| Abstract: |
| We study a nonisothermal phase-field system of Caginalp type that describes tumor growth under thermal therapy. 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 hyperthermia appears as a control variable. First, we prove well-posedness for the initial-boundary value problem and additional regularity. Then, we define a suitable cost functional and show the existence of optimal controls. Finally, we analyze the differentiability of the control-to-state operator and establish necessary first-order conditions of optimality. These results have been obtained in collaboration with Pierluigi Colli (University of Pavia) and Elisabetta Rocca (University of Pavia). |
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