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- ItemWell-posedness and regularity for a fractional tumor growth model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenIn this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalization of a phase field system of Cahn--Hilliard type modelling tumor growth that has been proposed in Hawkins-Daarud et al. (Int. J. Numer. Math. Biomed. Eng. 28 (2012), 3--24) and investigated in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. The generalization investigated in this paper is motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type. Under rather general assumptions, well-posedness and regularity results are shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth constributions of logarithmic or of double obstacle type to the energy density can be admitted.
- ItemOptimal control problems with sparsity for phase field tumor growth models involving variational inequalities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Colli, Pierluigi; Signori, Andrea; Sprekels, JürgenThis paper treats a distributed optimal control problem for a tumor growth model of Cahn--Hilliard type including chemotaxis. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the $L^1$--norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the so-called ``deep quench approximation'' in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, first-order necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls.
- ItemWell-posedness and long-time behavior for a nonstandard viscous Cahn-Hilliard system(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Colli, Pierluigi; Gilardi, Geanni; Podio-Guidugli, Paolo; Sprekels, JürgenWe study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of certain useful boundedness properties, we prove existence and uniqueness of a global-in-time smooth solution to the associated initial/boundary-value problem; moreover, we give a description of the relative $omega$-limit set.
- ItemA nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Colli, Pierluigi; Krejˇcí, Pavel; Rocca, Elisabetta; Sprekels, JürgenIn this paper, we prove the existence and global boundedness from above for a solution to an integrodifferential model for nonisothermal multi-phase transitions under nonhomogeneous third type boundary conditions. The system couples a quasilinear internal energy balance ruling the evolution of the absolute temperature with a vectorial integro-differential inclusion governing the (vectorial) phase-parameter dynamics. The specific heat and the heat conductivity $k$ are allowed to depend both on the order parameter $chi$ and on the absolute temperature $theta$ of the system, and the convex component of the free energy may or may not be singular. Uniqueness and continuous data dependence are also proved under additional assumptions.
- ItemOn a Cahn-Hilliard system with convection and dynamic boundary conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenThis paper deals with an initial and boundary value problem for a system coupling equation and boundary condition both of CahnHilliard type; an additional convective term with a forced velocity field, which could act as a control on the system, is also present in the equation. Either regular or singular potentials are admitted in the bulk and on the boundary. Both the viscous and pure CahnHilliard cases are investigated, and a number of results is proven about existence of solutions, uniqueness, regularity, continuous dependence, uniform boundedness of solutions, strict separation property. A complete approximation of the problem, based on the regularization of maximal monotone graphs and the use of a FaedoGalerkin scheme, is introduced and rigorously discussed.
- ItemDistributed optimal control of a nonstandard nonlocal phase field system(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenWe investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.
- ItemGlobal existence for a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenIn this paper, we study a model for phase segregation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter. This boundary condition models an additional nonconserving phase transition occurring on the surface of the domain. Different well-posedness results are shown, depending on the smoothness properties of the involved bulk and surface free energies.
- ItemDistributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenThis paper is concerned with a distributed optimal control problem for a nonlocal phase field model of CahnHilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors the nonlocal model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called deep quench approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.
- ItemOptimal boundary control of a nonstandard viscous Cahn-Hilliard system with dynamic boundary condition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Colli, Pierluigi; Gilardi, Gianni; Sprekels, JürgenIn this paper, we study an optimal boundary control problem for a model for phase separation taking place in a spatial domain that was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105118. The model consists of a strongly coupled system of nonlinear parabolic differential equations, in which products between the unknown functions and their time derivatives occur that are difficult to handle analytically. In contrast to the existing control literature about this PDE system, we consider here a dynamic boundary condition involving the Laplace-Beltrami operator for the order parameter of the system, which models an additional nonconserving phase transition occurring on the surface of the domain. We show the Fréchet differentiability of the associated control-to-state operator in appropriate Banach spaces and derive results on the existence of optimal controls and on first-order necessary optimality conditions in terms of a variational inequality and the adjoint state system.
- ItemGlobal existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Colli, Pierluigi; Gilardi, Gianni; Podio-Guidugli, Paolo; Sprekels, JürgenExistence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic [19]; in the balance equations of microforces and microenergy, the two unknowns are the order parameter $rho$ and the chemical potential $mu$. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for $rho$ is coupled with a genuinely nonlinear diffusion equation for $mu$. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of costant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.