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Similarity solutions of the porous medium equation with sign changes. (English) Zbl 0777.35034
We study similarity solutions with sign changes of the porous medium equation in one space dimension, \[ u_ t=(| u|^{m-1} u_ x)_ x,\quad x\in\mathbb{R},\;t>0,\tag{1} \] with \(m>1\). Here \(u\) is a function of \(x\) and \(t\). For nonnegative \(u\) eq. (1) arises in the theory of gas flow through a one-dimensional porous medium, and an extensive theory has been developed over the last decades.
For the development of the theory of (1) similarity solutions play an important role. Essentially these are solutions whose profiles remain the same as \(t\) varies. The most familiar of them are the Barenblatt-Pattle solutions and the so-called dipole solutions.
Our result concerns semilarity solutions of the form \(u(x,t)=t^ \alpha U(\eta)\), where \(\eta=xt^{-\beta}\), and \(\alpha\) and \(\beta\) are fixed reals. It is well-known that they have to be related by \(2\beta=(m- 1)\alpha+1\), and that \(U(\eta)\) has to satisfy the second order ordinary differential equation \[ (| U|^{m-1} U)''+m\beta\eta U'=m\alpha U. \] .

MSC:
35K65 Degenerate parabolic equations
35Q35 PDEs in connection with fluid mechanics
35K57 Reaction-diffusion equations
76S05 Flows in porous media; filtration; seepage
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