Boundary perturbation theorem for the essential spectral radius with application to a spatial tumor invasion with cell age

In this study, we investigate the essential growth rate of semigroups when subjected to boundary perturbations. Employing the variation of constants formula derived from regular system theory, alongside fundamental techniques from functional analysis, we construct an operator approximation of compact operators converging in the operator norm topology to the convolution outlined in the formula. Our approach improves the results of the existing literature using extrapolation methods while ignoring the norm continuity assumption and relaxing the compactness condition. To illustrate our outcomes, we treat a spatial diffusion within an age-structured population model of tumor invasion, where limitation in the supply of oxygen occurs for both proliferating and quiescent tumor cells. Furthermore, the asynchronous exponential growth behavior is analyzed.