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Subpotential lower bounds for nonnegative solutions to certain quasi-linear degenerate parabolic equations. (English) Zbl 1170.35054
The authors deal with the following quasi-linear degenerate parabolic equation
$u_t- \operatorname{div}\mathbf{A}(x,t,u,Du)=0, \quad\text{weakly in }E_T, \tag{1}$ where $$E$$ is an open set in $$\mathbb R^N$$, and for $$T>0$$, $$E_T$$ denote the cylindrical domain $$E\times(0,T]$$, $${\mathbf A}:E_T\times \mathbb R^{N+1}\to\mathbb R^N$$ is only assumed to be a measurable and subject to the structure conditions
$\begin{cases} {\mathbf A}( x,t,u,Du)\cdot Du\geq C_0|Du|^p,\\ |{\mathbf A}(x,t,u,Du)|\leq C_1 |Du|^{p-1}, \end{cases} \quad \text{a.e. in }E_T,$ where $$C_0$$ and $$C_1$$ are given constants. Let
$u\in C_{\text{loc}}(0,T;L_{\text{loc}}^2 (E))\cap L_{\text{loc}}^p (0,T;W_{\text{loc}}^{1,p} (E)),\;p>2,$ be a local nonnegative, weak solution of (1). The authors, also, consider the prototype equation
$u_t- \operatorname{div}|Du|^{p-2}Du=0, \quad\text{weakly in }E_T. \tag{2}$ This equation admits the family of subsolutions
$\Gamma_{\nu ,p} (x,t;\overline{x},\overline{t})= \frac{k\rho^\nu} {S^{\nu /\lambda}(t)} \left\{ 1-b(\nu ,p) \left(\frac{| x-\overline{x}|}{S^{1/\lambda}(t)} \right)^{p/(p-1)}\right\}_+^{(p-1)/(p-2)},$ where $$\nu \geq N$$, $$k$$ and $$p$$ are positive parameters, and
\begin{alignedat}{2} S(t)&= k^{p-2}\rho ^{\nu(p-2)} (t-\overline{t})+ \rho ^\lambda, &\quad t&\geq \overline{t},\\ b(\nu ,p)&= \frac {1}{\lambda^{1/(p-1)}} \frac{p-2}{p}, &\quad \lambda &=\nu (p-2)+p. \end{alignedat} Given the general quasi-linear structure of (1) and (2), the functions $$\Gamma _{\nu ,p}$$ are not subsolutions of (1) in any sense. Moreover, no comparison principle holds. The aim of the paper is to show that the fundamental subsolutions drive the structural behavior of nonnegative solutions of (1) and (2).
This implies the expansion of positivity from a ball $$B_\rho(\overline{x})$$ at time $$\overline{t}$$ to the ball $$B_{2\rho} (\overline{x})$$ at some larger time $$t$$ and a lower bound on time decay on $$u$$ (Section 2). The latter permits to the authors, to establish alternative, equivalent forms of the Harnack inequality (Section 3). Finally , nonnegative weak solutions of (1)–(3) are bounded below by one $$\Gamma_{\nu,p}$$ for some $$\nu>N$$, and thus they do not decay in space faster than these “subsolutions” (Section 4).

##### MSC:
 35K65 Degenerate parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 35B45 A priori estimates in context of PDEs 35K59 Quasilinear parabolic equations
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##### References:
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