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 \(\max_{0\leq r\leq 1}\big\{|P(rR(t),t)-P_*(rR_*)|+\)\(|Q(rR(t),t)-Q_*(rR_*)|\big\}< K\varepsilon e^{-\mu t}\), \(\sup_{0< r < 1}r(1-r)\big\{|d/dr\big(P(rR(t),t)-P_*(rR_*)\big)|\)+\(|d/dr\big( Q(rR(t),t) - Q_*(rR_*)\big)|\big\}< K\varepsilon e^{-\mu t} \), \(|R(t)-R_*|<K\varepsilon e^{-\mu t}\).


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
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