THE SMOOTHING PROPERTY FOR A CLASS OF DOUBLY NONLINEAR PARABOLIC EQUATIONS CARSTEN EBMEYER AND JOS ´ E MIGUEL URBANO Abstract. We consider a class of doubly nonlinear parabolic equations used in modeling free boundaries with a finite speed of propagation. We prove that nonnegative weak solutions satisfy a smoothing property; this is a well known feature in some particular cases like the porous medium equation or the parabolic p-Laplace equation. The result is obtained via regularization and a comparison theorem. 1. Introduction This paper deals with a class of partial differential equations - doubly nonlinear parabolic equations - that have recently attracted a lot of attention. They arise in many different physical contexts like, for instance, the description of turbulent filtration in porous media, or the flow of a gas through a porous medium in a tur- bulent regime; in general, doubly nonlinear parabolic equations are used to model processes obeying a nonlinear Darcy law (see [8], [14], and the references given therein). Typical examples of such parabolic PDEs are equations of the form u t =Δ p (|u| m-1 u), m(p - 1) > 1 , where Δ p is the p-Laplacian, which are used in modeling phenomena involving a free boundary with a finite speed of propagation. These degenerate equations exhibiting a doubly nonlinearity generalize the porous medium equation (p = 2) and the parabolic p-Laplace equation (m = 1). The aim of the paper is to show that nonnegative solutions of a class of doubly nonlinear parabolic equations satisfy the smoothing property, i. e., the estimate (1.1) u t ≥- c t u, where c is a constant depending only on the data. The smoothing property (1.1) implies the regularizing property u t  L 1 (R d ) ≤ 2c t u 0  L 1 (R d ) , as will be shown below, and plays a crucial role in the study of the finite speed of propagation of the free boundary (see, e. g., [5, 12], where the porous medium equation is treated) or the proof of regularity results for the solutions (cf. [13]). 2000 Mathematics Subject Classification. Primary 35K65; Secondary 35R35, 76S05. Key words and phrases. Degenerate parabolic equation, free boundary, finite speed of propa- gation, porous medium equation. The second author was supported in part by the Project FCT-POCTI/34471/MAT/2000 and CMUC/FCT. 1