The holomorphic solutions of the generalized Dhombres functional equation

We study holomorphic solutions f of the generalized Dhombres equation f(zf(z))=φ(f(z)), z∈C, where φ is in the class E of entire functions. We show, that there is a nowhere dense set E₀⊂E such that for every φ∈E?E₀, any solution f vanishes at 0 and hence, satisfies the conditions for local analytic solutions with fixed point 0 from our recent paper. Consequently, we are able to provide a characterization of solutions in the typical case where φ∈E?E₀. We also show that for polynomial φ any holomorphic solution on C?{0} can be extended to the whole of C. Using this, in special cases like φ(z)=zk+1, k∈N, we can provide a characterization of the analytic solutions in C.     

Local analytic solutions of the generalized Dhombres functional equation II

We study local analytic solutions f of the generalized Dhombres functional equation f(zf(z))=φ(f(z)), where φ is holomorphic at w₀≠0, f is holomorphic in some open neighborhood of 0, depending on f, and f(0)=w₀. After deriving necessary conditions on φ for the existence of nonconstant solutions f with f(0)=w₀ we describe, assuming these conditions, the structure of the set of all formal solutions, provided that w₀ is not a root of 1. If |w₀|≠1 or if w₀ is a Siegel number we show that all formal solutions yield local analytic ones. For w₀ with 0<|w₀|<1 we give representations of these solutions involving infinite products.     

On generalized Dhombres equations with nonconstant polynomial solutions in the complex plane

We characterize functional equations of the form ${f(zf(z))=f(z)^{k+1},z\in\mathbb {C}}$ , with ${k\in\mathbb N}$ , like those generalized Dhombres equations ${f(zf(z))=\varphi (f(z))}$ , ${z\in\mathbb C}$ , with given entire function ${\varphi}$ , which have a nonconstant polynomial solution f.     

The asymptotic behavior of solutions of the sine-gordon equation with singularities at zero

The problem of asymptotic analysis of radially symmetric solutions of the sine-Gordon equation reducible to the third Painlevé transcendent is posed. Solutions with singularities at the origin are studied. For finite values of the independent variable, an asymptotic expansion of such a solution is obtained; the leading term of this expansion is a modulated elliptic function. The corresponding modulation equation and phase shift are written out. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 329–342, March, 2000.     

Nontrivial solutions of elliptic boundary value problems with resonance at zero

Summary We consider a general nonlinear parabolic BVP (P) on a bounded and smooth domain R_n, the nonlinearity being given by a functionf: . We impose various hypotheses on f: « nonresonance » (with respect to the linearized BVP) at infinity, « nonresonance » or «resonance» at zero. Using an extension of Conley's index theory to noncompact spaces, we prove the existence of equilibria of (P) (i.e. solutions of a corresponding elliptic equation), as well as trajectories joining some of these equilibria. The results obtained generalize earlier results of Amann and Zehnder (who were the first to apply the Conley index to elliptic equations), of Peitgen and Schmitt, and of this author.Dedicated to Professor Jack K. Hale on his 55-th birthdayThis research was supported, in part, by a grant from the Deutsche Forschungsgemeinschaft (D.F.G.).     

Random generalized solutions to the heat equation

A solution for the heat conduction problem with random source term and random initial and boundary conditions is defined. Existence, uniqueness, properties, and asymptotic behavior of such a solution are investigated. Applications to one-dimensional problems are presented.     

Closed 1-forms with at most one zero

We prove that in any nonzero cohomology class ξ∈H¹(M;R) there always exists a closed 1-form having at most one zero.     

On the solutions of generalized discrete Poisson equation

The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a constant. The proof uses the Fourier transform as the main tool. The necessary condition for the existence of the solution is provided.     

Initial condition solutions of the generalized feller equation

Summary Solutions for given initial conditions are established for the generalized (autonomous parabolic) Feller equation in one positive space variable and one positive time variable. The coefficients of this equation are power functions of the space variable and depend on four parameters. In general, the equation is singular at the origin and at infinity. It contains as special cases the special Feller equation, the Kepinski equation, and the heat equation. Areas of application include biology, superradiant emission processes, heat propagation in solids (with special applications in the area of heat shield and ablation material design), and certain chemical reaction-diffusion processes. It is noteworthy that, for particular values of the parameters, the equation allows an evolution theoretic derivation of the fundamental distribution laws of Wien, Maxwell, Poisson, and Gauss. The general initial condition solution will be derived from a fundamental solution and will be given in terms of an integral transform for locally summable functions (singular integral). It is also shown that, for admissible parameter values, there always exist nontrivial solutions which approach zero as the time variable goes to zero and that, for particular parameter ranges, there exist singular solutions, conservative solutions, and delta function initial condition solutions.

---

Zusammenfassung Es werden Lösungen für gegebene Anfangsbedingungen für die verallgemeinerte (autonome, parabolische) Fellersche Gleichung in einer positiven Raumvariablen und einer positiven Zeitvariablen aufgestellt. Die Koeffizienten dieser Gleichung sind Potenzen der Raumvariablen und hängen von vier Parametern ab. Die Gleichung ist im allgemeinen singulär am Ursprung und im Unendlichen. Sie umfasst als Spezialfälle die spezielle Fellersche Gleichung, die Kepinskische Gleichung und die Wärmegleichung. Anwendung findet sie in der Biologie, in superstrahlenden Emissionsprozessen, in der Theorie der Wärmeausbreitung in Festkörpern (besonders beim Entwurf von Hitzeschilden und Abschmelzmaterialien) und im Gebiet gewisser chemischer Reaktions-Diffusionsprozesse. Est ist bemerkenswert, dass die Gleichung für besondere Parameterwerte eine evolutionstheoretische Herleitung der grundlegenden Verteilungsgesetze von Wien, Maxwell, Poisson und Gauss ermöglicht. Die allgemeine Lösung für gegebene Anfangsbedingung wird aus einer Grundlösung entwickelt und in der Form einer Integraltransformation gegeben für lokal summierbare Funktionen (singuläres Integral). Es wird weiterhin gezeigt, dass für zulässige Parameterwerte stets nichtriviale Lösungen existeren, die nach Null streben, wenn die Zeitvariable nach Null geht, und dass es für besondere Parameterbereiche singuläre und konservative Lösungen gibt und solche, die einer Deltafunktion als Anfangsbedingung entsprechen.

     

Asymptotics of solutions to the generalized Ostrovsky equation

We consider the Cauchy problem for the generalized Ostrovsky equation

_utx=u+_(f(u))xx,$u_{tx}=u+{(f(u))}_{xx},$

where f(u)=|u|^ρ−1u$f(u)={|u|}^{\rho -1}u$ if ρ   is not an integer and f(u)=u^ρ$f(u)=u^{\rho }$ if ρ   is an integer. We obtain the L^∞${\mathbf{L}}^{\infty }$ time decay estimates and the large time asymptotics of small solutions under suitable conditions on the initial data and the order of the nonlinearity.     

Small amplitude solutions of the generalized IMBq equation

In this paper, the global existence of small amplitude solution for the Cauchy problem of the multidimensional generalized IMBq equation is proved. Moreover, we obtain a nonlinear scattering result of the Cauchy problem of the IMBq equation for small initial data.     

Positive solutions to generalized Dickman equation

The paper investigates large-time behaviour of positive solutions to a generalized Dickman equation. The asymptotic behaviour of dominant and subdominant positive solutions is analysed and a structure formula describing behaviour of all solutions is proved. A criterion is also given for sufficient conditions on initial functions to generate positive solutions with prescribed asymptotic behaviour with values of their weighted limits computed.