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| In this talk we consider initial-Neumann boundary value problem of nonlinear evolution equations with strong dissipation and proliferation arising from mathematical biology and medicine formulated as
$(NE)\left\{\begin{array}{l}
\ u_{tt}=D\nabla^{2}u_{t}+\nabla\cdot(\chi(u_{t},e^{-u})e^{-u}\nabla u)+\mu_1 u_t(1-u_t) \ \ \\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ in\ \ (x,t)\in\mathbf{\Omega}\times(0,\infty) \ (1.1)\\
\ \displaystyle\frac{\partial}{\partial\nu}u|_{\partial\mathbf{\Omega}}=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ on\ \ \partial\mathbf{\Omega}\times(0,\infty)\ \ \ \ \ \ \ \ (1.2)\\
\ \ \ \ \ \ \\
\ u(x,0)=u_{0}(x),u_{t}(x,0)=u_{1}(x)\ \ \ \ \ \ \ \ \ \ \ in\ \ \mathbf{\Omega}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1.3)
\end{array}\right.$
where constants $D, \mu_1$ are positive, $\mathbf{\Omega}$ is a bounded domain in $R^{n}$\ with a smooth boundary $\partial\mathbf{\Omega}$\ and $\nu$\ is the outer unit normal vector. We show the existence and asymptotic behavior of the solution. Under some conditions of the coefficient $\chi(u_{t},e^{-u})$ of (1.1), we can derive the energy esitmate of (NE), which
enables us to show the global existence in time of the soluton and asymptotic behaviour. We will deal with an extended case of our result in this talk and apply it to Chaplain type of mathematical models of biology and medicine (cf. [1]).
Bibliography
[1] Chaplain, M.A.J., and Lolas, G., Mathematical modeling of cancer invasion of tissue:
Dynamic heterogeneity, Networks and Heterogeneous Media, vol 1, Issue 3, 399-439(2006). |
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