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Homogeneous diffusion in ℝ with power-like nonlinear diffusivity

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Abstract

We study the nonnegative solutions of the initial-value problem ut=(ur|ux|p-1ux)x,u(x, 0)∈L 1(ℝ), where p>0, r+p>0. The local velocity of propagation of the solutions is identified as V = -vx| vx|p-1 where v =cuα (with r +p - 1)/p and c (r +p/(r +p- 1)) is the nonlinear potential. Our main result is the a priori estimate (vx|vx|p-1)x≥-

$$\frac{1}{{(2p + r) t}}$$

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.

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  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Juan R. Esteban & Juan L. Vazquez

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  1. Juan R. Esteban
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  2. Juan L. Vazquez
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Communicated by Haim Brezis

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Esteban, J.R., Vazquez, J.L. Homogeneous diffusion in ℝ with power-like nonlinear diffusivity. Arch. Rational Mech. Anal. 103, 39–80 (1988). https://doi.org/10.1007/BF00292920

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