Symmetry-breaking bifurcation for a free-boundary tumor model with time delay

Large number of papers have been devoted to the study of tumor models. Wellpostedness as well as properties of solutions are systematically studied [1-3,5-21,25-40]. The properties include asymptotic behavior, stability, bifurcation analysis, etc. In this paper we study a tumor model with a time-delay, and establish a bifurcation result for all even mode n >= 2. The time delay represents the time taken for cells to undergo replication (approximately 24 hours). In contrast to some results in the literature where bifurcation is established for sufficiently large bifurcation parameter, our result includes the smallest bifurcation point, which is crucial as this is the point of stability changes under certain conditions as we have shown in [40]. The inclusion of a time delay, although biologically very reasonable, introduces two significant mathematical challenges: (a) the explicit solution utilized to verify the bifurcation theorem is no longer available; (b) the system becomes non-local because of the time-delay, which in term produces technical difficulties for the PDE estimates. To our knowledge, this is the first paper on the study of bifurcation for tumor models with a time delay. (C) 2020 Elsevier Inc. All rights reserved.