Compact almost automorphic solutions for semilinear parabolic evolution equations

ABSTRACT

In this paper, using the subvariant functional method due to Favard [Favard J. Sur les équations différentielles linéaires á coefficients presque-périodiques. ActaMathematica. 1928;51(1):31–81.], we prove the existence and uniqueness of compact almost automorphic solutions for a class of semilinear evolution equations in Banach spaces provided the existence of at least one bounded solution on the right half line. More specifically, we improve the assumptions in [Cieutat P, Ezzinbi K. Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities. J Funct Anal. 2011;260(9):2598–2634.], we show that the almost automorphy of the coefficients in a weaker sense (Stepanov almost automorphy of order $1\le p<\mathrm{\infty }$) is enough to obtain solutions that are almost automorphic in a strong sense (Bochner almost automorphy). For that purpose we distinguish two cases, $p=1$ and $p>1$. The main difficulty in this work, is to prove the existence of at least one solution with relatively compact range while the forcing term is not necessarily bounded. Moreover, we propose to study a large class of reaction-diffusion problems with unbounded forcing terms.

Keywords:

• Compact almost automorphic solutions
• $C_0$-semigroup
• nonlinear evolution equations
• reaction-diffusion equations
• subvariant functional method
• Stepanov almost automorphic functions
• unbounded forcing terms

2010 Mathematics subject classifications:

• Primary 46T20-47J35
• Secondary 34C27-35K58

Acknowledgements

The authors are very grateful to the anonymous referee for his attentive reading. His precious comments and valuable suggestions led to improve the final version of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.