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Well-posedness for a fourth-order nonisothermal tumor growth model of Caginalp type

Giulia Cavalleri, Pierluigi Colli, Elisabetta Rocca

TL;DR

This work establishes a rigorous well-posedness theory for a fourth-order, nonisothermal tumor growth model of Caginalp type that couples a (possibly viscous) Cahn–Hilliard equation for the phase field with a heat balance and a nutrient-diffusion equation. The authors develop a two-step approximation using a Moreau–Yosida regularization of the potential and a Faedo–Galerkin discretization, proving existence of weak solutions, higher regularity (strong solutions) under stronger initial data, and a continuous dependence result that yields uniqueness. A key contribution is the derivation of ε-independent a priori estimates and a careful limit process to pass from the approximate to the original problem, including the identification of the limit of the nonlinear term through maximal monotone operator theory. The results lay a solid analytical foundation for nonisothermal diffuse-interface tumor models and enable future work on optimal control and parameter identification in this context.

Abstract

We introduce a nonisothermal phase-field system of Caginalp type that describes tumor growth under hyperthermia. The model couples a possibly viscous Cahn-Hilliard equation, governing the evolution of the healthy and tumor phases, with an equation for the heat balance, and a reaction-diffusion equation for the nutrient concentration. The resulting nonlinear system incorporates chemotaxis and active transport effects, and is supplemented with no-flux boundary conditions. The analysis is carried out through a two-step approximation procedure, involving a regularization of the potential and a Faedo-Galerkin discretization scheme. Under stronger regularity assumptions, we further establish the existence of strong solutions and their uniqueness via a continuous dependence result.

Well-posedness for a fourth-order nonisothermal tumor growth model of Caginalp type

TL;DR

This work establishes a rigorous well-posedness theory for a fourth-order, nonisothermal tumor growth model of Caginalp type that couples a (possibly viscous) Cahn–Hilliard equation for the phase field with a heat balance and a nutrient-diffusion equation. The authors develop a two-step approximation using a Moreau–Yosida regularization of the potential and a Faedo–Galerkin discretization, proving existence of weak solutions, higher regularity (strong solutions) under stronger initial data, and a continuous dependence result that yields uniqueness. A key contribution is the derivation of ε-independent a priori estimates and a careful limit process to pass from the approximate to the original problem, including the identification of the limit of the nonlinear term through maximal monotone operator theory. The results lay a solid analytical foundation for nonisothermal diffuse-interface tumor models and enable future work on optimal control and parameter identification in this context.

Abstract

We introduce a nonisothermal phase-field system of Caginalp type that describes tumor growth under hyperthermia. The model couples a possibly viscous Cahn-Hilliard equation, governing the evolution of the healthy and tumor phases, with an equation for the heat balance, and a reaction-diffusion equation for the nutrient concentration. The resulting nonlinear system incorporates chemotaxis and active transport effects, and is supplemented with no-flux boundary conditions. The analysis is carried out through a two-step approximation procedure, involving a regularization of the potential and a Faedo-Galerkin discretization scheme. Under stronger regularity assumptions, we further establish the existence of strong solutions and their uniqueness via a continuous dependence result.

Paper Structure

This paper contains 10 sections, 3 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Hypotheses and main results
  4. Existence of weak solutions
  5. The approximated problem
  6. Existence of weak solutions for the approximated problem
  7. A priori estimate uniform in $\varepsilon$
  8. Passage to the limit as $\varepsilon \to 0$
  9. Regularity
  10. Continuous dependence

Key Result

Theorem 3.5

Assume that the set of hypotheses hyp:constants--hyp:beta_pi holds and that the initial data satisfy Then, the PDE system eq:problem--eq:initial_datum admits at least a weak solution $(\theta, \varphi, \mu, \sigma)$ with the additional regularity which satisfies the estimate for a constant $C_1>0$ that depends on $M$ and on the other problem's data.

Theorems & Definitions (8)

  • Remark 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 3.4
  • Theorem 3.5: Existence of weak solutions
  • Theorem 3.6: Regularity
  • Remark 3.7
  • Theorem 3.8: Continuous dependence