Bifurcation analysis for a modified quasilinear equation with negative exponent

In this paper, we consider the following modified quasilinear problem: − Δ u − κ u Δ u 2 = λ a ( x ) u − α + b ( x ) u β i n Ω , u > 0 i n Ω , u = 0 o n ∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a, b are two bounded continuous functions, α > 0, 1 < β ≤ 22* − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.

On a nonclassical problem for the heat equation and the Feller semigroup generated by it

The initial boundary value problem for the equation of heat conductivity with the Wenzel conjugation condition is studied. It does not fit into the general theory of parabolic initial boundary value problems and belongs to the class of conditionally correct ones. In space of bounded continuous functions by the method of boundary integral equations its classical solvability under some conditions is established. In addition, it is proved that the obtained solution is a Feller semigroup, which represents some homogeneous generalized diffusion process in the area considered here.

Towards a Koopman theory for dynamical systems on completely regular spaces

The Koopman linearization of measure-preserving systems or topological dynamical systems on compact spaces has proven to be extremely useful. In this article, we look at dynamics given by continuous semiflows on completely regular spaces, which arise naturally from solutions of PDEs. We introduce Koopman semigroups for these semiflows on spaces of bounded continuous functions. As a first step we study their continuity properties as well as their infinitesimal generators. We then characterize them algebraically (via derivations) and lattice theoretically (via Kato’s equality). Finally, we demonstrate—using the example of attractors—how this Koopman approach can be used to examine properties of dynamical systems. This article is part of the theme issue ‘Semigroup applications everywhere’.

The Riesz representation theorem and weak∗ compactness of semimartingales

We show that the sequential closure of a family of probability measures on the canonical space of càdlàg paths satisfying Stricker’s uniform tightness condition is a weak∗ compact set of semimartingale measures in the dual pairing of bounded continuous functions and Radon measures, that is, the dual pairing from the Riesz representation theorem under topological assumptions on the path space. Similar results are obtained for quasi- and supermartingales under analogous conditions. In particular, we give a full characterisation of the strongest topology on the Skorokhod space for which these results are true.

Generalized Numerical Index of Function Algebras

Let X be a complex Banach space and Cb(Ω:X) be the Banach space of all bounded continuous functions from a Hausdorff space Ω to X, equipped with sup norm. A closed subspace A of Cb(Ω:X) is said to be an X-valued function algebra if it satisfies the following three conditions: (i) A≔{x⁎∘f:f∈A, x⁎∈X⁎} is a closed subalgebra of Cb(Ω), the Banach space of all bounded complex-valued continuous functions; (ii) ϕ⊗x∈A for all ϕ∈A and x∈X; and (iii) ϕf∈A for every ϕ∈A and for every f∈A. It is shown that k-homogeneous polynomial and analytic numerical index of certain X-valued function algebras are the same as those of X.

On rings of Baire one functions

<p>This paper introduces the ring of all real valued Baire one functions, denoted by B₁(X) and also the ring of all real valued bounded Baire one functions, denoted by B^∗₁(X). Though the resemblance between C(X) and B₁(X) is the focal theme of this paper, it is observed that unlike C(X) and C^∗(X) (real valued bounded continuous functions), B^∗₁ (X) is a proper subclass of B₁(X) in almost every non-trivial situation. Introducing B₁-embedding and B^∗₁-embedding, several analogous results, especially, an analogue of Urysohn’s extension theorem is established.</p>

Nonlocal eigenvalue problems with variable exponent

We consider the nonlocal eigenvalue problem of the following form$$(\mathcal{P}k)\left\{ {\matrix{ {\mathcal{L}_K^{p(x)}u(x) + {{\left| {u(x)} \right|}^{\bar p(x) - 2}}u(x)} \hfill & = \hfill & {\lambda {{\left| {u(x)} \right|}^{r(x) - 2}}u(x)} \hfill & {in} \hfill & {\Omega ,} \hfill \cr u \hfill & = \hfill & 0 \hfill & {in} \hfill & {{{\rm\mathbb{R}}^N}\backslash \Omega ,} \hfill \cr } } \right.$$where Ω is a smooth open and bounded set in 𝕉N (N ⩾ 3), λ > 0 is a real number, K is a suitable kernel and p, r are two bounded continuous functions on ̄Ω. The main result of this paper establishes that any λ > 0 sufficiently small is an eigenvalue of the above nonhomogeneous nonlocal problem. The proof relies on some variational arguments based on Ekeland's variational principle.