×

Asymptotic stability of the stationary solution for a hyperbolic free boundary problem modeling tumor growth. (English) Zbl 1181.35345

Let \(\Omega(t):=\{x\in \mathbb{R}^3:\,r=|x|<R(t)\}\), \(t\geq 0\). It is required to find free boundary \(r=R(t)\) and radially symmetric functions \(C(r,t)\), \(P(r,t)\), \(Q(r,t)\) and \(u(r,t)\) as a solution of the problem: \(\nabla^2C=F(C)\), \(\partial P/\partial t\) + \(\nabla\cdot(\vec{u} P)\) \(=[K_B(C)-K_Q(C)]P\) \(+K_P(C)Q\), \(\partial Q/\partial t\) + \(\nabla\cdot(\vec{u} Q)\) \(=K_Q(C)P-[K_D(C)+K_P(C)]Q\), \(P+Q=N=\) const, \(x\in \Omega(t)\), \(t\geq 0\); \(R|_{t=0}=R_0\), \(P|_{t=0}=P_0\), \(Q|_{t=0}=P_0\); \(C|_{\partial\Omega(t)}=C_0=\) const \(>0\), \(t\geq 0\), \(dR/dt=\vec{u}\cdot\vec \nu|_{\partial\Omega(t)}\), \(\vec{u}=u(r,t)r^{-1}x\), \(t\geq 0\), where \(\vec{\nu}\) is a unit normal to \(\partial\Omega(t)\), \(K_B\), \(K_D\), \(K_P\) are given functions. This problem is a mathematical model of a tumor growth, in which \(C\) is a concentration of nutrient, \(P\) and \(Q\) denote the density of proliferating and quiescent cells respectively, \(\vec{u}\) is a velocity of cell movement.
Let \((C_{*},\, P_{*}, \, Q_{*},\, \vec{u}_{*},\,R_{*})\) be the unique stationary solution of the problem. The author proves under some conditions on given functions that there exist positive constants \(\mu, \varepsilon, K\) such that if \(\max_{0\leq r\leq 1}\big\{|P_0(rR_0)-P_*(rR_*)|+\)\(|Q_0(rR_0)-Q_*(rR_*)|\big\}< \varepsilon\), \(\sup_{0< r < 1}r(1-r)\big\{|d/dr\big(P_0(rR_0)-P_*(rR_*)\big)|+\) \(|d/dr\big(Q_0(rR_0)-Q_*(rR_*)\big)|\big\}< \varepsilon\), \(|R_0-R_*|<\varepsilon\), then for all \(t\geq 0\) the solution \((P,\,Q,\,R)\) of the problem satisfies the estimates \(\max_{0\leq r\leq 1}\big\{|P(rR(t),t)-P_*(rR_*)|+\)\(|Q(rR(t),t)-Q_*(rR_*)|\big\}< K\varepsilon e^{-\mu t}\), \(\sup_{0< r < 1}r(1-r)\big\{|d/dr\big(P(rR(t),t)-P_*(rR_*)\big)|\)+\(|d/dr\big( Q(rR(t),t) - Q_*(rR_*)\big)|\big\}< K\varepsilon e^{-\mu t} \), \(|R(t)-R_*|<K\varepsilon e^{-\mu t}\).

MSC:

35R35 Free boundary problems for PDEs
35L50 Initial-boundary value problems for first-order hyperbolic systems
35B40 Asymptotic behavior of solutions to PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92C15 Developmental biology, pattern formation
PDF BibTeX XML Cite
Full Text: DOI arXiv