<p>In this work we are interested in the study of a class of anisotropic porous medium-type equations whose prototype is <Equation ID="Equ1"> <EquationNumber>1.1</EquationNumber> <MediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2979_Article_Equ1.gif" Format="GIF" Height="51" Rendition="HTML" Resolution="72" Type="Linedraw" Width="405" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} u_t =\sum _{i=1}^N \left( m_i u^{m_i-1} u_{x_i} \right) _{x_i} \ , \qquad 0<m_1 \le \cdots \le m_N <1 \ , \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mfenced close=")" open="("> <msub> <mi>m</mi> <mi>i</mi> </msub> <msup> <mi>u</mi> <mrow> <msub> <mi>m</mi> <mi>i</mi> </msub> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>u</mi> <msub> <mi>x</mi> <mi>i</mi> </msub> </msub> </mfenced> <msub> <mi>x</mi> <mi>i</mi> </msub> </msub> <mspace width="4pt" /> <mo>,</mo> <mspace width="2em" /> <mn>0</mn> <mo><</mo> <msub> <mi>m</mi> <mn>1</mn> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>m</mi> <mi>N</mi> </msub> <mo><</mo> <mn>1</mn> <mspace width="4pt" /> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for which we derive several estimates, namely two Harnack-type inequalities; and, when considering the associated Dirichlet problem, we determine the finite time of extinction and thereby present a decay rate of extinction.</p>

Harnack-type estimates and extinction in finite time for a class of anisotropic porous medium type equations

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2025年3月24日

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Simone Ciani,
Eurica Henriques

摘要

In this work we are interested in the study of a class of anisotropic porous medium-type equations whose prototype is 1.1 $\begin{aligned} u_t =\sum _{i=1}^N \left( m_i u^{m_i-1} u_{x_i} \right) _{x_i} \ , \qquad 0<m_1 \le \cdots \le m_N <1 \ , \end{aligned}$ $\begin{matrix} u_{t} = \sum_{i = 1}^{N} {(m_{i}, u^{m_{i} - 1}, u_{x_{i}})}_{x_{i}}, 0 < m_{1} \leq \dots \leq m_{N} < 1, \end{matrix}$ for which we derive several estimates, namely two Harnack-type inequalities; and, when considering the associated Dirichlet problem, we determine the finite time of extinction and thereby present a decay rate of extinction.