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Homogeneous diffusion in \({\mathbb{R}}\) with power-like nonlinear diffusivity. (English) Zbl 0683.76073

Summary: We study the nonnegativ solutions of the initial-value problem \[ u_ t = (u^ r | u_ x|^{p-1} u_ x)_ x, \qquad u(x,0)\in L^ 1(\mathbb{R}),\tag{1} \] where \(p>0\), \(r+p>0\). The local velocity of propagation of the solutions is identified as \(V=-v_ x| v_ x|^{p-1}\), where \(v=cu^{\alpha}\) (with \(\alpha = (r+p-1)/p\) and \(c=(r+p)/(r+p-1)\)) is the nonlinear potential. Our main result is the a priori estimate \((v_ x| v_ x|^{p-1})_ x \geq -1/(2p+r)t\) which we use to establish:
i) existence and uniqueness of a solution of (1),
ii) regularity of the free boundaries that appear when \(r+p>1\), and
iii) asymptotic behavior of solutions and free boundaries for initial data with compact support.

MSC:

76R50 Diffusion
35K57 Reaction-diffusion equations
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