<p>In this paper, we study the initial-boundary value problem for the semilinear parabolic equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(b(u))_t - \Delta _X u = \vert u \vert ^{p-2}u\log (\vert u \vert )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </msub> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>X</mi> </msub> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mi>u</mi> <mo>log</mo> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$X = (X_1, X_2, \cdots , X_{m-1}, X_m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>X</mi> <mi>m</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is a system of real smooth vector fields that satisfy Hörmander’s condition, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Delta _X = \sum \nolimits _{i=1}^nX_j^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>X</mi> </msub> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <msubsup> <mi>X</mi> <mi>j</mi> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> is a finitely degenerate elliptic operator. By using the Galerkin approximation, potential wells, and concavity methods, we show the global existence, exponential decay, and blow-up in finite time of solutions with subcritical or critical initial energy.</p>

Global existence, blow-up, and exponential decay for a class of finitely degenerate semilinear parabolic equations with logarithmic nonlinearity

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发表日期：

2025年11月14日

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Abdelkader El Minsari,
Anass Ourraoui

摘要

In this paper, we study the initial-boundary value problem for the semilinear parabolic equation $(b(u))_t - \Delta _X u = \vert u \vert ^{p-2}u\log (\vert u \vert )$ ${(b (u))}_{t} - Δ_{X} u = {| u |}^{p - 2} u log (| u |)$ , where $X = (X_1, X_2, \cdots , X_{m-1}, X_m)$ $X = (X_{1}, X_{2}, \dots, X_{m - 1}, X_{m})$ is a system of real smooth vector fields that satisfy Hörmander’s condition, and $\Delta _X = \sum \nolimits _{i=1}^nX_j^2$ $Δ_{X} = \sum_{i = 1}^{n} X_{j}^{2}$ is a finitely degenerate elliptic operator. By using the Galerkin approximation, potential wells, and concavity methods, we show the global existence, exponential decay, and blow-up in finite time of solutions with subcritical or critical initial energy.