## 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
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