We consider a system of partial differential equations modeling tumors. The system under consideration describes the spatial dynamics of the tumor cells, extracellular matrix, and matrix degrading enzymes. We first carry out a complete group classification of the Lie point symmetries of this model. Next, we use symmetry techniques to construct invariant solutions for it. In addition, we consider a second system of partial differential equations, coupling to the original one the concentration of oxygen, and we find several analytical solutions to this system. Most of the solutions are biologically relevant and consistent with the evolution of such tumors.

中文翻译：

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径向对称无血管癌模型的分组分类及解析解

我们考虑模拟肿瘤的偏微分方程系统。所考虑的系统描述了肿瘤细胞、细胞外基质和基质降解酶的空间动态。我们首先对该模型的李点对称性进行完整的群分类。接下来，我们使用对称技术为其构建不变解。此外，我们考虑了第二个偏微分方程组，将氧浓度耦合到原始方程组，我们找到了这个系统的几个解析解。大多数解决方案具有生物学相关性，并且与此类肿瘤的进化一致。