Differential inequalities
containing functionals are assuming an increasing importance in problems of
biomathematics, mathematical medicine, chemistry, heat flow and population growth.
Many of these applications lead to an equation which is of parabolic structure, in the
sense that the equation would be parabolic if the functional in it were replaced by a
known function. One way in which a functional arises in such equations is through a
Volterra type memory term, which takes account of the past history of the
process.
We shall present a number of comparison inequalities for parabolic functional
operators. These can be used to answer questions pertaining to uniqueness,
monotonicity, stability and qualitative behavior with the same simplicity
and directness as has long been available in the purely parabolic case. As
an application, we obtain new results on the behavior of strongly coupled
systems.
