| Abstract: |
| In this talk we investigate the global existence in time and asymptotic profile of the solution of a nonlinear evolution equation:
$$\ w_{tt}= D\Delta w_{t} +\nabla\cdot(\alpha(w_{t})e^{-l(t)}\chi[w])
+ \mu(1-w_{t})w_{t}+\beta(1-w_t),\ \mbox{in}\ {\Omega}\times(0,T)\,$$
arising from non-local mathematical models in biology with proliferation and re-establishment, for $l(t)=a+dt, a,d>0,$ where $D, \mu$ are positive constants, $\alpha(\cdot)$ and $ \beta(\cdot)$ are polynomial functions with respect to $\cdot$, $\Omega$ is a bounded domain in $R^n$ with smooth boundary $\partial \Omega$ and $\nu$ is the outer unit normal vector on $\partial \Omega$, $\chi[w]$ is a non-local term.
By making use of the related singular integral operator, we consider the initial and zero-Neumann boundary value problem and derive estimates required for our desired result.
We will apply our result to the original non-local mathematical model (by Gerisch and Chaplain), which reflects the mathematical understanding of the biological processes
described cell migration in vivo. |
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