Mathematical Sciences: Elliptic and Parabolic Systems of    Nonlinear Equations
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
This award provides support for mathematical research on                   two issues of current interest, the question of stability of                    travelling waves in diffusive systems and symmetry breaking                     bifurcations in unbounded cylindrical domains.                                       In the first instance, work will be done analyzing the                     stability of travelling wave solutions of reaction-diffusion                    systems of partial differential equations related to specific                   model equations in ecology, combustion theory and                               neurophysiology.  The technique will involve efforts to                         understand the distribution of eigenvalues of the linearized                    parabolic systems about a travelling wave solution.  General                    topological methods may then be applied to gain insight into                    the qualitative nature of the true solution.  At the present                    time very little is known about systems of such equations.                      Initial work will involve studies of important specific                         examples and bringing new tools to bear.                                             The second line of investigation is related to the first                   in that solutions of the corresponding steady-state                             equations, which are elliptic, will be studied.  Questions                      here focus on the bifurcation of branches of solutions.                         Specifically, work will be done on semilinear equations                         defined in infinite cylinders (depending on a parameter)                        which admit positive radial solutions.  Non-radial solutions                    bifurcate out as the parameter varies.  Several questions                       arise in connection with the symmetry breaking.  One is to                      understand the inherited symmetries of the associated vector                    field obtained by reduction of the problem to systems of                        ordinary differential equations.  A second is to consider the                   effects of changing the geometry of the cylinder from                           circular to elliptic and finally, one wants to find more                        complex bifurcations than the expected periodic ones.                                This research has potential for use in many areas of                       applied science where reaction-diffusion and Navier-Stokes                      equations are used in modeling.  It may also lead to new                        insights into chaotic dynamics in solutions of scalar                           diffusion equations.
{'FirstName': 'Robert', 'LastName': 'Gardner', 'PI_MID_INIT': 'A', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Robert A Gardner', 'EmailAddress': 'gardner@math.umass.edu', 'NSF_ID': '000180518', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of Massachusetts Amherst', 'CityName': 'AMHERST', 'ZipCode': '010039252', 'PhoneNumber': '4135450698', 'StreetAddress': '101 COMMONWEALTH AVE', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Massachusetts', 'StateCode': 'MA', 'CONGRESSDISTRICT': '02', 'CONGRESS_DISTRICT_ORG': 'MA02', 'ORG_UEI_NUM': 'VGJHK59NMPK9', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF MASSACHUSETTS', 'ORG_PRNT_UEI_NUM': 'VGJHK59NMPK9'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8802468.xml'}