Electron. J. Diff. Equ., Vol. 2012 (2012), No. 212, pp. 1-11.

Almost automorphic mild solutions of hyperbolic evolution equations with stepanov-like almost automorphic forcing term

Indira Mishra, Dhirendra Bahuguna

Abstract:
This article concerns the existence and uniqueness of almost automorphic solutions to the semilinear parabolic boundary differential equations
$$\displaylines{ x'(t)=A_mx(t)+f(t,x(t)), \quad t\in \mathbb{R}, \cr Lx(t)=\phi(t,x(t)), \quad t\in \mathbb{R}, }$$
where $A:=A_m|_{\ker L}$ generates a hyperbolic analytic semigroup on a Banach space $X$, with Stepanov-like almost automorphic nonlinear term, defined on some extrapolated space $X_{\alpha-1}$, for $0<\alpha<1$ and $\phi$ takes values in the boundary space $\partial X$.

Submitted December 1, 2011. Published November 27, 2012.
Math Subject Classifications: 34K06, 34A12, 37L05.
Key Words: Almost automorphic; evolution equation; hyperbolic semigroups; extrapolation spaces; interpolation spaces; neutral differential equation; mild solution.

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Indira Mishra
Department of Mathematics & Statistics
Indian Institute of Technology-Kanpur
Kanpur - 208016, India
email: indiram@iitk.ac.in
Dhirendra Bahuguna
Department of Mathematics & Statistics
Indian Institute of Technology-Kanpur
Kanpur - 208016, India
email: dhiren@iitk.ac.in

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