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<p>For superlinear heat equations with the Dirichlet boundary condition, the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation> estimates of radially symmetric solutions are studied. In particular, the uniform boundedness of global solutions and the non-existence of solutions with type II blow-up are proved. For the space dimension greater than 9, our results are shown under the condition that an exponent representing the growth rate of a nonlinear term is between the Sobolev exponent and the Joseph-Lundgren exponent. In the case where the space dimension is greater than 2 and smaller than 10, our results are applicable for nonlinear terms growing extremely faster than the exponential function.</p>

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Uniform boundedness and blow-up rate of solutions in non-scale-invariant superlinear heat equations

  • 期刊论文

  • 发表日期：

    2025年5月5日

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  • Yohei Fujishima,
  • Toru Kan

摘要

For superlinear heat equations with the Dirichlet boundary condition, the \(L^\infty \) L ∞ estimates of radially symmetric solutions are studied. In particular, the uniform boundedness of global solutions and the non-existence of solutions with type II blow-up are proved. For the space dimension greater than 9, our results are shown under the condition that an exponent representing the growth rate of a nonlinear term is between the Sobolev exponent and the Joseph-Lundgren exponent. In the case where the space dimension is greater than 2 and smaller than 10, our results are applicable for nonlinear terms growing extremely faster than the exponential function.