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Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. (English) Zbl 1126.47060

The author considers an evolution family $𝒰={\left(U\left(t,s\right)\right)}_{t\ge s\ge 0}$ on ${ℝ}_{+}$ and the integral equation

$u\left(t\right)=U\left(t,s\right)u\left(s\right)+{\int }_{s}^{t}U\left(t,\xi \right)f\left(\xi \right)\phantom{\rule{0.166667em}{0ex}}d\xi ·$

He characterizes the exponential dichotomy of the evolution family through the solvability of this integral equation in admissible function spaces which contain function spaces of ${L}_{p}$ type, the Lorentz spaces ${L}_{p,q}$, and many other function spaces occurring in interpolation theory.

The paper applies the technique of choosing test functions related to integral equations. This technique allows to use the Banach isomorphism theorem applied to an abstract differential operator for obtaining explicit dichotomy estimates.

Using the characterization of exponential dichotomy, the author proves the robustness of exponential dichotomy of evolution families on ${ℝ}_{+}$ under small perturbations.

##### MSC:
 47N20 Applications of operator theory to differential and integral equations 47D06 One-parameter semigroups and linear evolution equations 34D09 Dichotomy, trichotomy 34G10 Linear ODE in abstract spaces 35B35 Stability of solutions of PDE 35B40 Asymptotic behavior of solutions of PDE 35K20 Second order parabolic equations, initial boundary value problems 35K55 Nonlinear parabolic equations