The notion of (homogeneous) exponential Besov spaces is introduced
and the infinite smoothness of such spaces is shown. Moreover, we
consider some applications of exponential Besov spaces to a class
of evolution equations involving dissipative terms, such as
Cauchy-Riemann equations, semi-linear parabolic equations and
semi-linear viscoelastic equations. The existence, uniqueness and
regularity of solutions for the Cauchy problem of these equations
will be established with rough initial data.