快扩散方程的一些性质及其多维精确解
Some properties of the equation of fast diffusion and its multidimensional exact solutions
摘要：
证明了快速扩散方程在二维坐标空间中的不变性，并证明了它的一维（相对于空间变量）模拟的减少。在这些结果的基础上，构造了新的精确多维解，这些解依赖于任意的调和函数。其结果是，新的精确解的著名的刘维尔方程-稳态模拟的快速扩散方程与线性源-已获得。拟线性抛物方程组，以及系统的椭圆方程与泊松相互作用，这是应用在半导体理论，被认为是一些推广。
英文摘要：
The invariance for the equation of fast diffusion in the 2D coordinate space has been proved, and its reduction to the 1D (with respect to the spatial variable) analog is demonstrated. On the basis of these results, new exact multi-dimensional solutions, which are dependent on arbitrary harmonic functions, are constructed. As a result, new exact solutions of the well-known Liouville equation - the steady-state analog for the fast diffusion equation with the linear source - have been obtained. Some generalizations for the systems of quasilinear parabolic equations, as well as systems of elliptic equations with Poisson interaction, which are applied in the theory of semiconductors, are considered.
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