Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor

Abstract

We consider a free boundary problem modeling the growth of angiogenesis tumor with inhibitor, in which the tumor aggressiveness is modeled by a parameter µ. The existences of radially symmetric stationary solution and symmetry-breaking stationary solution are established. In addition, it is proved that there exist a positive integer m * * and a sequence of µm, such that for each µm(m > m * *), the symmetry-breaking stationary solution is a bifurcation branch of the radially symmetric stationary solution.

Stationary solutions of a free boundary problem modeling the growth of vascular tumors with a necrotic core

“…The asymptotic stability of radially symmetric stationary solutions was also analyzed in [55,56]. For cases in the presence of inhibitor, Wang et al [44] obtained the existence of symmetric-breaking stationary solutions for µ n (even n > n * * ); also see a very recent paper [41] for the discussion on the existence of radially symmetric stationary solutions and the asymptotic behavior of radially symmetric transient solutions. Assuming β(t) = ∞, i.e., (1.8) holds, tumor models have been intensively studied; we refer the reader to [12-14, 16, 19, 23, 26, 27, 29-31, 43, 49, 50, 52, 53] and the reference therein.…”

Analysis of a free boundary tumor model with time-dependent in the presence of inhibitors

“…Tumorigenesis is a complex multi-step process triggered by gene mutations, immune deficiency, etc., transforming normal cells into malignant cells. Over the past forty years, numerous researchers have extensively studied the partial differential mathematical model of tumor growth and have made many interesting results; see [3][4][5][6][7][8][9][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and references therein.…”

“…The literatures [13,[15][16][17][18][19]24,25] all consider α(t) as a positive constant. However, in reality, the vascularization process is dynamic, in other words, it is not a state of equilibrium, but is constantly changing and evolving.…”