Vol. 85, No. 2, 1979

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Comparison theorems for parabolic functional inequalities

Raymond Moos Redheffer and Wolfgang V. Walter

Vol. 85 (1979), No. 2, 447–470

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.

Mathematical Subject Classification 2000
Primary: 35R45
Secondary: 35K60
Received: 17 July 1978
Published: 1 December 1979
Raymond Moos Redheffer
Wolfgang V. Walter