- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 14:04:08
32
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=ZAA&vol=25&update_since=2024-03-29
Zeitschrift für Analysis und ihre Anwendungen
Z. Anal. Anwend.
ZAA
0232-2064
1661-4534
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
10.4171/ZAA
http://www.ems-ph.org/doi/10.4171/ZAA
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2006)
25
2006
1
Low-Frequency Stability Analysis of Periodic Traveling-Wave Solutions of Viscous Conservation Laws in Several Dimensions
Myunghyun
Oh
University of Kansas, LAWRENCE, UNITED STATES
Kevin
Zumbrun
Indiana University, BLOOMINGTON, UNITED STATES
Stability, WKB, periodic traveling-waves
We generalize the work of Oh & Zumbrun and Serre on spectral stability of spatially periodic traveling waves of systems of viscous conservation laws from the one-dimensional to the multi-dimensional setting. Specifically, we extend to multi-dimensions the connection observed by Serre between the linearized dispersion relation near zero frequency of the linearized equations about the wave and the homogenized system obtained by slow modulation (WKB) approximation. This may be regarded as partial justification of the WKB expansion; an immediate consequence is that hyperbolicity of the multi-dimensional homogenized system is a necessary condition for stability of the waves. As pointed out by Oh & Zumbrun in one dimension, the description of the low-frequency dispersion relation is also a first step in the determination of time-asymptotic behavior.
Partial differential equations
Numerical analysis
General
1
21
10.4171/ZAA/1275
http://www.ems-ph.org/doi/10.4171/ZAA/1275
Compactness and Sobolev-Poincaré Inequalities for Solutions of Kinetic Equations
Myriam
Lecumberry
Mathematik in den Naturwissenschaften, LEIPZIG, GERMANY
Kinetic equations, Sobolev spaces, Littlewood-Paley decomposition
In this paper, we prove a regularity result on the velocity averages of the solution of a kinetic equation whose data have a Sobolev regularity \vspace{-0.05cm} $W^{s,p}$, $0
Partial differential equations
General
23
49
10.4171/ZAA/1276
http://www.ems-ph.org/doi/10.4171/ZAA/1276
Long Time Behavior of Solutions to the Caginalp System with Singular Potential
Maurizio
Grasselli
Politecnico di Milano, MILANO, ITALY
Hana
Petzeltová
Czech Academy of Sciences, PRAGUE 1, CZECH REPUBLIC
Giulio
Schimperna
Università di Pavia, PAVIA, ITALY
Phase-field models, maximal monotone operators, comparison principle, asymptotic behavior, Lojasiewicz-Simon inequality
We consider a nonlinear parabolic system which governs the evolution of the (relative) temperature $\teta$ and of an order parameter $\chi$. This system describes phase transition phenomena like, e.g., melting-solidification processes. The equation ruling $\chi$ is characterized by a singular potential $W$ which forces $\chi$ to take values in the interval $[-1,1]$. We provide reasonable conditions on $W$ which ensure that, from a certain time on, $\chi$ stays uniformly away from the pure phases $1$ and $-1$. Combining this separation property with the {\L}ojasiewicz-Simon inequality, we show that any smooth and bounded trajectory uniformly converges to a stationary state and we give an estimate of the decay rate.
Partial differential equations
Ordinary differential equations
Classical thermodynamics, heat transfer
General
51
72
10.4171/ZAA/1277
http://www.ems-ph.org/doi/10.4171/ZAA/1277
Non-Compact and Sharp Embeddings of Logarithmic Bessel Potential Spaces into Hölder-Type Spaces
David
Edmunds
University of Sussex, BRIGHTON, UNITED KINGDOM
Petr
Gurka
Czech University of Life Sciences, PRAGUE 6, CZECH REPUBLIC
Bohumír
Opic
Czech Academy of Sciences, PRAGUE 1, CZECH REPUBLIC
Lorentz-Zygmund spaces, logarithmic Bessel potential spaces, Hölder-continuous functions, embeddings
In our recent paper [Compact and continuous embeddings of logarithmic Bessel potential spaces. Studia Math.~168 (2005), 229 -- 250] we have proved an embedding of a logarithmic Bessel potential space with order of smoothness $\sigma$ less than one into a space of $\lambda(\cdot)$-H\"older-continuous functions. We show that such an embedding is not compact and that it is sharp.
Functional analysis
General
73
80
10.4171/ZAA/1278
http://www.ems-ph.org/doi/10.4171/ZAA/1278
The Factorization Method for Real Elliptic Problems
Bastian
Gebauer
Johannes Gutenberg-Universität Mainz, MAINZ, GERMANY
Elliptic partial differential equations, inverse problems, factorization method
The Factorization Method localizes inclusions inside a body from measurements on its surface. Without a priori knowing the physical parameters inside the inclusions, the points belonging to them can be characterized using the range of an auxiliary operator. The method relies on a range characterization that relates the range of the auxiliary operator to the measurements and is only known for very particular applications. In this work we develop a general framework for the method by considering symmetric and coercive operators between abstract Hilbert spaces. We show that the important range characterization holds if the difference between the inclusions and the background medium satisfies a coerciveness condition which can immediately be translated into a condition on the coefficients of a given real elliptic problem. We demonstrate how several known applications of the Factorization Method are covered by our general results and deduce the range characterization for a new example in linear elasticity.
Partial differential equations
Numerical analysis
General
81
102
10.4171/ZAA/1279
http://www.ems-ph.org/doi/10.4171/ZAA/1279
A Parabolic Integro-Differential Identification Problem in a Barrelled Smooth Domain
Jaan
Janno
Tallin Technical University, TALLINN, ESTONIA
Alfredo
Lorenzi
Università degli Studi di Milano, MILANO, ITALY
Inverse problem, memory kernel, parabolic equation
We consider the problem of recovering a space- and time-dependent kernel in a parabolic integro-differential equation. The related domain is assumed to be smooth and provided with two bases. Global existence and uniqueness results are proved.
Partial differential equations
Integral equations
General
103
130
10.4171/ZAA/1280
http://www.ems-ph.org/doi/10.4171/ZAA/1280
2
Nonexistence of Solutions to a Hyperbolic Equation with a Time Fractional Damping
Mokhtar
Kirane
Lab. et Dép. de Mathématiques, LA ROCHELLE CEDEX 1, FRANCE
Nasser-edine
Tatar
Dept. of Mathematics, DHAHRAN, SAUDI ARABIA
Fractional damping, non-existence, nonlinear hyperbolic equations
We consider the nonlinear hyperbolic equation \begin{align*} u_{tt}-\Delta u+D_{+}^{\alpha }u=h(t,x)\left| u\right| ^{p} \end{align*} posed in $Q:=(0,\infty )\times \mathbb{R}^{N},$ where $D_{+}^{\alpha }u$, $% 0
Partial differential equations
Real functions
Ordinary differential equations
General
131
142
10.4171/ZAA/1281
http://www.ems-ph.org/doi/10.4171/ZAA/1281
A Gauss-Bonnet Formula for Metrics with Varying Signature
Michael
Steller
Universität Stuttgart, STUTTGART, GERMANY
Gauss-Bonnet formula, singular metric, pseudo-geodesic, generic metric
A Gauss-Bonnet formula for compact orientable connected Riemannian or Lorentz\-ian 2-manifolds is well-known. We investigate singular metrics on 2-manifolds with varying signature. Such metrics are necessarily degenerate at some points of $M$ where most of the usual definitions for geometric quantities break down. We prove that under some additional assumptions there is a Gauss--Bonnet formula for compact orientable connected 2-manifolds with a singular metric. Some examples are given.
Differential geometry
General
143
162
10.4171/ZAA/1282
http://www.ems-ph.org/doi/10.4171/ZAA/1282
Approximative Compactness and Full Rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces
Henryk
Hudzik
Adam Mickiewicz University, POZNAN, POLAND
Wojciech
Kowalewski
Adam Mickiewicz University, POZNAN, POLAND
Grzegorz
Lewicki
Jagiellonian University, KRAKOW, POLAND
Musielak-Orlicz spaces, Lorentz-Orlicz spaces, Luxemburg norm, Amemyia norm, approximative compactness, reflexivity, Kadec-Klee property, drop property, full rotundity
We prove that approximative compactness of a Banach space $X$ is equivalent to the conjunction of reflexivity and the Kadec-Klee property of $X$. This means that approximative compactness coincides with the drop property defined by Rolewicz in {\it Studia Math.} 85 (1987), 25 -- 35. %\cite{RO}. Using this general result we find criteria for approximative compactness in the class of Musielak--Orlicz function and sequence spaces for both (the Luxemburg norm and the Amemiya norm) as well as critria for this property in the class of Lorentz--Orlicz spaces. Criteria for full rotundity of Musielak-Orlicz spaces are also presented in the case of the Luxemburg norm. An example of a reflexive strictly convex K\"othe function space which is not approximatively compact and some remark concerning the compact faces property for Musielak--Orlicz spaces are given.
Functional analysis
General
163
192
10.4171/ZAA/1283
http://www.ems-ph.org/doi/10.4171/ZAA/1283
The Plancherel and Hausdorff–Young Type Theorems for an Index Transformation
Semyon
Yakubovich
Faculdade de Ciências do Porto, PORTO, PORTUGAL
Bessel functions, index transform, Mellin transform, Kontorovich-Lebedev transform, Plancherel theorem, Hausdorff-Young inequality, Parseval equality
The Plancherel and Hausdorff-Young type theorems are proved for an integral transformation, which is associated with the product of the modified Bessel functions of different arguments. The transform essentially generalizes Lebedev's transformation involving squares of the modified Bessel functions as kernels.
Integral transforms, operational calculus
Special functions
General
193
204
10.4171/ZAA/1284
http://www.ems-ph.org/doi/10.4171/ZAA/1284
Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem of General Form. I
N.A.
Chernyavskaya
Ben Gurion University of the Negev, BEER-SHEBA, ISRAEL
L.A.
Shuster
Bar-Ilan University, RAMAT-GAN, ISRAEL
First order linear differential equation, correct solvability
We consider the singular boundary value problem %\eqref{1} -- \eqref{2} %\begin{equation}\label{1} %$$-r(x)y'(x)+q(x)y(x)=f(x),\quad x\in R$$ %\end{equation} %\begin{equation}\label{2} %$$\lim_{|x|\to\iy}y(x)=0,$$ %\end{equation} \begin{align*} -r(x)y'(x)+q(x)y(x)&=f(x),\quad x\in R \\ \lim_{|x|\to\iy}y(x)&=0, \end{align*} where $f \in L_p(\mathbb R),$\ $p\in[1,\iy]$ $(L_\iy(\mathbb R):=C(\mathbb R)),$\ $r $ is a continuous positive function on $\mathbb R$, \ $ 0\le q \in L_1^{\loc}.$ A solution of this problem is, by definition, any absolutely continuous function $y $ satisfying the limit condition and almost everywhere the differential equation. This problem is called correctly solvable in a given space $L_p(\mathbb R)$ if for any function $f\in L_p(\mathbb R)$ it has a unique solution $y\in L_p(\mathbb R)$ and if the following inequality holds with an absolute constant $c_p\in (0,\iy):$ %\begin{equation}\label{3} $$\|y\|_{L_p(\mathbb R)}\le c_p\|f\|_{L_p(\mathbb R)},\quad \ f\in L_p(\mathbb R) . %\end{equation} $$ We find minimal requirements for $r $ and $q$ under which the above problem is correctly solvable in $L_p(\mathbb R).$
Ordinary differential equations
General
205
235
10.4171/ZAA/1285
http://www.ems-ph.org/doi/10.4171/ZAA/1285
Existence of Periodic Solutions of a Class of Planar Systems
Xiaojing
Yang
Tsinghua University, BEIJING, CHINA
Periodic solutions, resonance, planar systems
In this paper, we consider the existence of periodic solutions for the following planar system: $$ J u'=\D H(u)+ G(u)+h(t)\,, $$ where the function $H(u)\in C^3(\R^2\backslash \{0\},\,\R)$ is positive for $u\ne 0$ and positively $(q,\,p)$-quasi-homogeneous of quasi-degree $pq,\, \,G: \R^2\to \R^2$ is local Lipschitz and bounded, $h\in L^\infty(0,\,2\pi)$ is $2\pi$-periodic and $J$ is the standard symplectic matrix.
Ordinary differential equations
General
237
248
10.4171/ZAA/1286
http://www.ems-ph.org/doi/10.4171/ZAA/1286
Fewer Convergence Conditions for the Halley Method
José Antonio
Ezquerro
Universidad de la Rioja, LOGRONO - LA RIOJA, SPAIN
M.A.
Hernández
Universidad de la Rioja, LOGRONO - LA RIOJA, SPAIN
Nonlinear equations in Banach spaces, Halley's method, semilocal convergence theorem, nonlinear integral equation
We present a new semilocal convergence result of Newton-Kantorovich type for Halley's method, where fewer convergence conditions are required than all the existing ones until now.
Operator theory
Integral equations
Numerical analysis
General
249
255
10.4171/ZAA/1287
http://www.ems-ph.org/doi/10.4171/ZAA/1287
An Existence Result for a Class of Extended Inclusion Problems with Applications to Equilibrium Problems
Ya-ping
Fang
Sichuan University, SICHUAN, CHINA
Nan-jing
Huang
Sichuan University, SICHUAN, CHINA
Extended inclusion problem, equilibrium problem, variational inequality, existence
Let $X$ be a real reflexive Banach space, $K\subset X$ a nonempty, closed and convex set, and $F:K\times K\to 2^X$ (the family of all the subsets of $X$) be a multi-valued mapping. In this paper, we consider the following extended inclusion problem: find $x^*\in K$ such that $K\subset F(x^*,x^*)$. Under suitable conditions, we establish an existence result for the extended inclusion problem. As applications, we give some existence theorems for equilibrium problems.
Calculus of variations and optimal control; optimization
Operator theory
General
257
264
10.4171/ZAA/1288
http://www.ems-ph.org/doi/10.4171/ZAA/1288
3
Local Growth Envelopes of Besov Spaces of Generalized Smoothness
António
Caetano
Universidade de Aveiro, AVEIRO, PORTUGAL
Walter
Farkas
University of Zurich, ZÜRICH, SWITZERLAND
Function space, generalized smoothness, atomic decomposition, growth envelope, sharp inequality, sharp embedding
The concept of local growth envelope $(\mathcal{E}_{LG}A,u)$ of the quasi-normed function space $A$ is applied to the Besov spaces of generalized smoothness $B_{p,q}^{\sigma,N}(\mathbb{R}^n)$.
Functional analysis
General
265
298
10.4171/ZAA/1289
http://www.ems-ph.org/doi/10.4171/ZAA/1289
Regularization of Hyperfunctions
Ricardo
Estrada
Louisiana State University, BATON ROUGE, UNITED STATES
Hyperfunctions, regularizations
We show that there are no continuous regularization procedures for the extension of hyperfunctions. We also show that there is a continuous projection operator from the space of hyperfunctions with support in a given compact set onto the subspace of hyperfunctions with support on a given \textsl{closed} subset if and only if the subset is a countable intersection of sets that are closed and open.
Functional analysis
General
299
310
10.4171/ZAA/1290
http://www.ems-ph.org/doi/10.4171/ZAA/1290
Controllability Results for Nondensely Defined Semilinear Functional Differential Equations
Mouffak
Benchohra
Université de Sidi Bel-Abbès, SIDI BEL-ABBES, ALGERIA
L.
Gorniewicz
Nicholas Copernicus University, TORUN, POLAND
S. K.
Ntouyas
University of Ioannina, IOANNINA, GREECE
A.
Ouahab
Université de Sidi Bel-Abbès, SIDI BEL-ABBES, ALGERIA
Controllability, functional semilinear differential equations, nondensely defined operator, fixed point, semigroup, measurable, Banach space
In this paper we investigate the controllability of first order semilinear functional and neutral functional differential equations in Banach spaces.
Systems theory; control
General
311
325
10.4171/ZAA/1291
http://www.ems-ph.org/doi/10.4171/ZAA/1291
Asymptotic and Pseudo Almost Periodicity of the Convolution Operator and Applications to Differential and Integral Equations
Dariusz
Bugajewski
Adam Mickiewicz University, POZNAN, POLAND
Toka
Diagana
Howard University, WASHINGTON, UNITED STATES
Crépin
Mahop
Howard University, WASHINGTON, UNITED STATES
Almost periodic function, asymptotically almost periodic function, Banach fixed-point principle, convolution operator, differential equation, integral equation, functional equation, pseudo almost periodic function, Zima's fixed-point theorem
We examine conditions which do ensure the asymptotic almost periodicity (respectively, pseudo almost periodicity) of the convolution function $f \ast h$ of $f$ with $h$ whenever $f$ is asymptotically almost periodic (respectively, pseudo almost periodic) and $h$ is a (Lebesgue) measurable function satisfying some additional assumptions. Next we make extensive use of those results to investigate on the asymptotically almost periodic (respectively, pseudo almost periodic) solutions to some differential, functional, and integral equations.
Integral transforms, operational calculus
Fourier analysis
General
327
340
10.4171/ZAA/1292
http://www.ems-ph.org/doi/10.4171/ZAA/1292
Spectral Properties of a Fourth Order Differential Equation
Manfred
Möller
University of Witwatersrand, WITS, SOUTH AFRICA
Vyacheslav
Pivovarchik
South-Ukrainian State Pedagogical University, ODESSA, UKRAINE
Fourth-order differential equation, pure imaginary eigenvalues, eigenvalue distribution
The eigenvalue problem $y^{(4)}(\lambda,x)-(gy')'(\lambda,x)= \lambda^2y(\lambda,x)$ with boundary conditions $y(\lambda,0)=0$, $y''(\lambda,0)=0$, $y(\lambda,a)=0$, $y''(\lambda,a)+i \alpha\lambda y'(\lambda,a)=0$ is considered, where $g\in C^1[0,a]$ and $\alpha >0$. It is shown that the eigenvalues lie in the closed upper half-plane and on the negative imaginary axis. A formula for the asymptotic distribution of the eigenvalues is given and the location of the pure imaginary spectrum is investigated.
Ordinary differential equations
General
341
366
10.4171/ZAA/1293
http://www.ems-ph.org/doi/10.4171/ZAA/1293
Computational Aspects of a Method of Stochastic Approximation
Konstantin
Runovski
Lomonosov State University, SEVASTOPOL, UKRAINE
Igor
Rystsov
National Technical University, KIEV, UKRAINE
Hans-Jürgen
Schmeisser
Friedrich-Schiller-University, JENA, GERMANY
Fast Fourier transform, random numbers, families of linear polynomial operators, approximation algorithms
A method of stochastic approximation is studied in the framework of the general convergence theory for families of linear polynomial operators of interpolation type. The description of the corresponding computational procedure, in particular, its input parameters, is given. Some optimization problems and aspects of implementation of the algorithm by means of {\it Maple} are discussed. It is shown that the algorithm can be applied not only to problems of "pure approximation" in the spaces $\,L_p\,$ with $\,0
Approximations and expansions
Fourier analysis
Numerical analysis
General
367
383
10.4171/ZAA/1294
http://www.ems-ph.org/doi/10.4171/ZAA/1294
Associated Spaces Defined by Ordinary Differential Equations
Le Thu
Hoai
Faculty of Applied Mathematics and Informatics, HANOI, VIETNAM
Wolfgang
Tutschke
Technische Universität Graz, GRAZ, AUSTRIA
Initial value problem, contraction-mapping principle, separation of variables, conservation law
The paper deals with initial value problems for desired functions $u(t,x)$ depending on the time $t$ and one spacelike variable $x$. In case the initial function $\varphi (x)$ satisfies an associated (ordinary) differential equation, the solution $u(t,x)$ satisfies the associated differential equation for each $t$. If the general solution of the associated differential equation is known, one gets a system of ordinary differential equations for the desired coefficients depending on $t$. In any case, the solution can be obtained as fixed point of a related integro-differential operator.
Partial differential equations
General
385
392
10.4171/ZAA/1295
http://www.ems-ph.org/doi/10.4171/ZAA/1295
On the Summation of Series in Terms of Bessel Functions
Slobodan
Trickovic
University of Nis, NIS, SERBIA
Mirjana
Vidanovic
University of Nis, NIS, SERBIA
Miomir
Stankovic
University of Nis, NIS, SERBIA
Bessel functions, Riemann $\z$-function, Poisson formula, Fourier transform
In this article we deal with summation formulas for the series %(\ref{1}), $ \sum_{n=1}^\infty\frac{J_\mu(nx)}{n^\nu}\,, $ referring partly to some results from our paper in %\cite{jmaa}. J. Math. Anal. Appl. 247 (2000) 15 -- 26. We show how these formulas arise from different representations of Bessel functions. In other words, we first apply Poisson's or Bessel's integral, then in the sequel we define a function by means of the power series representation of Bessel functions and make use of Poisson's formula. Also, closed form cases as well as those when it is necessary to take the limit have been thoroughly analyzed.
Special functions
Number theory
Numerical analysis
General
393
406
10.4171/ZAA/1296
http://www.ems-ph.org/doi/10.4171/ZAA/1296
Corrigendum to: On Harmonic Potential Fields and the Structure of Monogenic Functions, Z. Anal. Anw. 22 (2003)(2), 261-27
Fred
Brackx
Ghent University, GENT, BELGIUM
R.
Delanghe
Universiteit Gent, GENT, BELGIUM
Special functions
General
407
410
10.4171/ZAA/1297
http://www.ems-ph.org/doi/10.4171/ZAA/1297
4
What do we Learn from the Discrepancy Principle?
Peter
Mathé
Angewandte Analysis und Stochastik, BERLIN, GERMANY
Regularization, general source condition, saturation, variable Hilbert scale
The author analyzes the discrepancy principle when smoothness is given in terms of general source conditions. As it turns out, this framework is particularly well suited to reveal the mechanism under which this principle works. For general source conditions there is no explicit way to compute rates of convergence. Instead arguments must be based on geometric properties. Still this approach allows to generalize previous results. The analysis is accomplished with a result showing why this discrepancy principle inherently has the early saturation for a large class of regularization methods of bounded qualification.
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
411
420
10.4171/ZAA/1298
http://www.ems-ph.org/doi/10.4171/ZAA/1298
Best Possible Maximum Principles for Fully Nonlinear Elliptic Partial Differential Equations
G.
Porru
Univ. Studi Cagliari, CAGLIARI, ITALY
A.
Safoui
University of Marrakesh, MARRAKESH, MOROCCO
S.
Vernier-Piro
Univ. Studi Cagliari, CAGLIARI, ITALY
Fully nonlinear elliptic equations, Weingarten surfaces, best possible maximum principles
We investigate a class of equations including generalized Monge--Ampere equations as well as Weingarten equations and prove a maximum principle for suitable functions involving the solution and its gradient. Since the functions which enjoy the maximum principles are constant for special domains, we have a so called best possible maximum principle that can be used to find accurate estimates for the solution of the corresponding Dirichlet problem. For these equations we also give a variational form which may have its own interest.
Partial differential equations
Ordinary differential equations
Integral equations
Numerical analysis
421
434
10.4171/ZAA/1299
http://www.ems-ph.org/doi/10.4171/ZAA/1299
Lipschitz Stability of Solutions to Some State-Constrained Elliptic Optimal Control Problems
Roland
Griesse
Austrian Academy of Sciences, LINZ, AUSTRIA
Optimal control, elliptic equations, state constraints, Lipschitz stability
In this paper, optimal control problems with pointwise state constraints for linear and semilinear elliptic partial differential equations are studied. The problems are subject to perturbations in the problem data. Lipschitz stability with respect to perturbations of the optimal control and the state and adjoint variables is established initially for linear--quadratic problems. Both the distributed and Neumann boundary control cases are treated. Based on these results, and using an implicit function theorem for generalized equations, Lipschitz stability is also shown for an optimal control problem involving a semilinear elliptic equation.
Calculus of variations and optimal control; optimization
Operations research, mathematical programming
General
435
455
10.4171/ZAA/1300
http://www.ems-ph.org/doi/10.4171/ZAA/1300
Composition Operators between H∞ and α-Bloch Spaces on the Polydisc
Stevo
Stevic
Serbian Academy of Science, BEOGRAD, SERBIA
Composition operators, alpha-Bloch space, unit polydisc, compactness, boundedness
Let $U^n$ be the unit polydisc of ${\mathbb C}^n$ and $\vp(z)=(\vp_1(z),\ldots,\vp_n(z))$ a holomorphic self-map of $U^n.$ Let $H(U^n)$ denote the space of all holomorphic functions on $U^n,$ $H^\infty(U^n)$ the space of all bounded holomorphic functions on $U^n,$ and ${\cal B}^a(U^n),$ $a>0,$ the $a$-Bloch space, i.e.,\hspace{-0.3cm} $$ {\cal B}^a(U^n)=\bigg\{ f\in H(U^n)\, |\, \|f\|_{{\cal B}^a}=|f(0)|+\sup\limits_{z\in U^n}\sum\limits^n_{k=1} \left|\frac{\partial f} {\partial z_k}(z)\right|\left(1- |z_k|^2\right)^a
Operator theory
General
457
466
10.4171/ZAA/1301
http://www.ems-ph.org/doi/10.4171/ZAA/1301
Colombeau Generalized Functions and Solvability of Differential Operators
Khaled
Benmeriem
Centre Universitaire de Mascara, MASCARA, ALGERIA
Chikh
Bouzar
University of Oran, ESSENIA, ALGERIA
Colombeau generalized functions, regularized derivatives, Mizohata type operators, solvability of differential operators
The aim of this paper is to prove that the well known non solvable Mizohata type partial differential equations have Colombeau generalized solutions which are distributions if and only if they are solvable in the space of Schwartz distributions. Therefore the Colombeau generalized solvability includes both a new solution concept and new mathematical objects as solutions.
Functional analysis
Partial differential equations
General
467
477
10.4171/ZAA/1302
http://www.ems-ph.org/doi/10.4171/ZAA/1302
Blow-up of Solutions for a Class of Nonlinear Parabolic Equations
Zhang
Lingling
Taiyuan University of Technology, TAIYUAN, SHANXI, CHINA
Nonlinear parabolic equations, blow-up solutions, maximum principles
In this paper, the blow up of solutions for a class of nonlinear parabolic equations $$ u_t(x,t)=\nabla _{x}(a(u(x,t))b(x)c(t)\nabla _{x}u(x,t))+g(x,|\nabla _{x}u(x,t) |^2,t)f(u(x,t)) $$ with mixed boundary conditions is studied. By constructing an auxiliary function and using Hopf's maximum principles, an existence theorem of blow-up solutions, upper bound of ``blow-up time" and upper estimates of ``blow-up rate" are given under suitable assumptions on $a, b,c, f, g$, initial data and suitable mixed boundary conditions. The obtained result is illustrated through an example in which $a, b,c, f, g$ are power functions or exponential functions.
Partial differential equations
General
479
486
10.4171/ZAA/1303
http://www.ems-ph.org/doi/10.4171/ZAA/1303
Integral Equations with Diagonal and Boundary Singularities of the Kernel
Arvet
Pedas
Tartu University, TARTU, ESTONIA
Gennadi
Vainikko
Tartu University, TARTU, ESTONIA
Fredholm integral equation, Volterra integral equation, weakly singular integral equation, boundary singularities, smoothness of the solution, compact operators
We study the smoothness and the singularities of the solution to Fredholm and Volterra integral equations of the second kind on a bounded interval. The kernel of the integral operator may have diagonal and boundary singularities, information about them is given through certain estimates. The weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described. Examples show that the results cannot be improved.
Integral equations
General
487
516
10.4171/ZAA/1304
http://www.ems-ph.org/doi/10.4171/ZAA/1304
Elaboration of Some Results of Srivastava and Choi
Li
Hailong
Weinan Normal College, WEINAN, SHAANXI, CHINA
Masayuki
Toda
Saga University, SAGA, JAPAN
Hurwitz zeta-function, partial sum, digamma function, generalized Euler constant
In this paper we shall utilize some recent results of %\cite{KKSY} S. Kanemitsu, H.~Kumagai, H. M. Srivastava and M. Yoshimoto in {\it Appl. Math. Comput.} 154 (2004) %\cite{KKSY} on an asymptotic as well as an integral formula for the partial sum of the Hurwitz zeta-function, to elaborate on some results of Srivastava and Choi in {\it Series Associated with the Zeta and Related Functions} (Kluwer 2001), %\cite{SC}, and in some cases to give improved generalizations thereof. More specifically, we shall give an asymptotic expansion of the sum of the values derivative of the digamma function. We shall also re-establish Bendersky--Adamchik's result and Elizalde's result.
Number theory
Special functions
General
517
533
10.4171/ZAA/1305
http://www.ems-ph.org/doi/10.4171/ZAA/1305
The Hilbert Problem for Generalized Q-Holomorphic Functions
Sezayi
Hizliyel
Uludag University, GORUKLE BURSA, TURKEY
Generalized Beltrami systems, Q-Holomorphic functions, Hilbert boundary value problem
In this work, we extended classical Hilbert boundary value problem to generalized $Q$-holomorphic functions by replacing the condition that the solution vanishes at infinity by that the solution has a finite order of growth at infinity.
Functions of a complex variable
Partial differential equations
General
535
554
10.4171/ZAA/1306
http://www.ems-ph.org/doi/10.4171/ZAA/1306