Pseudo-almost automorphic mild solutions to nonautonomous differential equations and applications (Q1015843)

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.