Exponential and polynomial decay for first order linear Volterra evolution equations

Mainini, Edoardo; Mola, Gianluca

doi:10.1090/S0033-569X-09-01145-X

Authors: Edoardo Mainini and Gianluca Mola
Journal: Quart. Appl. Math. 67 (2009), 93-111
MSC (2000): Primary 35B41, 37L30, 45J05, 80A22
DOI: https://doi.org/10.1090/S0033-569X-09-01145-X
Published electronically: January 7, 2009
MathSciNet review: 2495073
Full-text PDF Free Access

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Abstract: We consider, in an abstract setting, an instance of the Coleman-Gurtin model for heat conduction with memory, that is, the Volterra integro-differential equation \[ \partial _t u(t) - \beta \Delta u(t) - \int _{0}^{t}k(s)\Delta u(t-s)ds = 0. \] We establish new results for the exponential and polynomial decay of solutions, by means of conditions on the convolution kernel which are weaker than the classical differential inequalities.

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