Harnack Estimates and Extinction Profile for Weak Solutions of Certain Singular Parabolic Equations

Abstract

Abstract. We establish an intrinsic Harnack estimate for nonnegative weak solutions of the singular equation (1.1) below, for m in the optimal range ((N -2)+/N, 1). Intrinsic means that, due to the singularity, the space-time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself. Consequences are, sharp supestimates on the solutions and decay rates as t approaches the extinction time. Analogous results are shown for p-laplacian type equations.

Interior Harnack Estimates: The State-of-the-Art for Quasilinear Singular Parabolic Equations

“…The same method (intrinsic rescaling) is at the basis of a new version of Harnack inequality for nonnegative solutions of the p-Laplacian and Porous Medium equations, as established in the late 1980s and early 1990s in [15] and [24], respectively. The new Harnack inequality reads as follows.…”

On quasilinear parabolic equations which admit global solutions for initial data with unrestricted growth

“…Now we consider cylinder-type estimates based on the method in DiBenedetto-Kwong [3]. We shall establish them for the following equation…”

Initial trace for a doubly nonlinear parabolic equation

“…Conversely, if there exists a nonnegative solution of the heat equation in R N × (0, T ) for some T > 0, then the initial trace of the solution is uniquely given as a nonnegative σ -finite Borel measure μ in R N satisfying (1.4) for some λ > 0 (see [1,2,21] and [24]). The Cauchy problem related to nonlinear parabolic equations has been studied, for example, in [3][4][5][6][7][8]12,17], and [25]. Among others, the first author of this paper considered in [12] the doubly nonlinear Cauchy problem…”

“…The case (I) can be obtained by the same arguments as in [4] and [6] and the case (III) as in [7] and [8]. Moreover, for the case (I), it is known that if there exists a nonnegative solution in R N × (0, T ) for some T > 0, then the initial trace of the solution is uniquely given as a nonnegative σ -finite Borel measure μ in R N satisfying (1.6).…”

“…Our approach is based on the scale and location invariant Harnack inequality. This is the same starting point as in [2,3], and [8], but our method is completely different from theirs. We obtain a pointwise estimate of nonnegative solution u of (1.1) and prove that there exists a positive constant λ for which…”