<p>We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L^\delta $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>δ</mi> </msup> </math></EquationSource> </InlineEquation> with low dimension <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0< \delta < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>δ</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L^\delta $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>δ</mi> </msup> </math></EquationSource> </InlineEquation> is a local operator, whose drift component is the derivative of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$x \mapsto \log (\vert x\vert )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>↦</mo> <mo>log</mo> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>: in particular it is a Schwartz distribution, which is not the derivative of a continuous real function. The solutions are intended in a duality <i>(weak)</i> sense with respect to state space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L^2(\mathbb R_+, d\mu ),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo>,</mo> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> being an invariant measure for the Bessel semigroup.</p>

About semilinear low dimension Bessel PDEs

期刊论文
发表日期：

2025年8月29日

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Alberto Ohashi,
Francesco Russo,
Alan Teixeira

摘要

We prove existence and uniqueness of solutions of a semilinear PDE driven by a Bessel type generator $L^\delta $ $L^{δ}$ with low dimension $0< \delta < 1$ $0 < δ < 1$ . $L^\delta $ $L^{δ}$ is a local operator, whose drift component is the derivative of $x \mapsto \log (\vert x\vert )$ $x \mapsto log (| x |)$ : in particular it is a Schwartz distribution, which is not the derivative of a continuous real function. The solutions are intended in a duality (weak) sense with respect to state space $L^2(\mathbb R_+, d\mu ),$ $L^{2} (R_{+}, d μ),$ $\mu $ $μ$ being an invariant measure for the Bessel semigroup.