Regularity of solutions and interfaces of a generalized porous medium equation inR
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Authors: | Arturo De Pablo Juan Luis Vazquez |
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Institution: | (1) Present address: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain |
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Abstract: | Summary We consider the Cauchy problem for the generalized porous medium equation ut=(u) where u=u(x, t), xRn and t>0, and the initial datum u(x, 0) is assumed to be nonnegative, integrable mid to nave compact support. The nonlinearity (u) is a C1 function defined for uO which grows like a power of u. Our assumptions generalize the porous medium case, (u)=um, m>1, and also include the equation of the Marshak waves. This problem has finite speed of propagation. We estimate the rate of growth of the support of the solution with precise estimates for t 0 and t. Our main result deals with the regularity of the solutions. We show that after a certain time t0 the pressure, defined by v=(u), with (u)=(u)/u and (0)=0, is a Lipschitz-continuous function of x and t and the interface is a Lipschitz-continuous surface in RN+1; the solution u is Hölder continuous for all times t> 0.Both authors partially supported by CAICYT, Project 2805-83. The second author also supported by USA-Spain Joint Research Grant CCB-8402023. |
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