SYMMETRY PROPERTIES OF POSITIVE SOLUTIONS OF PARABOLIC EQUATIONS: A SURVEY

P. Poláčik
2009 Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions  
This survey is concerned with positive solutions of nonlinear parabolic equations. Assuming that the underlying domain and the equation have certain reflectional symmetries, the presented results show how positive solutions reflect the symmetries. Depending on the class of solutions considered, the symmetries for all times or asymptotic symmetries are established. Several classes of problems, including fully nonlinear equations on bounded domains, quasilinear equations on R N , asymptotically
more » ... , asymptotically symmetric equations, and cooperative parabolic systems, are examined from this point of view. Applications of the symmetry results in the study of asymptotic temporal behavior of solutions are also shown. 6 On the proofs: a comparison of bounded and unbounded domains 25 7 Applications and some open problems 29
doi:10.1142/9789812834744_0009 fatcat:yzk4a5m6oza5jhwgrwdjp43xea