Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 212, pp. 1-11.
Title: Almost automorphic mild solutions of hyperbolic evolution equations
with stepanov-like almost automorphic forcing term
Authors: Indira Mishra (Indian Institute of Tech., Kanpur, India)
Dhirendra Bahuguna (Indian Institute of Tech., Kanpur, India)
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.