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A hyperbolic free boundary problem modeling tumor growth: asymptotic behavior. (English) Zbl 1082.35166

The authors consider the following system \[ \frac{1}{r^2} \frac{\partial}{\partial r} (r^2\frac{\partial \bar{c}}{\partial r})=\lambda \bar{c}\quad (0<r<R(t),\;t\geq 0), \]
\[ \frac{\partial\bar{c}}{\partial r}(0,t)=0,\quad \bar{c}(R(t),t)=1\quad (t\geq 0), \]
\[ \frac{\partial\bar{p}}{\partial t}+u\frac{\partial\bar{p}}{\partial r}=K_P(\bar{c})+[K_M(\bar{c})-K_N(\bar{c})]\bar{p}-K_M(\bar{c})\bar{p}^2\quad (0\leq r\leq R(t),\;t>0), \]
\[ \frac{1}{r^2}\frac{\partial}{\partial r}(r^2\bar{u})=-K_D(\bar{c})+K_M(\bar{c})\bar{p}\quad (0<r\leq R(t),\;t\geq0), \]
\[ \bar{u}(0,t)=0\quad (t\geq 0), \]
\[ \frac{dR(t)}{dt}=R(t)\bar{u}(R(t),t)\;(t>0), \]
\[ \bar{p}(r,0)=\bar{p}_0(r)\quad (0\leq r\leq R_0), \]
\[ R(0)=R_0, \] which they associate to the evolution of a spherical tumour in which dead cells are instantly removed. Existence and uniqueness have been proved elsewhere. Also it is known that a unique stationary solution exists. The aim of the paper is to study the asymptotic behaviour showing that
(i) for \(t\to +\infty\) non-equilibrium solutions tend to the stationary solution,
(ii) the stationary solution is linearly asymptotically stable.
The first result is proved within the class of solutions satisfying the condition \[ \lim_{T\to + \infty}\int^{T+1}_T| \dot{R}(t)| \,dt=0. \]

MSC:

35R35 Free boundary problems for PDEs
35B40 Asymptotic behavior of solutions to PDEs
92C50 Medical applications (general)
35L70 Second-order nonlinear hyperbolic equations
35Q80 Applications of PDE in areas other than physics (MSC2000)
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