞ I ( ξ ) > 0 , then at each appropriately large mass level m, there exist radial initial data u 0 such that ∫ Ω u 0 = m , and that the associated solution blows up either in finite or in infinite time. This especially reveals the presence of critical mass phenomena whenever lim ξ → ∞ I ( ξ ) ∈ ( 0 , ∞ ) exists.

Optical Bio-Inspired Synaptic Devices.

The traditional computer with von Neumann architecture has the characteristics of separate storage and computing units, which leads to sizeable time and energy consumption in the process of data transmission, which is also the famous "von Neumann storage wall" problem. Inspired by neural synapses, neuromorphic computing has emerged as a promising solution to address the von Neumann problem due to its excellent adaptive learning and parallel capabilities. Notably, in 2016, researchers integrated light into neuromorphic computing, which inspired the extensive exploration of optoelectronic and all-optical synaptic devices. These optical synaptic devices offer obvious advantages over traditional all-electric synaptic devices, including a wider bandwidth and lower latency. This review provides an overview of the research background on optoelectronic and all-optical devices, discusses their implementation principles in different scenarios, presents their application scenarios, and concludes with prospects for future developments.

A nonvariational form of the Neumann problem for the Poisson equation

We present a nonvariational setting for the Neumann problem for the Poisson equation for solutions that are Hölder continuous and that may have infinite Dirichlet integral. We introduce a distributional normal derivative on the boundary for the solutions that extends that for harmonic functions which has been introduced in a previous paper and we solve the nonvariational Neumann problem for data in the interior with a negative Schauder exponent and for data on the boundary that belong to a certain space of distributions on the boundary.

Infinitely Many Weak Solutions for a Neumann Problem Involving $$p(\cdot )$$-Kirchhoff Triharmonic Operator

Infinitely Many Weak Solutions for a Neumann Problem Involving $$p(\cdot )$$-Kirchhoff Triharmonic Operator

Numerical solution to the Neumann problem in a Lipschitz domain, based on random walks

We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer methods.

Sharp boundary concentration for a two-dimensional nonlinear Neumann problem*

Sharp boundary concentration for a two-dimensional nonlinear Neumann problem*

Existence of solutions for critical Neumann problem with superlinear perturbation in the half‐space

AbstractIn this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem where , , , , , is the outward unit normal vector at the boundary , is the usual critical exponent for the Sobolev embedding and is the critical exponent for the Sobolev trace embedding . By establishing an improved Pohozaev identity, we show that problem () has no nontrivial solution if . Applying the mountain pass theorem without the condition and the delicate estimates for the mountain pass level, we obtain the existence of a positive solution for all and the different values of the parameters and . Particularly, for , , , we prove that problem () has a positive solution if and only if . Moreover, the existence of multiple solutions for () is also obtained by dual variational principle for all and suitable .

Multiple solutions to nonlocal Neumann boundary problem with sign‐changing coefficients

We consider the following Neumann boundary problem with a nonlocal Kirchhoff term where is a bounded smooth domain and and are sign‐changing coefficients which play important impact on the study. By using variational method and dividing corresponding Nehari manifold into two disjoint subsets, we study the existence of two positive solutions. To show the multiplicity of positive solutions, we find a set whose elements can be mapped to two subsets of Nehari manifold, which is also the key to proving strong convergence for minimizing sequences. Finally, the asymptotic behavior of solutions is studied as parameter tends to 0. Different from Dirichlet boundary problem, we show that the two solutions of Neumann problem tend to two different positive constants, respectively.

Radial solutions for Neumann problems involving prescribed mean curvature operator in a ball and in an annular domain

Using topological transversality method together with barrier strip technique and cut-off technique, we obtain new existence and uniqueness results of radial solutions to the Neumann problems involving prescribed mean curvature operator div∇v1+|∇v|2=f|x|,v,dvdrinΩ,∂v∂n=0on∂Ω,where Ω={x∈RN:R1<|x|<R2}(0≤R1<R2,N≥2), f:[R1,R2]×R2→R is continuous. Meanwhile, we demonstrate the importance of our results through an illustrative example.

On a Neumann Problem with an Intrinsic Operator

On a Neumann Problem with an Intrinsic Operator

Finite volume scheme and renormalized solutions for nonlinear elliptic Neumann problem with $$L^1$$ data

Finite volume scheme and renormalized solutions for nonlinear elliptic Neumann problem with $$L^1$$ data

Uniform boundary estimates for Neumann problems in parabolic homogenization

For a family of second-order parabolic systems with periodic, oscillating and time-dependent coefficients, we establish the uniform boundary Hölder and Lipschitz estimates for Neumann problems in [Formula: see text] and [Formula: see text] cylinders, respectively, by using the convergence rate method. Moreover, we establish the uniform [Formula: see text] estimates for initial-Neumann problems in [Formula: see text] cylinders for [Formula: see text] by using the real-variable method. As a by-product, we also obtain the Gaussian estimates for Neumann function and its derivatives.

Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones

We consider semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. The aim of the paper is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones.

Green’s Function of the Riquier–Neumann Problem for the Polyharmonic Equation in the Unit Ball

Green’s Function of the Riquier–Neumann Problem for the Polyharmonic Equation in the Unit Ball

The Tricomi–Neumann Problem for a Three-Dimensional Mixed-Type Equation with Singular Coefficients

The Tricomi–Neumann Problem for a Three-Dimensional Mixed-Type Equation with Singular Coefficients

Subellipticity, compactness, $H^{\epsilon}$ estimates and regularity for $\bar{\partial}$ on weakly $q$-pseudoconvex/concave domains

For a weakly q -pseudoconvex (resp. q -pseudoconcave) domain \Omega in a Stein manifold X of dimension n , we give a sufficient condition for subelliptic estimates for the \bar{\partial} -Neumann problem. Moreover, we study the compactness of the \bar{\partial} -Neumann operator N on \Omega . Such compactness estimates immediately lead to smoothness of solutions, the closed range property, the L^{2} -setting and the Sobolev estimates of N on \Omega for any \bar{\partial} -closed (r, k) -form with k \geqslant q (resp. k \leqslant q ). Furthermore, we study the \bar{\partial} -problem with support conditions in \Omega for forms of type (r, k) , with values in a holomorphic vector bundle. Applications to the \bar{\partial}_{b} -problem for smooth forms on boundaries of \Omega are given.

Stochastic Model of the Cauchy–Neumann Problem for Nonlinear Parabolic Equations

Stochastic Model of the Cauchy–Neumann Problem for Nonlinear Parabolic Equations

Stability and bifurcation diagram for a shadow Gierer–Meinhardt system in one spatial dimension

We are concerned with a Neumann problem of a shadow system of the Gierer–Meinhardt model in an interval I=(0,1) . A stationary problem is studied, and we consider the diffusion coefficient ɛ > 0 as a bifurcation parameter. Then a complete bifurcation diagram of the stationary solutions is obtained, and a stability of every stationary solution is determined. In particular, for each n⩾1 , two branches of n-mode solutions emanate from a trivial branch. All 1-mode solutions are stable for small τ > 0, and all n-mode solutions, n⩾2 , are unstable for all τ > 0, where τ > 0 is a time constant. The system is known for having stationary spiky patterns with large amplitude for small ɛ > 0. Then, asymptotic expansions of maximum and minimum values of a stationary solution as ɛ → 0 are also obtained.

Global structure of positive solutions for p-Laplacian Neumann problem with indefinite weight

Global structure of positive solutions for p-Laplacian Neumann problem with indefinite weight

The Neumann problem for fully nonlinear SPDE

The Neumann problem for fully nonlinear SPDE