Harnack type inequalities for the parabolic logarithmic p-Laplacian equation (original) (raw)
Related papers
2009
Non-negative solutions to quasi-linear, degenerate or singular parabolic partial differential equations, of p-Laplacian type for p > 2N Nþ1 , satisfy Harnack-type estimates in some intrinsic geometry ([2, 3]). Some equivalent alternative forms of these Harnack estimates are established, where the supremum and the infimum of the solutions play symmetric roles, within a properly redefined intrinsic geometry. Such equivalent forms hold for the non-degenerate case p ¼ 2 following the classical work of Moser ([5, 6]), and are shown to hold in the intrinsic geometry of these degenerate and/or parabolic p.d.e.'s. Some new forms of such an estimate are also established for 1 < p < 2.
Harnack type inequalities for some doubly nonlinear singular parabolic equations
Discrete and Continuous Dynamical Systems, 2015
We prove Harnack type inequalities for a wide class of parabolic doubly nonlinear equations including ut = div(|u| m−1 |Du| p−2 Du). We will distinguish between the supercritical range 3 − p N < p + m < 3 and the subcritical 2 < p + m ≤ 3 − p N range. Our results extend similar estimates holding for general equations having the same structure as the parabolic p-Laplace or the porous medium equation and recently collected in [6]. 1. Introduction. Consider an open set E ⊂ R N , T > 0, and quasi-linear parabolic differential equations of the form u t − divA x, t, u, D(|u| m−1 p−1 u) = 0 (1) in E T = E × (0, T ], with p + m > 2. The function A : E T × R N +1 → R N is assumed to be measurable and subject to the structure conditions A(x, t, z, η) • η ≥ C 0 |η| p |A(x, t, z, η)| ≤ C 1 |η| p−1 (2) for almost all (x, t) ∈ E T , for all z ∈ R and η ∈ R N , with C 0 , C 1 positive constants. Assume also that the function A is monotone in the variable η (A(x, t, z, η 1) − A(x, t, z, η 2)) • (η 1 − η 2) ≥ 0 (3) and Lipschitz continuous in the variable |z| m−1 p−1 z in the following sense |A(x, t, z 1 , η) − A(x, t, z 2 , η)| ≤ Λ |z 1 | m−1 p−1 z 1 − |z 2 | m−1 p−1 z 2 (1 + |η| p−1) (4)
Rendiconti Lincei - Matematica e Applicazioni, 2009
Non-negative solutions to quasi-linear, degenerate or singular parabolic partial differential equations, of p-Laplacian type for p > 2N Nþ1 , satisfy Harnack-type estimates in some intrinsic geometry ([2, 3]). Some equivalent alternative forms of these Harnack estimates are established, where the supremum and the infimum of the solutions play symmetric roles, within a properly redefined intrinsic geometry. Such equivalent forms hold for the non-degenerate case p ¼ 2 following the classical work of Moser ([5, 6]), and are shown to hold in the intrinsic geometry of these degenerate and/or parabolic p.d.e.'s. Some new forms of such an estimate are also established for 1 < p < 2.
Intrinsic Harnack inequalities for quasi-linear singular parabolic partial differential equations
Rendiconti Lincei - Matematica e Applicazioni, 2000
Intrinsic Harnack estimates for non-negative solutions of singular, quasilinear, parabolic equations, are established, including the prototype p-Laplacean equation (1.4) below. For p in the super-critical range 2N N+1 < p < 2, the Harnack inequality is shown to hold in a parabolic form, both forward and backward in time, and in a elliptic form at fixed time. These estimates fail for the heat equation (p → 2). It is shown by counterexamples, that they fail for p in the sub-critical range 1 < p ≤ 2N N+1 . Thus the indicated super-critical range is optimal for a Harnack estimate to hold. The novel proofs are based on measure theoretical arguments, as opposed to comparison principles and are sufficiently flexible to hold for a large class of singular parabolic equation including the porous medium equation and its quasi-linear versions.
Annali Della Scuola Normale Superiore Di Pisa-classe Di Scienze, 2010
Forward, backward and elliptic Harnack inequalities for non-negative solutions of a class of singular, quasi-linear, parabolic equations, are established. These classes of singular equations include the p-Laplacean equation and equations of the porous medium type. Key novel points include form of a Harnack estimate backward in time, that has never been observed before, and measure theoretical proofs, as opposed to comparison principles. These Harnack estimates are established in the super-critical range (1.5) below. Such a range is optimal for a Harnack estimate to hold. Mathematics Subject Classification (2010): 35K65 (primary); 35B65, 35B45 (secondary). 1. Main results Let E be an open set in RN and for T > 0 let ET = E × (0, T ]. Let u be a weak solution u ∈ Cloc ( 0, T ; Lloc(E) ) ∩ L p loc(0, T ; W 1,p loc (E)) 1 < p < 2 (1.1) of a quasi-linear, singular parabolic equation of the type ut − div A(x, t, u, Du) = B(x, t, u, Du) weakly in ET (1.2) where the functions A : E...
Harnack and Pointwise Estimates for Degenerate or Singular Parabolic Equations
Contemporary Research in Elliptic PDEs and Related Topics, 2019
In this paper we give both a historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical consequences. The linear parabolic Harnack inequality of Moser is discussed extensively, together with its link to two-sided kernel estimates and to the Li-Yau differential Harnack inequality. Then we overview the more recent developments of the theory for nonlinear degenerate/singular equations, highlighting the differences with the quadratic case and introducing the so-called intrinsic Harnack inequalities. Finally, we provide complete proofs of the Harnack inequalities in some paramount case to introduce the reader to the expansion of positivity method.
Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations
2006
We establish the intrinsic Harnack inequality for nonnegative solutions of the parabolic p-Laplacian equation by a proof that uses neither the comparison principle nor explicit self-similar solutions. The significance is that the proof applies to quasilinear p-Laplacian-type equations, thereby solving a long-standing problem in the theory of degenerate parabolic equations.