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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
##### Keywords:
Dipole solution; conservation law
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##### References:
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