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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 ' (t)=A(t)x(t)+f(t,x(t)),t,
x ' (t)=A(t)x(t)+f(t,x(t-h)),t,
x ' (t)=A(t)x(t)+f(t,x(t),ϰα(t,x(t))),t·

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.

34G20Nonlinear ODE in abstract spaces
43A60Almost periodic functions on groups, etc.; almost automorphic functions
35K05Heat equation