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Communications on Pure & Applied Analysis

Open Access Articles

T. Ogawa and  Tohru Ozawa
2015, 14(4): i-iii doi: 10.3934/cpaa.2015.14.4i +[Abstract](27) +[PDF](162.5KB)
This special issue of Discrete and Continuous Dynamical Systems is dedicated to Professor Gustavo Ponce on the occasion of his sixtieth birthday.

Gustavo Ponce was born on April 20, 1952, in Venezuela. He received his B.A. in 1976 from Universidad Central de Venezuela and his Ph. D. in 1982 with the dissertation entitled ``Long time stability of solutions of nonlinear evolution equations" under the supervision of Sergiu Klainerman and Louis Nirenberg at Courant Institute, New York University. After professional experiences at University of California at Berkely (1982-1984), Universidad Central de Venezuela (1984-1986), University of Chicago (1986-1989), and Pennsylvania State University (1989-1991), he was appointed to a full professorship at Department of Mathematics, University of California at Santa Barbara in 1991, where he has remained up until now.
Bernd Kawohl and  Juan J. Manfredi
2015, 14(1): i-i doi: 10.3934/cpaa.2015.14.1i +[Abstract](25) +[PDF](91.2KB)
This special issue Emerging Trends in Nonlinear PDE of this journal was conceived during the Fall 2013 research semester “Evolutionary Problems” held at the Mittag-Leffler-Institute and its colophon conference Quasilinear PDEs and Game Theory held at Uppsala University in early December. The editors of this special issue participated in these activities. Following several conversations with other participants, we solicited manuscripts from participants of the Mittag-Leffler special semester, the Uppsala conference, as well as from colleagues working in closely related fields. Seventeen papers (out of twenty-one) are authors by participants in the research program at the Mittag-Leffler or the conference at Uppsala.
Alain Miranville and  Vladimir V. Chepyzhov
2014, 13(5): i-x doi: 10.3934/cpaa.2014.13.5i +[Abstract](29) +[PDF](169.8KB)
-- Mark Iosifovich, how do You think, which scientists have influenced You in the very beginning of Your academic career?

If viewed chronologically, there were, first of all, my teachers at Lvov State University, which I entered in 1939. The University formerly bore the name of king Kazimir and then became the Ivan Franko University, where the Dean of the Mathematics Department Stefan Banach, a brilliant mathematician, worked. We were taught by the most outstanding professors of the Banach's school: Bronislaw Knaster -- analytical geometry, Yuliush Schauder -- theoretical mechanics, Professor Stanislaw Mazur -- differential geometry. Professor Vladislav Orlicz gave lectures on algebra. All this teaching was in Polish. Only the Deputy Dean Professor Myron Zaritsky gave lectures in Ukrainian.
Noureddine Alaa , Marc Dambrine , Antoine Henrot and  Alain Miranville
2012, 11(6): i-ii doi: 10.3934/cpaa.2012.11.6i +[Abstract](20) +[PDF](92.0KB)
The present volume is dedicated to Michel Pierre. An international Workshop for his 60th birthday, entitled "Partial Di erential Equations and Applications", was organized in Vittel (France) from October 22 to October 24, 2009.
Jacques Demongeot , Danielle Hilhorst , Hiroshi Matano and  Masayasu Mimura
2012, 11(1): i-i doi: 10.3934/cpaa.2012.11.1i +[Abstract](55) +[PDF](72.5KB)
This volume deals with the mathematical modeling and the analysis of reaction-diffusion systems, as well as their applications in a number of different fields. It grew from a workshop organized by ReaDiLab, a Japan-France research collaboration unit of CNRS (Laboratoire International Associé du CNRS). This workshop took place at the University of Paris-Sud in June, 2009, bringing together many members of ReaDiLab with researchers from other French and Japanese laboratories. ReaDiLab is composed of 33 Japanese and 36 French researchers in the fields of mathematics, biology, medicine, and chemistry. Its goal is to develop mathematical modeling, analysis and numerical methods for reaction-diffusion systems arising in all those fields.

In order to understand the problems occurring in these areas of application, one should not only apply known methods, but also develop novel mathematical tools. Because of this, many results corresponding to new approaches are given in the main topics of this CPAA Special Volume, including demography and travelling waves in epidemics modelling, structured populations growth, propagation in inhomogeneous media, ecology and dry land vegetation, formation of stationary spatio-temporal patterns in reaction-diffusion systems both from a mathematical and an experimental view point, spatio-temporal dynamics of cooperation, cell migration and bacterial suspensions. This issue also includes more mathematically oriented topics such as interface dynamics, stability of non-constant stationary solutions, heterogeneity-induced spot dynamics, boundary spikes, appearance of anomalous singularities in parabolic equations, finite time blow-up, a multi-parameter inverse problem, and the numerical approximation of parabolic equations and chemotactic systems. We hope these advanced results will be useful to the community of researchers working in the domain of partial differential equations, and that they will serve as examples of mathematical modelling to those working in the different areas of application mentioned above.
Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis
Norikazu Saito
2012, 11(1): 339-364 doi: 10.3934/cpaa.2012.11.339 +[Abstract](100) +[PDF](486.1KB)
We are concerned with the finite-element approximation for the Keller-Segel system that describes the aggregation of slime molds resulting from their chemotactic features. The scheme makes use of a semi-implicit time discretization with a time-increment control and Baba-Tabata's conservative upwind finite-element approximation in order to realize the positivity and mass conservation properties. The main aim is to present error analysis that is an application of the discrete version of the analytical semigroup theory.
Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system
Yasumasa Nishiura , Takashi Teramoto and  Xiaohui Yuan
2012, 11(1): 307-338 doi: 10.3934/cpaa.2012.11.307 +[Abstract](49) +[PDF](1388.2KB)
Spatially localized patterns form a representative class of patterns in dissipative systems. We study how the dynamics of traveling spots in two-dimensional space change when heterogeneities are introduced in the media. The simplest but fundamental one is a line heterogeneity of jump type. When spots encounter the jump, they display various outputs including penetration, rebound, and trapping depending on the incident angle and its height. The system loses translational symmetry by the heterogeneity, but at the same time, it causes the emergence of various types of heterogeneity-induced-ordered-patterns (HIOPs) replacing the homogeneous constant state. We study these issues by using a three-component reaction-diffusion system with one activator and two inhibitors. The above outputs can be obtained through the interaction between the HIOPs and the traveling spots. The global bifurcation and eigenvalue behavior of HISPs are the key to understand the underlying mechanisms for the transitions among those dynamics. A reduction to a finite dimensional system is presented here to extract the model-independent nature of the dynamics. Selected numerical techniques for the bifurcation analysis are also provided.
Zhaosheng Feng and  Wei Feng
2011, 10(5): i-ii doi: 10.3934/cpaa.2011.10.5i +[Abstract](18) +[PDF](104.4KB)
This issue of Communications on Pure and Applied Analysis, comprises a collection in the general area of nonlinear systems and analysis, and related applications in mathematical biology and engineering. During the past few decades people have seen an enormous growth of the applicability of dynamical systems and the new developments of related dynamical concepts. This has been driven by modern computer power as well as by the discovery of advanced mathematical techniques. Scientists in all disciplines have come to realize the power and beauty of the geometric and qualitative techniques developed during this period. More importantly, they have been able to apply these techniques to a various nonlinear problems ranging from physics and engineering to biology and ecology, from the smallest scales of theoretical particle physics up to the largest scales of cosmic structure. The results have been truly exciting: systems which once seemed completely intractable from an analytical point of view can now be studied geometrically and qualitatively. Chaotic and random behavior of solutions of various systems is now understood to be an inherent feature of many nonlinear systems, and the geometric and numerical methods developed over the past few decades contributed significantly in those areas.
Roberta Fabbri and  Carmen Núñez
2011, 10(3): i-iii doi: 10.3934/cpaa.2011.10.3i +[Abstract](47) +[PDF](110.5KB)
This special issue collects eleven papers in the general area of nonautonomous dynamical systems. They contain a rich selection of new results on pure and applied aspects of the eld.
Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion
Andrea L. Bertozzi and  Dejan Slepcev
2010, 9(6): 1617-1637 doi: 10.3934/cpaa.2010.9.1617 +[Abstract](48) +[PDF](276.2KB)
We present an energy-methods-based proof of the existence and uniqueness of solutions of a nonlocal aggregation equation with degenerate diffusion. The equation we study is relevant to models of biological aggregation.
B. Brighi , Michel Chipot , A. Corbo Esposito , G. Mingione , C. Sbordone , I. Shafrir , V. Valente and  G. Vergara Caffarelli
2010, 9(5): i-i doi: 10.3934/cpaa.2010.9.5i +[Abstract](33) +[PDF](18.8KB)
The 6th european conference on elliptic and parabolic problems took place in Gaeta from May 25 to May 29, 2009. It brought together more than 170 participants. This volume collects some of the papers presented there.
    This meeting could not have been possible without the support of the Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, the Università di Cassino, the Accademia Pontaniana (Napoli), the Istituto Italiano per gli Studi Filosoci (Napoli), the GNAMPA, the Université de Haute Alsace (Mulhouse), the Universität Zürich, the MeMoMat Sapienza Università di Roma, the IAC CNR, the Comune di Gaeta and the partial support of the ERC grant 207573-2 Vectorial Problems. We thank all these institutions for their help.
    We would like also to thank DCDS and especially Professor Shouchuan Hu for having accepted to publish these articles.
Tao Qiang and  Yuesheng Xu
2007, 6(3): i-iii doi: 10.3934/cpaa.2007.6.3i +[Abstract](15) +[PDF](49.5KB)
This special issue contains a selection of 19 papers from two separate sources. About half of the total is from submissions to the international conference on Wavelet Analysis and Applications, 2005, University of Macau. The conference had 170 submissions from scholars and engineers from 22 different countries and areas, including Australia, Belgium, Brazil, China, Ethiopia, France, Germany, India, Iran, Hong Kong, Japan, Korea, Macao, Malaysia, Maxico, Portugal, Russia, Taiwan, Thailand, Tunisia, United Kingdom and United States. Papers selected for this issue are of top quality among the conference submissions and all contain substantial new results. The second source of this issue is from invited contributions from world-wide experts in the relevant areas. All the papers that appear in this volume are strictly refereed. We sincerely thank the referees for their extremely valuable assistance in creating this volume.

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T. Diogo , P. Lima and  Tao Tang
2006, 5(2): i-ii doi: 10.3934/cpaa.2006.5.2i +[Abstract](19) +[PDF](40.0KB)
Over the past decades there have been very rapid developments of analysis and numerical approximations for singular problems. To review the recent developments and to explore exciting new directions in this area, the International Workshop on Analysis and Numerical Approximation of Singular Problems was held at Instituto Superior Técnico, Lisbon, Portugal, from 10-12 November 2004. The aim of this workshop was to bring together active scientists working on singular problems in physics and engineering, and to provide a forum so that they would meet and exchange ideas in a stimulating environment. The conference was attended by more than forty participants from over ten countries, including 14 invited talks, 13 contributed talks and a poster session. The detailed information of the workshop can be found in

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