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

(1)利用扩散项证明了某些数学模型经典解的存在性和解的渐近性，并证明了当时间趋于无穷时，我们的解强收敛于相关logistic方程的解。(2)为了涵盖比以往更广泛的肿瘤侵袭现象，我们表征和研究了非线性进化方程<tt D'<tf (x, t; u)，这些方程从我们的数学模型中得到，包括(1)在更一般的框架工作中。通过改进chapplain等人提出的肿瘤生长和侵袭的局部数学模型序列的数学方法，我们可以证明该问题经典解的存在性和解的渐近性，这类问题属于更一般的一类。(3)我们研究了一些有效的数学方法来确定行波的波前与胶质瘤肿瘤细胞侵袭范围之间的关系。(4)局部模型与非局部模型最关键的区别在于后者具有与广义微分算子相似的性质。

(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.