This paper deals with homogeneous Dirichlet boundary value problem to a class of porous medium equations with viscoelastic term <Equation ID="Equa"> <EquationSource Format="TEX">${{\partial u} \over {\partial t}} - \Delta {u^m} + \int_0^t {g(t - s)} \Delta {u^m}(x,s){\rm{d}}s = {u^p}, \quad x \in \Omega ,\;t \ge 0,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mrow> <msup> <mi>u</mi> <mi>m</mi> </msup> </mrow> <mo>+</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>t</mi> </msubsup> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mrow> <msup> <mi>u</mi> <mi>m</mi> </msup> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow> <mrow> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>s</mi> <mo>=</mo> <mrow> <msup> <mi>u</mi> <mi>p</mi> </msup> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </math></EquationSource> </Equation> where p > m (m > 0). We prove that the weak solutions of the above problem blow up in finite time when the initial energy is positive and the function g satisfies suitable conditions. Our result generalizes that of S.A. Messaoudi in [1].

The blow-up of solutions for porous medium equations with viscoelastic term under positive initial energy

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2026年3月1日

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Xiu-lan Wu,
Ya-xin Zhao,
Xiao-xin Yang

摘要

This paper deals with homogeneous Dirichlet boundary value problem to a class of porous medium equations with viscoelastic term ${{\partial u} \over {\partial t}} - \Delta {u^m} + \int_0^t {g(t - s)} \Delta {u^m}(x,s){\rm{d}}s = {u^p}, \quad x \in \Omega ,\;t \ge 0,$ $\frac{\partial u}{\partial t} - Δ u^{m} + \int_{0}^{t} g (t - s) Δ u^{m} (x, s) d s = u^{p}, x \in Ω, t \geq 0,$ where p > m (m > 0). We prove that the weak solutions of the above problem blow up in finite time when the initial energy is positive and the function g satisfies suitable conditions. Our result generalizes that of S.A. Messaoudi in [1].