Asymptotic analysis of a tumor growth model with fractional operators. (English) Zbl 1476.35284
The authors perform the asymptotic analysis of an evolutionary system generalizing a system of partial differential equations which model tumor growth. The system under consideration is a coupling of three semilinear evolution equations involving fractional powers of three (possibly different) selfadjoint monotone unbounded linear operators having compact resolvents. The coupled equations also contain two (small) positive relaxation parameters. (When the three involved fractional powers coincide with the minus Laplacian operator with zero Neumann boundary condition, the evolutionary system becomes a relaxed version of a phase field system of Cahn-Hilliard type modelling tumor growth.) The authors essentially show that, as the two relaxation parameters approach zero, either separately or simultaneously, the evolutionary system approaches a meaningful limit system for which the existence of solutions can be proved. To be able to do so, they use the (previously established) well-posedness and regularity results for the considered system. The asymptotic analysis requires some specific assumptions on the admissible nonlinearities.
Reviewer: Catalin Popa (Iaşi)
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 92C37 | Cell biology |
| 92C17 | Cell movement (chemotaxis, etc.) |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
Keywords:
fractional operators; Cahn-Hilliard systems; well-posedness; regularity of solutions; tumor growth models; asymptotic analysisReferences:
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