Abstract
We consider a diffuse-interface tumor-growth model which has the form of a phase-field system. We characterize the singular limit of this problem. More precisely, we formally prove that as the coefficient of the reaction term tends to infinity, the solution converges to the solution of a novel free boundary problem. We present numerical simulations which illustrate the convergence of the diffuse-interface model to the identified sharp-interface limit.
Highlights
We consider a di↵use-interface tumor-growth model which has the form of a phase-field system
We note that the singular limit corresponds to a moving boundary problem which is similar to other sharp-interface tumor-growth models.[26,8,20,7,10,39,18]
The interesting characteristic of the current singular limit is that the reactive terms of the phase-field model collapse to the interface in the limit, which is di↵erent than in some other models where the reactive terms remain as bulk contributions
Summary
We have to deal with a fourth order equation, at least if we substitute (1.3b) into (1.3a), which is a generalization of the Cahn-Hilliard equation. ↵Vn = (N 1) + Cμ together with the boundary and initial conditions in Q+T [ QT , on Q+T [ QT , on T , on T , on T , on T , In this case, we say that ( T , μ, ) is a classical solution of Problem (P0) on the time interval [0, T ]. We note that the singular limit corresponds to a moving boundary problem which is similar to other sharp-interface tumor-growth models.[26,8,20,7,10,39,18] The interesting characteristic of the current singular limit is that the reactive terms of the phase-field model collapse to the interface in the limit, which is di↵erent than in some other models where the reactive terms remain as bulk contributions.
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