A parabolic–hyperbolic system modeling the tumor growth with angiogenesis

2022 ◽  
Vol 64 ◽  
pp. 103456
Author(s):  
Haishuang Shen ◽  
Xuemei Wei
Author(s):  
Min Tang ◽  
Nicolas Vauchelet ◽  
Ibrahim Cheddadi ◽  
Irene Vignon-Clementel ◽  
Dirk Drasdo ◽  
...  

2020 ◽  
pp. 1-33
Author(s):  
K. Sakthivel ◽  
A. Arivazhagan ◽  
N. Barani Balan

2020 ◽  
Vol 75 (7) ◽  
pp. 637-648
Author(s):  
Martin O. Paulsen ◽  
Henrik Kalisch

AbstractConsideration is given to the shallow-water equations, a hyperbolic system modeling the propagation of long waves at the surface of an incompressible inviscible fluid of constant depth. It is well known that the solution of the Riemann problem associated to this system may feature dry states for some configurations of the Riemann data. This article will discuss various scenarios in which the Riemann problem for the shallow water system arises in a physically reasonable sense. In particular, it will be shown that if certain physical assumptions on the disposition of the Riemann data are made, then dry states can be avoided in the solution of the Riemann problem.


Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 792 ◽  
Author(s):  
Pierluigi Colli ◽  
Gianni Gilardi ◽  
Jürgen Sprekels

In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn–Hilliard type phase field system modeling tumor growth that has been proposed by Hawkins–Daarud, van der Zee and Oden. The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in a recent work by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.


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