Asymptotic stability of the stationary solution for a parabolic-hyperbolic free boundary problem modeling tumor growth (Q2870593)

There is considered the boundary value problem with unknowns \(c(r,t)\), \(p(r,t)\), \(v(r,t)\) and free boundary \(r=R(t)\): NEWLINE\[NEWLINE\begin{cases}\varepsilon c_t = c_{rr} + \frac{2}{r} c_r - F(c), \;p_t +v p_r = f(c,p), \;v_r = \frac{2}{r} v= g(c,p), \;0 < r < R(t), \;t>0, \\ c_r|_{r=0} = 0, \;c|_{r=R(t)} = 1, \;v|_{r=0} = 0, \;R'(t) = v|_{r=R(t)}, \;t > 0, \end{cases}\tag{1}NEWLINE\]NEWLINE where \(0 < \varepsilon \ll 1\). The three-dimensional free boundary problem modeling tumor growth with spherically symmetric unknown functions in the domain \(\{x\in \mathbb{R}^3: r=|x| <R(t)\}\) is reduced to the problem (1).NEWLINENEWLINEAfter substitution \(r=\bar{r} e^{z(t)}\) and notations \(\bar{c}(\bar{r},t), \bar{p}(\bar{r},t) = c(r,t), p(r,t)|_{r=\bar{r} e^{z(t)}}\), \(\bar{v}(\bar{r},t) = v(r,t)|_{r=\bar{r} e^{z(t)}} e^{-z(t)}\), \(R(t) = e^{z(t)}\), where \(\bar{r}\) is the new independent variable, the author obtains the problem (which we denote by \((P)\)) in the fixed domain \((0,1)\) remaining the previous notations without all bars ``\(\;{\bar{\;}}\;\)''. The main result proved for this problem \((P)\) is as follows.NEWLINENEWLINELet \(c^*(r), p^*(r), v^*(r), z^*\) be the stationary solution and \(c_0(r), p_0(r), z(0)\) -- initial data of the problem \((P)\). There exist positive constants \(\varepsilon_0, \;\mu^*\) such that for all \(\varepsilon E (0, \varepsilon_0), \mu E (0, \mu^*)\) there exists \(\delta > 0, \;C > 0\), such that, if the stationary solution and initial data for all \(0\leq r \leq 1\) satisfy the conditions \(|c_0(r)-c^*(r) \leq \delta,\) \(|c_0'(r)-{c^*}'(r)| \leq \delta,\) \(|p_0(r)-p^*(r) \leq \delta,\) \( r\leq 1r(1-r)|p_0'(r)-{p^*}'(r)| \leq \delta\) and \(|z_0 - z^*| \leq \delta,\) then for the solution of the problem \((P)\) there hold the inequalities NEWLINE\[NEWLINE|c(r,t)-c^*(r)+ |c_r(r,t)-{c^*}'(r)+ |c_t(r,t) \leq C\delta e^{-\mu t},NEWLINE\]NEWLINE NEWLINE\[NEWLINE|p(r,t)-p^*(r)+ |r(1-r)(p_r(r,t)-{p^*}'(r))+ |p_t(r,t) \leq C\delta e^{-\mu t},NEWLINE\]NEWLINE NEWLINE\[NEWLINE|z(t)-z^*| + |\dot{z}(t)| \leq C\delta e^{-\mu t}NEWLINE\]NEWLINE for all \(t \geq 0\) and \(0 \leq r \leq 1\).