Long-time dynamics in semilinear parabolic problems with autocatalysis. (English) Zbl 1181.35136

Ackermann, Nils

Du, Yihong (ed.) et al., Recent progress on reaction-diffusion systems and viscosity solutions. Based on the international conference on reaction-diffusion systems and viscosity solutions, Taichung, Taiwan, January 3–6, 2007. Hackensack, NJ: World Scientific (ISBN 978-981-283-473-7/hbk). 1-30 (2009).

Summary: The discovery of a priori bounds for semilinear parabolic equations with superlinear nonlinearity enables the use of the parabolic semiflow in proving existence results for equilibria, as an alternative to minimax methods from the calculus of variations. We present an overview of results that use this approach, discuss its shortcomings and advantages, and outline possible future research directions.
For the entire collection see [Zbl 1165.35002].

MSC:

35K58	Semilinear parabolic equations
35K20	Initial-boundary value problems for second-order parabolic equations
35B40	Asymptotic behavior of solutions to PDEs
35B33	Critical exponents in context of PDEs

Keywords:

superlinear nonlinearity; nodal properties; connecting orbit