With the help of caloric transformation, we construct nonzero solutions to the heat equation on the entire space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {R}^{n+1},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> which vanish (i.e., <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u\equiv 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>≡</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) on a family of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> or 2n hyperplanes, including the initial hyperplane <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t=0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> With the help of both caloric and Appell transformations, we also construct nonzero solutions to the heat equation on the space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R}^{n}\times \left( \mathbb {R}\backslash \left\{ 0\right\} \right) ,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo>×</mo> <mfenced close=")" open="("> <mrow> <mi mathvariant="double-struck">R</mi> <mo stretchy="true">\</mo> </mrow> <mfenced close="}" open="{"> <mn>0</mn> </mfenced> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> which are singular at <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$t=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and vanish on a family of infinitely many non-parallel hyperplanes converging to the hyperplane <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$t=0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Finally, we also discuss some interesting properties of caloric and Appell transformations.

Using caloric and Appell transformations to find nonzero solutions of the heat equation which vanish on hyperplanes

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2025年8月15日

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Dong-Ho Tsai

摘要

With the help of caloric transformation, we construct nonzero solutions to the heat equation on the entire space $\mathbb {R}^{n+1},$ $R^{n + 1},$ which vanish (i.e., $u\equiv 0$ $u \equiv 0$ ) on a family of $n+1$ $n + 1$ or 2n hyperplanes, including the initial hyperplane $t=0.$ $t = 0 .$ With the help of both caloric and Appell transformations, we also construct nonzero solutions to the heat equation on the space $\mathbb {R}^{n}\times \left( \mathbb {R}\backslash \left\{ 0\right\} \right) ,$ $R^{n} \times (R \, \{0\}),$ which are singular at $t=0$ $t = 0$ and vanish on a family of infinitely many non-parallel hyperplanes converging to the hyperplane $t=0.$ $t = 0 .$ Finally, we also discuss some interesting properties of caloric and Appell transformations.