Time-periodic solutions to semilinear parabolic equations

1986 ◽  
Vol 104 (3-4) ◽  
pp. 329-342 ◽  
Author(s):  
Peter Grindrod ◽  
Bryan P. Rynne
Keyword(s):  
Periodic Solutions ◽  
Parabolic Equations ◽  
Evolution Equations ◽  
Periodic Forcing ◽  
Linear Diffusion ◽  
Linear Evolution ◽  
Time Periodic Solutions ◽  
Semilinear Parabolic ◽  
Time Periodic ◽  
Linear Evolution Equations

SynopsisWe consider a class of non-linear evolution equations subject to a periodic forcing term. Using bifurcation theory we obtain results on the existence and number of periodic solutions. The theory applies to semi-linear diffusion equations defined on bounded or unbounded domains.

Periodic Solutions of Non-Linear Evolution Equations in Banach Spaces

1971 ◽  
Vol 23 (1) ◽  
pp. 189-196 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Periodic Solutions ◽  
Banach Spaces ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Inner Product ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linear Evolution ◽  
Non Linear ◽  
Linear Evolution Equations

In this paper the theory of Browder [2] and of Lions [3] on periodic solutions of non-linear evolution equations in Banach spaces is put in a more general framework so as to include the Navier-Stokes equations and their variants.An abstract existence theorem is proved in § 1. Applications are given in § 2. The existence of periodic solutions of the Navier-Stokes equations without any restriction on the dimension of the space domain is established. Application of the abstract theorem to the following problem is given:1. Let H be a Hilbert space and (., .)H the inner product in H. Let V and W be two reflexive separable Banach spaces with W ⊂ V ⊂ H. W is dense in V and V is dense in H.

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