Study on partial differential equations describing local・non-local phenomena of life

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(1)We could show the existence of classical solutions and the asymptotic behavior of solutions of some mathematical models with the proliferation term and that our solution strongly converges to the solution of the related logistic equation as time goes to infinity. (2) To cover a wider range of phenomena of tumour invasion than ever, we characterized and studied nonlinear evolution equations<tt D'<tf (x, t; u)reduced from our mathematical models including (1) in much moregeneral frame work. We could show the existence of classical solutions and the asymptotic behavior of solutions of this problem, which belong to a more general class, by improving our mathematical way used in a sequence of local mathematical models of tumour growth and invasion proposed by Chaplain and et al.. (3) We studied some effective mathematical approach to confirm some evaluation of the relationship between the wave front of the traveling wave and the invasive range of tumour cells of Glioma. (4) It is concluded that the most crucial difference between local and non-local models could become more focus on the point that the latter possesses similar property to a generalized differential operator.