Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

✍ Scribed by Stepan Tersian; Julia Chaparova

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 132 KB
• Volume: 260
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2001.7470

✦ Synopsis

In this paper we study the existence of periodic solutions of the fourth-order equations u iv -pu -a x u + b x u 3 = 0 and u iv -pu + a x u -b x u 3 = 0, where p is a positive constant, and a x and b x are continuous positive 2Lperiodic functions. The boundary value problems P 1 and P 2 for these equations are considered respectively with the boundary conditions u 0 = u L = u 0 = u L = 0. Existence of nontrivial solutions for P 1 is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for P 2 is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation u iv + pu + a x ub x u 2 -c x u 3 = 0, where p is a constant, and a x , b x , and c x are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentrationcompactness arguments are used.