<p>We prove expansion of positivity and reduction of the oscillation results to the local weak solutions to a doubly nonlinear anisotropic class of parabolic differential equations with bounded and measurable coefficients, whose prototype is <Equation ID="Equ82"> <EquationSource Format="TEX">$\begin{aligned} u_t-\sum \limits _{i=1}^N \left( u^{(m_i-1)(p_i-1)} \ |u_{x_i}|^{p_i-2} \ u_{x_i} \right) _{x_i}=0 , \end{aligned}$</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 movablelimits="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msub> <mfenced close=")" open="("> <msup> <mi>u</mi> <mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mi>i</mi> </msub> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </msup> <mspace width="4pt" /> <msup> <mrow> <mo stretchy="false">|</mo> <msub> <mi>u</mi> <msub> <mi>x</mi> <mi>i</mi> </msub> </msub> <mo stretchy="false">|</mo> </mrow> <mrow> <msub> <mi>p</mi> <mi>i</mi> </msub> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mspace width="4pt" /> <msub> <mi>u</mi> <msub> <mi>x</mi> <mi>i</mi> </msub> </msub> </mfenced> <msub> <mi>x</mi> <mi>i</mi> </msub> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for a restricted range of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>s and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$m_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>m</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>s, that reflects their competition for the diffusion. The positivity expansion relies on an exponential shift and is presented separately for singular and degenerate cases. Finally we present a study of the local oscillation of the solution for some specific ranges of exponents, within the singular and degenerate cases.</p>

Qualitative properties of solutions to parabolic anisotropic equations: part I—expansion of positivity

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2026年6月23日

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Simone Ciani,
Eurica Henriques,
Mariia Savchenko,
Igor I. Skrypnik

摘要

We prove expansion of positivity and reduction of the oscillation results to the local weak solutions to a doubly nonlinear anisotropic class of parabolic differential equations with bounded and measurable coefficients, whose prototype is $\begin{aligned} u_t-\sum \limits _{i=1}^N \left( u^{(m_i-1)(p_i-1)} \ |u_{x_i}|^{p_i-2} \ u_{x_i} \right) _{x_i}=0 , \end{aligned}$ $\begin{matrix} u_{t} - \sum_{i = 1}^{N} {(u^{(m_{i} - 1) (p_{i} - 1)}, {| u_{x_{i}} |}^{p_{i} - 2}, u_{x_{i}})}_{x_{i}} = 0, \end{matrix}$ for a restricted range of $p_i$ $p_{i}$ s and $m_i$ $m_{i}$ s, that reflects their competition for the diffusion. The positivity expansion relies on an exponential shift and is presented separately for singular and degenerate cases. Finally we present a study of the local oscillation of the solution for some specific ranges of exponents, within the singular and degenerate cases.