On certain nonlinear parabolic equations with singular diffusion and data in L^1

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September 2005, 4(3): 589-612. doi: 10.3934/cpaa.2005.4.589

On certain nonlinear parabolic equations with singular diffusion and data in $L^1$

C. García Vázquez ^1, and Francisco Ortegón Gallego ^1,

1.	Departamento de Matemáticas, Universidad de Cádiz, CASEM, Campus del Río San Pedro, 11510 Puerto Real, Cádiz, Spain, Spain

Received August 2004 Revised April 2005 Published June 2005

Abstract
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An existence result is established for a class of nonlinear parabolic equations having a coercive diffusion matrix blowing-up for a finite value of the unknown, a second hand $f\in L^1(Q)$, and an initial data $u_0\in L^1(\Omega)$. We develop a technique which relies on the notion of a renormalized solution and an adequate regularization in time for certain truncation functions. Some uniqueness results are also shown under additional hypotheses.

Keywords: singular diffusion, Sobolev spaces., Nonlinear parabolic equations, renormalized solutions.

Mathematics Subject Classification: 35K55, 35K6.

Citation: C. García Vázquez, Francisco Ortegón Gallego. On certain nonlinear parabolic equations with singular diffusion and data in $L^1$. Communications on Pure & Applied Analysis, 2005, 4 (3) : 589-612. doi: 10.3934/cpaa.2005.4.589

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