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Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications. (English) Zbl 1175.34076
The authors investigate the existence and uniqueness of pseudo-almost automorphic solutions to the following nonautonomous evolution equations in a Banach space \(X:\)
\[ x^{\prime}(t)=A(t)x(t)+f(t,x(t)),\;t\in\mathbb{R}, \]
\[ x^{\prime}(t)=A(t)x(t)+f(t,x(t-h)),\;t\in\mathbb{R}, \]
\[ x^{\prime}(t)=A(t)x(t)+f(t,x(t),\varkappa\left[ \alpha(t,x(t))\right] ),\;t\in\mathbb{R}. \]
They introduce a new concept of bi-almost automorphic functions, in order to study the existence of pseudo-almost automorphic solutions to the above equations. They also establish some new existence and uniqueness results for pseudo-almost automorphic mild solutions. As applications, one studies two heat equations with Dirichlet boundary conditions.

34G20 Nonlinear differential equations in abstract spaces
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
35K05 Heat equation
Full Text: DOI
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