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The dipole solution for the porous medium equation in several space dimensions. (English) Zbl 0832.35082
We address the existence of special solutions \(u(x, t)\) of the porous medium equation \(u_t= \Delta(|u|^{m- 1} u)\), \(m> 1\), \(x\in \mathbb{R}^N\), \(t> 0\). We work in any space dimension \(N\geq 1\) and consider solutions with changing sign. Our main interest is to find a solution to the initial value problem with data \(u(x, 0)= -{\partial\over \partial x_1} \delta(x)\), where \(\delta\) is the Dirac delta function in \(\mathbb{R}^N\). This is called a dipole solution. We prove that such a solution exists and has the self-similar form \(\overline u(x, t)= t^{- \alpha} U(xt^{- \beta})\), with \[ \alpha= {N+ 1\over (N+ 1)m+ (1- N)}\qquad\text{and} \qquad \beta= {1\over (N+ 1)m+ (1- N)}. \] A second part of our paper concerns the use of the dipole solution to describe the asymptotic behaviour of the solutions of the initial and boundary problem \[ u_t= \Delta(|u|^{m- 1} u)\quad\text{in}\quad Q^+= H\times (0, \infty),\tag{P} \] \[ u(x, 0)= u_0(x)\quad\text{for}\quad x\in \overline H,\;u(x, t)= 0\quad\text{on}\quad \Sigma= \partial H\times [0, \infty). \] There is a conservation law associated to the solutions \(u\) of this problem, namely the invariance in time of the first moment \[ \int_H xu(x, t) dx= \text{constant}. \] This allows us to prove that any solution of problem (P) with nonnegative and integrable initial data with compact support converges as \(t\to \infty\) to the dipole solution having the same moment.

MSC:
35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
76S05 Flows in porous media; filtration; seepage
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