## 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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### References:

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