<p>We study the fully nonlinear heat equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$b(\partial _tu)\partial _tu=\Delta u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> posed in a bounded domain with Dirichlet boundary conditions. Here <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$b(s)=b^-$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>b</mi> <mo>-</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$s<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$b(s)=b^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>b</mi> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$s>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$b^-\ne b^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>b</mi> <mo>-</mo> </msup> <mo>≠</mo> <msup> <mi>b</mi> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> being two positive constants. This equation models the flow of an elastic fluid in an elasto-plastic porous medium. We are interested in the existence and uniqueness of viscosity solutions and in their asymptotic behaviour as <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$t\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and when <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$b^-\rightarrow 0^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>b</mi> <mo>-</mo> </msup> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$b^+\rightarrow +\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>b</mi> <mo>+</mo> </msup> <mo stretchy="false">→</mo> <mo>+</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. We also characterize solutions of the problem as limits of a minimization dynamic game.</p>

On the elasto-plastic filtration equation

期刊论文
发表日期：

2026年6月27日

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Arturo de Pablo,
Fernando Quirós,
Julio D. Rossi

摘要

We study the fully nonlinear heat equation $b(\partial _tu)\partial _tu=\Delta u$ $b (\partial_{t} u) \partial_{t} u = Δ u$ posed in a bounded domain with Dirichlet boundary conditions. Here $b(s)=b^-$ $b (s) = b^{-}$ if $s<0$ $s < 0$ , $b(s)=b^+$ $b (s) = b^{+}$ if $s>0$ $s > 0$ , $b^-\ne b^+$ $b^{-} \neq b^{+}$ being two positive constants. This equation models the flow of an elastic fluid in an elasto-plastic porous medium. We are interested in the existence and uniqueness of viscosity solutions and in their asymptotic behaviour as $t\rightarrow \infty $ $t \to \infty$ and when $b^-\rightarrow 0^+$ $b^{-} \to 0^{+}$ or $b^+\rightarrow +\infty $ $b^{+} \to + \infty$ . We also characterize solutions of the problem as limits of a minimization dynamic game.