APMAdvances in Pure Mathematics2160-0368Scientific Research Publishing10.4236/apm.2016.612066APM-72061ArticlesPhysics&Mathematics Nonlinear Evolution Equations and Its Application to a Tumour Invasion Model AkisatoKubo1YutoMiyata2HidetoshiKobayashi3HirokiHoshino1NaokiHayashi4Faculty of Radiological Technology, School of Health Sciences, Fujita Health University, Toyoake, JapanDepartment of Mathematics, School of Health Sciences, Fujita Health University, Toyoake, JapanDepartment of Radiation Oncology, School of Medicine, Fujita Health University, Toyoake, JapanMedical Radiation Sciences, Graduate School of Health Sciences, Fujita Health University, Toyoake, Japan081120160612878893September 12, 2016Accepted: November 14, November 17, 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

We consider nonlinear evolution equations with logistic term satisfying initial Neumann-boundary condition and show global existence in time of solutions to the problem in arbitrary space dimension by using the method of energy. Applying the result to a mathematical model of tumour invasion, we discuss the property of the rigorous solution to the model. Finally we will show the time depending relationship and interaction between tumour cells, the surrounding tissue and matrix degradation enzymes in the model by computer simulations. It is seen that our mathematical result of the existence and asymptotic behaviour of solutions verifies our simulations, which also confirm the mathematical result visibly.

Nonlinear Evolution Equation Mathematical Analysis Tumour Invasion Proliferation Re-Establishment1. Introduction

In this paper we consider the initial Neumann-boundary value problem of nonlinear evolution equations with logistic term, arising from tumour invasion models with proliferation and re-establishment: (NE)

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