Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

The chemotaxis system u t = Δ u − ∇ ⋅ ( u ∇ v ) , v t = Δ v − u v , is considered under the boundary conditions ∂ u ∂ ν − u ∂ v ∂ ν = 0 and v = v ⋆ on ∂Ω, where Ω ⊂ R n is a ball and v ⋆ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case n = 2, while global weak solutions are constructed when n ∈ {3, 4, 5}. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.

Multiple solutions for superlinear double phase Neumann problems

We study a double phase Neumann problem with a superlinear reaction which need not satisfy the Ambrosetti-Rabinowitz condition. Using the Nehari manifold method, we show that the problem has at least three nontrivial bounded ground state solutions, all with sign information (positive, negative and nodal).

Continuous operator method application for direct and inverse scattering problems

We describe the continuous operator method for solution nonlinear operator equations and discuss its application for investigating direct and inverse scattering problems. The continuous operator method is based on the Lyapunov theory stability of solutions of ordinary differential equations systems. It is applicable to operator equations in Banach spaces, including in cases when the Frechet (Gateaux) derivative of a nonlinear operator is irreversible in a neighborhood of the initial value. In this paper, it is applied to the solution of the Dirichlet and Neumann problems for the Helmholtz equation and to determine the wave number in the inverse problem. The internal and external problems of Dirichlet and Neumann are considered. The Helmholtz equation is considered in domains with smooth and piecewise smooth boundaries. In the case when the Helmholtz equation is considered in domains with smooth boundaries, the existence and uniqueness of the solution follows from the classical potential theory. When solving the Helmholtz equation in domains with piecewise smooth boundaries, the Wiener regularization is carried out. The Dirichlet and Neumann problems for the Helmholtz equation are transformed by methods of potential theory into singular integral equations of the second kind and hypersingular integral equations of the first kind. For an approximate solution of singular and hypersingular integral equations, computational schemes of collocation and mechanical quadrature methods are constructed and substantiated. The features of the continuous method are illustrated with solving boundary problems for the Helmholtz equation. Approximate methods for reconstructing the wave number in the Helmholtz equation are considered.

On the convergence of the barycentric method in solving internal Dirichlet and Neumann problems in $\mathbb{R}^2$ for the Helmholtz equation

The application of the barycentric method for the numerical solution of Dirichlet and Neumann problems for the Helmholtz equation in the bounded simply connected domain $\Omega\subset\mathbb{R}^2$ is considered. The main assumption in the solution is to set the $\Omega$ boundary in a piecewise linear representation. A distinctive feature of the barycentric method is the order of formation of a global system of vector basis functions for $\Omega$ via barycentric coordinates. The existence and uniqueness of the solution of Dirichlet and Neumann problems for the Helmholtz equation by the barycentric method are established and the convergence rate estimate is determined. The features of the algorithmic implementation of the method are clarified.