<p>This work examines the singular nonlinear heat equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u_t-\Delta _\alpha u=|x|^{-\gamma } u^q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>-</mo> <msub> <mi mathvariant="normal">Δ</mi> <mi>α</mi> </msub> <mi>u</mi> <mo>=</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mo>-</mo> <mi>γ</mi> </mrow> </msup> <msup> <mi>u</mi> <mi>q</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathbb {R}}_{+}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </math></EquationSource> </InlineEquation>, governed by the Bessel operator <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Delta _\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$0<q<1,0<\gamma <\gamma _\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>q</mi> <mo><</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo><</mo> <mi>γ</mi> <mo><</mo> <msub> <mi>γ</mi> <mi>α</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, and non-negative initial data in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathscr {C}}_{, 0}({\mathbb {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi mathvariant="script">C</mi> <mrow> <mrow /> <mo>∗</mo> <mo>,</mo> <mn>0</mn> </mrow> <mrow /> </mmultiscripts> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Employing fixed-point arguments and semigroup theory, we establish existence, uniqueness, and a comparison principle in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L_\alpha ^{\infty }\left( {\mathbb {R}}_{+}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>α</mi> <mi>∞</mi> </msubsup> <mfenced close=")" open="("> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. Solutions exhibit strict positivity and satisfy sharp lower bounds via explicit subsolutions. A critical threshold <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma ^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>γ</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> ensures uniqueness for small <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation>, bridging harmonic analysis and singular PDE theory in Bessel frameworks.</p>

Positivity of global solution for a singular nonlinear heat equation associated with the Bessel operator

期刊论文
发表日期：

2025年10月21日

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Youssef Bettaibi

摘要

This work examines the singular nonlinear heat equation $u_t-\Delta _\alpha u=|x|^{-\gamma } u^q$ $u_{t} - Δ_{α} u = {| x |}^{- γ} u^{q}$ on ${\mathbb {R}}_{+}$ $R_{+}$ , governed by the Bessel operator $\Delta _\alpha $ $Δ_{α}$ , with $0<q<1,0<\gamma <\gamma _\alpha $ $0 < q < 1, 0 < γ < γ_{α}$ , and non-negative initial data in ${\mathscr {C}}_{*, 0}({\mathbb {R}})$ $C_{*, 0} (R)$ . Employing fixed-point arguments and semigroup theory, we establish existence, uniqueness, and a comparison principle in $L_\alpha ^{\infty }\left( {\mathbb {R}}_{+}\right) $ $L_{α}^{\infty} (R_{+})$ . Solutions exhibit strict positivity and satisfy sharp lower bounds via explicit subsolutions. A critical threshold $\gamma ^*$ $γ^{*}$ ensures uniqueness for small $\gamma $ $γ$ , bridging harmonic analysis and singular PDE theory in Bessel frameworks.