LINEAR STABILITY ANALYSIS FOR A FREE BOUNDARY PROBLEM MODELING MULTILAYER TUMOR GROWTH WITH TIME DELAY\ast

In this paper, we consider a mixed system of free boundary problem modeling multilayer tumor growth with time delay \tau . The concept of a time delay goes back to Byrne [Math. Biosci., 144 (1997), pp. 83--117] and the formulation of a multilayer tumor model can be found in Zhou, quasi-steady state approximation of the multilayer tumor growth model with a time delay, He, Xing, and Hu [Math. Methods Appl. Sci., to appear] established a critical value mu* for which the system is linearly stable if mu < mu* and linearly unstable if mu > mu* under nonflat perturbations. We consider the full system without quasi-steady state simplification in this paper. This is a free boundary system of mixed elliptic, parabolic, and hyperbolic equations. A key difference to our earlier work is the consideration of the full parabolic equation for the nutrient in the model, which makes the system more complex and more challenging. By applying series expansion and Laplace transform methods, we found a threshold mu* > 0 such that the unique flat stationary solution is linearly stable if mu < mu* and linearly unstable if mu > mu* under nonflat perturbations. This is accomplished with the inverse Laplace transform and a careful study of the spectrum of the system. Some of our results are presented in a more useful abstract form, which would be applicable to other systems. As far as we know, this is the first paper for the study of stability for multilayer tumor models with a time delay and with a parabolic equation for the nutrient.