Well-posedness Of A Phase-field Model For Tumor Growth In The Avascular Phase

Malignant tumor(cancer)has become one of the major public health problems that seriously threatens the health of Chinese people.In order to decrease the mortality of cancer,tumor study becomes more and more important.In this paper,the phase-field method is used to study tumor growth.Phase-field method is an important tool for studying the evolution of microstructure in the process of phase transition,originated from material science.Based on the theory of phase-field method,phase-field model is a model to solve interface problems,which can simulate and predict the evolution of mesoscopic scale morphology and microstructure.There are tumor cells and normal ones in the tissue,and they separate from each other due to different physiological mechanisms.Therefore,we can apply the Cahn-Hilliard equation to describe tumor growth at mesoscopic scale,leading a deeper understanding of the dynamic behavior of tumors.In this paper,we study the weak solutions to an initial-boundary value problem for phase-field model of tumor growth,which is the Allen-Cahn equation for the nutrient concentration n coupled with the Cahn-Hilliard equation for the order parameter ?,and describes the diffusion process of nutrient and the evolution process of tumor.We prove the existence of global solutions by the Aubin-Lions Lemma and the method of continuation of local solutions.Specially,we prove the uniqueness of global solution by making use of fundamental inequalities in one dimension.Finally,numerical experiments are carried out on tumor growth during the phase transition by using the model.We simulate the evolution of tumor under the two conditions of sufficient and deficient nutrients.