On a Cahn-Hilliard type phase field system related to tumor growth

Colli, Pierluigi; Gilardi, Gianni; Hilhorst, Danielle

doi:10.3934/dcds.2015.35.2423

Cited by 93 publications

(135 citation statements)

References 12 publications

Supporting

Mentioning

133

Contrasting

Order By: Relevance

“…A formal asymptotic analysis is carried out in [22] to identify the corresponding sharp interface problem. The growth assumptions on the potential F and the proliferation function are more general than the ones in [8]. Our first main result is a well-posedness theorem for weak solutions which are just energy bounded.…”

Section: Discussionmentioning

confidence: 97%

“…The same contribution contains a numerical scheme as well as some numerical simulations. The first theoretical results concerning existence and uniqueness of solutions are proven in [8]. The first theoretical results concerning existence and uniqueness of solutions are proven in [8].…”

Section: Discussionmentioning

confidence: 99%

“…Numerical simulations of diffuse-interface model for tumour growth have been carried out in several papers (see, for instance, [11,Chap. Moreover, a very recent contribution (see [8]) is devoted to analyze an approximation of a model recently proposed in [21] (see also [22,35]). Nonetheless, a rigorous mathematical analysis of the resulting systems of differential equations is still in its infancy.…”

Section: Introductionmentioning

confidence: 99%

“…However, in this paper we suppose p to be, at least, Lipschitz continuous, but we allow it to satisfy a suitable growth condition (cf. (1.5) In [8], the authors consider a relaxed model in which the chemical potential μ contains a viscous term αϕ t , α > 0 and equation (1.1) has an additional term αμ t which requires a further initial condition. Also, it is worth observing that more general potentials F, possibly depending on ψ as well, might be taken into account since they are relevant from the modelling viewpoint (cf.…”

Section: Introductionmentioning

confidence: 99%

“…The assumptions on F and p are more general than the ones in [8] for the case α = 0. This is our first result, namely, existence of a weak solution of finite energy.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On a diffuse interface model of tumour growth

2015

View full text Add to dashboard Cite

Link to this article: http://journals.cambridge.org/abstract_S0956792514000436How to cite this article: SERGIO FRIGERI, MAURIZIO GRASSELLI and ELISABETTA ROCCA (2015). On a diffuse interface model of tumour growth. We consider a diffuse interface model of tumour growth proposed by A. Hawkins-Daruud et al. ((2013) J. Math. Biol. 67 1457-1485). This model consists of the Cahn-Hilliard equation for the tumour cell fraction ϕ nonlinearly coupled with a reaction-diffusion equation for ψ, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function p(ϕ) multiplied by the differences of the chemical potentials for ϕ and ψ. The system is equipped with no-flux boundary conditions which give the conservation of the total mass, that is, the spatial average of ϕ + ψ. Here, we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential F and p satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that p satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.

show abstract

Section: Discussionmentioning

confidence: 97%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The assumptions on F and p are more general than the ones in [8] for the case α = 0. This is our first result, namely, existence of a weak solution of finite energy.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On a diffuse interface model of tumour growth

2015

View full text Add to dashboard Cite

show abstract

Asymptotic analysis for Cahn–Hilliard type phase‐field systems related to tumor growth in general domains

Kurima

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

This article considers a limit system by passing to the limit in the following Cahn–Hilliard type phase‐field system related to tumor growth as β↘0: α∂tμβ+∂tφβ−normalΔμβ=pfalse(σβ−μβfalse)in1emnormalΩ×false(0,Tfalse),μβ=β∂tφβ+false(−normalΔ+1false)φβ+ξβ+πfalse(φβfalse),1emξβ∈Bfalse(φβfalse)in1emnormalΩ×false(0,Tfalse),∂tσβ−normalΔσβ=−pfalse(σβ−μβfalse)in1emnormalΩ×false(0,Tfalse), in a bounded or an unbounded domain normalΩ⊂RN with smooth‐bounded boundary. Here, N∈double-struckN, T > 0, α > 0, β > 0, p ≥ 0, B is a maximal monotone graph, and π is a Lipschitz continuous function. In the case that Ω is a bounded domain, p and −Δ + 1 are replaced with p(φβ) and −Δ, respectively, and p is a Lipschitz continuous function; Colli, Gilardi, Rocca, and Sprekels (Discrete Contin Dyn Syst Ser S 2017; 10:37–54) have proved existence of solutions to the limit problem with this approach by applying the Aubin–Lions lemma for the compact embedding H1(Ω)↪L2(Ω) and the continuous embedding L2(Ω)↪(H1(Ω))∗. However, the Aubin–Lions lemma cannot be applied directly when Ω is an unbounded domain. The present work establishes existence of weak solutions to the limit problem along with uniqueness and error estimates in terms of the parameter β↘0. To this end, we construct an applicable theory by noting that the embedding H1(Ω)↪L2(Ω) is not compact in the case that Ω is an unbounded domain.

show abstract

Sliding mode control for a diffuse interface tumor growth model coupling a Cahn–Hilliard equation with a reaction–diffusion equation

Colturato

2020

Math Methods in App Sciences

View full text Add to dashboard Cite

We consider the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a Cahn–Hilliard equation with a reaction–diffusion equation perturbed by a maximal monotone nonlinearity. We prove existence and regularity of strong solutions and, under further assumptions, a uniqueness result. Then, we show that the chosen SMC law forces the system to reach within finite time a sliding manifold that we chose in order that the tumor phase remains constant in time.

show abstract

On a Cahn-Hilliard type phase field system related to tumor growth

Cited by 93 publications

References 12 publications

On a diffuse interface model of tumour growth

On a diffuse interface model of tumour growth

Asymptotic analysis for Cahn–Hilliard type phase‐field systems related to tumor growth in general domains

Sliding mode control for a diffuse interface tumor growth model coupling a Cahn–Hilliard equation with a reaction–diffusion equation

Contact Info

Product

Resources

About