Backward estimates for nonnegative solutions to a class of singular parabolic equations

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• 作者：Marco Vinicio Calahorrano Recalde^a ; ^{marco.calahorrano@epn.edu.ec" class="auth_mail" title="E-mail the corresponding author} ; Vincenzo Vespri^b ; ¹ ; ^{vincenzo.vespri@unifi.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35K67 ; secondary ; 35K92 ; 35K20
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：144
• 期：Complete
• 页码：194-203
• 全文大小：646 K

文摘

In this paper we improve a recent result of the same Authors (Recalde and Vespri, 2015) by using the same approach in a more tricky way. More precisely, we study a class of quasilinear singular parabolic equations with L^∞$L^{\infty }$ coefficients, whose prototypes are the p$p$-Laplacian ($\frac{2N}{N+1}<p<2$) equations. Let us consider nonnegative solutions defined in R⁺×R^N${\mathbb{R}}^{+}\times {\mathbb{R}}^N$. We recall that in Recalde and Vespri (2015), starting from the value attained in a point (x₀,t₀)$(x_0,t_0)$ by the solution, we proved sharp estimates in the stripe ((1−ε)t₀,∞)×R^N$((1-\epsilon )t_0,\infty )\times {\mathbb{R}}^N$. In this note we show that, quite surprisingly, sharp estimates hold also for the remote past i.e. also for the stripe (0,t₀)×R^N$(0,t_0)\times {\mathbb{R}}^N$.

In the last section we briefly show how these results can be adapted to equations of Porous Medium type in the fast diffusion range i.e. ${\left(\frac{N-2}{N}\right)}_{+}<m<1$.