## Asymptotic Behaviors of Solutions to One-dimensional Tumor Invasion Model with Quasi-variational Structure

J. Math. Sci. Univ. Tokyo
Vol. 22 (2015), No. 2, Page 569–612.

Ito, Akio
Asymptotic Behaviors of Solutions to One-dimensional Tumor Invasion Model
with Quasi-variational Structure

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Abstract:
We consider a one-dimensional tumor invasion model of Chaplain--Anderson type with quasi-variational structure, which is originally proposed in \cite{Chaplain-Anderson-2003}. One object is to show the existence of global-in-time solutions by using the limit procedure for suitable approximate solutions. The other is to consider the asymptotic behaviors of global-in-time solutions as time goes to $\infty$. Actually, we construct at least one global-in-time solution, which enables us to consider the convergence to a certain constant steady-state solution as time goes to $\infty$ whenever the initial data satisfy suitable conditions.

Keywords: Asymptotic behavior, tumor invasion, quasi-variational inequality.

Mathematics Subject Classification (2010): Primary 35B40; Secondary 35Q92, 35A01.
Mathematical Reviews Number: MR3287269