Asymptotic behaviors of solutions to one-dimensional tumor invasion model with quasi-variational structure (Q2788960)

This paper is concerned with the following one-dimensional model of tumour growth in the unknowns \((n,f,m)\): NEWLINE\[NEWLINE n_t=[d_1 n_x-\lambda (f)nf_x]_x+\mu_p n(1-n-f)-\mu_d n,NEWLINE\]NEWLINE NEWLINE\[NEWLINEf_t=-amf,\qquad m_t=d_2 m_{xx}+bn-cm,NEWLINE\]NEWLINE in \((-L,L)\times (0,T)\), subject to the constraint, boundary and initial conditions NEWLINE\[NEWLINEn\geq 0,\quad f\geq 0,\quad n+f\leq \alpha\quad\text{in }(-L,L)\times (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINE[d_1 n_x-\lambda (f)nf_x](\pm L,t)=0,\quad m_x(\pm L,t)=0\quad\text{for }t\in (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINE(n(0),f(0),m(0))=(n_0,f_0,m_0)\quad\text{in }(-L,L).NEWLINE\]NEWLINE In this model of Chaplain-Anderson type [\textit{M. A. J. Chaplain} and \textit{A. R. A. Anderson}, in: Cancer Modelling and Simulation. Boca Raton: Chapman\&Hall/CRC, Mathematical Biology and Medicine Series, 269--297 (2003; Zbl 1337.92103)], \(n(x,t)\) and \(f(x,t)\) denote the ratio of density of tumour cells and the extracellular matrix, respectively, whereas \(m(x,t)\) corresponds to the concentration of a matrix-degrading enzyme.NEWLINENEWLINEThe main results of this article are twofold. First, the authors show -- under suitable regularity and integrability conditions of the initial conditions and the parameter functions \(\lambda\), \(\mu_d\) and \(\mu_p\) -- the existence of nonnegative global-in-time weak solutions to the system above. Second, under more refined conditions, it is proved that there exists at least one solution which converges as \(t\to\infty\) to a constant steady state corresponding to complete saturation of tumour cells.NEWLINENEWLINEThe methods of proof are based on a quasi-variational structure of this system, see Kano, Murase and Kenmochi [\textit{R. Kano} et al., Banach Cent. Publ. 86, 175--194 (2009; Zbl 1178.35224)].