Dynamic behavior of a heat equation with memory

This paper addresses the spectrum-determined growth condition for a heat equation with exponential polynomial kernel memory. By introducing some new variables, the time-variant system is transformed into a time-invariant one. The detailed spectral analysis is presented. It is shown that the system demonstrates the property of hyperbolic equation that all eigenvalues approach a line that is parallel to the imaginary axis. The residual spectral set is shown to be empty and the set of continuous spectrum is exactly characterized. The main result is the spectrum-determined growth condition that is one of the most difficult problems for infinite-dimensional systems. Consequently, a strong exponential stability result is concluded.