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<p>In this paper, we show global existence and non-existence results for the heat equation with sum of the squares of smooth vector fields on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^{n}\)</EquationSource> </InlineEquation> satisfying Hörmander’s rank condition with a non-linearity of the form <i>f</i>(<i>u</i>), where <i>f</i> is a suitable function and <i>u</i> is the solution. In particular, when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(f(u)=u^p\)</EquationSource> </InlineEquation>, we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varphi (t)f(u)\)</EquationSource> </InlineEquation>.</p>

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Fujita exponent for heat equation with Hörmander vector fields

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月17日

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  • Marianna Chatzakou,
  • Aidyn Kassymov,
  • Michael Ruzhansky

摘要

In this paper, we show global existence and non-existence results for the heat equation with sum of the squares of smooth vector fields on \(\mathbb {R}^{n}\) satisfying Hörmander’s rank condition with a non-linearity of the form f(u), where f is a suitable function and u is the solution. In particular, when \(f(u)=u^p\) , we calculate the critical Fujita exponent. We also give necessary conditions for blow-up or, alternatively, a sufficient condition for the existence of positive global solutions for time-dependent nonlinearities of the type \(\varphi (t)f(u)\) .