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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

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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