Abstract
We show that the long time asymptotic solutions of initial value problems for linear and nonlinear mathematical models of tumor angiogenesisare self-similar spreading solutions. The symmetries of the governing equationsyield three-parameter families of these solutions given in terms of their mass,center of mass, and variance. Unlike the mass and center of mass, the variance,or ”time-shift,” of a solution is not a conserved quantity for the non linear problem
Highlights
In [7] a code has been used to investigate the validity of the semi-in...nite tumor assumption in an epithelial tissue model
The epithelial tissue model consists of three layers, which includes the top epithelium, the middle tumor and the bottom stroma
The porous medium equation is proposed in order to describe the distribution of the density of a substance that ‡ows through a uniformly distributed porous medium
Summary
Let us calculate self similar solutions of the porous medium equation (1.5). It can be shown that its total mass Rn u(x; t)dx is conserved for evolution of time in the same way as for the heat equation. This preserves the total mass and is a generalization of the scaling transformation for the heat equation. Let u be a function invariant under the scaling transformation (1.7), and which preserves the total mass.
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