Sub-Potential Lower Bounds for Non-negative local solutions to certain quasi lilear degenerate parabolic equations.

ABSTRACT The first parabolic version of the Harnack inequality is due to Hadamard ([Ha]) and Pini ([Pi]). Their result is the following: Let $u$ be a non-negative solution of the heat equation in a cylindric domain $\Omega_T$. Let $(x_0,t_0) \in \Omega_T$ and assume that the cylinder $(x_0,t_0) + Q_{2\rho} \subset \Omega_T$ where $Q_\rho \equiv B_\rho \times (-\rho^2,0)$. Then there exists a constant $\gamma$, depending only upon $N$, such that $$u(x_0,t_0) \geq \gamma \sup_{B_\rho(x_0)} u(x, t_0 -\rho^2) $$ The proof is based on local representations by means of heat potentials. A breakthrough in the theory is due to Moser, who in his celebrated paper [MO1] proved that the Harnack estimates continues to hold for nonnegative weak solution of linear parabolic equations with $L^\infty$ coefficients. The result of Moser can be extended (see [AS] and [TR]) to nonnegative weak solutions of quasi-linear parabolic equation. The proof of Moser's result is based on suitable integral estimates for powers and logarithm of the solution $u$; the general structure follows the same one Moser used in his earlier work on Harnack's inequality in the elliptic case ([MO]). It is worth saying that the most difficult part of Moser's proof is an adaptation to the parabolic case of the exponential decay of the distribution function of a function with bounded mean oscillation. Going from the elliptic to the parabolic situation the difficulty lies in the special role played by the time variable. Indeed Moser's proof is hard to follow and the need for a possible simplification was immediately felt. Moser himself published a new proof of Harnack inequality in 1971 ([MO2]) with the expressed purpose to avoid the use of his parabolic John - Nirenberg Lemma ([JN]); through estimates on the logarithm of the solution via a measure lemma based on a result of Bombieri ([BO], [BG]). Quite surprisingly Moser's method does not work in the case of the $p$-Laplacean and this is not simply a matter of technique. As the $p$-Laplacean is invariant by the scaling $x\rightarrow hx$ and $t \rightarrow h^p t$, one would guess that Harnack estimates would hold in the cylinder $[(x_0, t_0) + B_\rho \times (\rho^p,0)]$, but this is not true as one can easily check considering the explicit solution introduced by Barenblatt in [BA]. DiBenedetto overcame this difficulty realizing that a specific Harnack estimate holds with an intrinsic time scale exactly of the order $u(x_0,t_0)^{2-p}$. More precisely the following result is proved in [DB], and [DBK]: Let $u$ be a nonnegative weak solution of the $p$-Laplacean and let $p> \frac{2N}{N+1}$. Fix $(x_0,t_0) \in \Omega_T$ and assume that $u(x_0,t_0) >0$. There exists constants $\gamma>1$ and $C>0$, depending only upon $N$ and $p$, such that $$u(x_0,t_0) \leq \gamma \inf_{B_\rho(x_0)}u(\cdot, t_0 +\theta)$$ where $$\theta \equiv \frac {C\rho^p}{[u(x_0,t_0)]^{p-2}}$$ provided that the cylinder $(x_0,t_0) + B_{4\rho}\times (-4\theta,4\theta)$ is contained in $\Omega_T$. The proof given by DiBenedetto is based on explicit representation of solutions and cannot be extended to the general case, In [DBGV] this gap was fullifed in the case of degenerate equations The proof is based on DeGiorgi techniques ([DG]) and on a theoretical measure lemma([DBGV1]). With this new approach we were able to give also a new proof by Moser only based on geometrical approaches and not on BMO properties or on Bombieri’s lemma. In this paper we consider the Harnack estimates proved in [DBGV]. We prove that non–negative weak solutions of quasilinear degenerate parabolic equations of p–Laplacian type are locally bounded below by Barenblatt type sub–potentials. As a consequence non–negative solutions expand their positivity set. That is, a quantitative lower bound on a ball $B_\rho$ at time $t_0$ yields a quantitative lower bound on a ball $B_{2\rho} $ at some further time$ t_1$ These lower bounds also permit one to recast the Harnack inequality of [DBGV] in a family of alternative, equivalent forms. Such kind of result is very useful to prove asymptotic limit for a class of general degenerate parabolic equations (see, for instance, [RVV]). From Google Scholar database it results that this paper was quoted 8 times, for ISI database 4 times, for Scopus database 4 times, for MathSciNet database 7. REFERENCES [AS ] D.G. Aronson - J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Rat. Mech. Anal. 25 (1967), pp. 81-123. [BA] G.I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Mech. 16 (1952), pp. 67-78. [BO] E. Bombieri, Theory of minimal surfaces and a counterexample to the Bernstein conjecture in high dimension, Mimeographed Notes of Lectures held at Courant Institut, New York University, 1970. [BG] E. Bombieri - E. Giusti, Harnack’s inequality for elliptic differential equations on minimal surfaces, Inventiones Math. 15 (1972), pp. 24-46. [DG] E. DeGiorgi, Sulla differenziabilita’ e l’analiticita’ delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino, Cl. Sci. Fis. Mat. Nat., Ser. 3, 3 (1957), pp. 25-43. [DB] E. DiBenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Rat. Mech. Anal. 100 (1988), pp. 129-147. [DBGV] E. DiBenedetto; U. Gianazza; V. Vespri; Harnack estimates for quasi linear degenerate parabolic differential equations Acta Math 200 (2008), no. 2, 181-209. [DBGV1] E. DiBenedetto; U. Gianazza; V. Vespri; Local clustering of the non-zero set of functions in $W^ {1,1}(E)$. Atti Accad. Naz. Lincei Cl. 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