<p>We consider the energy critical four dimensional semi-linear heat equation <Equation ID="Equ267"> <EquationSource Format="TEX">$ \partial _{t}v-\Delta v-v^{3}=0, \quad (t,x)\in \mathbb {R}\times \mathbb {R}^4. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>v</mi> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>-</mo> <msup> <mi>v</mi> <mn>3</mn> </msup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Formal computation of Filippas et al. (R. Soc. Lond. Proc. 2000) conjectures the existence of a sequence of type II blow-up solutions with various blow-up rates <Equation ID="Equ268"> <EquationSource Format="TEX">$ \Vert v(t)\Vert _{L^\infty (\mathbb {R}^4)}\approx \frac{|\log (T-t)|^{\frac{2L}{2L-1}}}{(T-t)^L} ,\quad L=1,2,\cdots .$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mrow> <mo stretchy="false">‖</mo> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≈</mo> <mfrac> <msup> <mrow> <mo stretchy="false">|</mo> <mo>log</mo> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>-</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mfrac> <mrow> <mn>2</mn> <mi>L</mi> </mrow> <mrow> <mn>2</mn> <mi>L</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> </msup> <msup> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>-</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>L</mi> </msup> </mfrac> <mo>,</mo> <mspace width="1em" /> <mi>L</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>⋯</mo> <mo>.</mo> </mrow> </math></EquationSource> </Equation>Schweyer (J. Funct. Anal. 2012) rigorously constructs a type II blow-up solution for the case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we show the existence of type II blow-up solution for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>L</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. It is inspired by Raphaël and Schweyer’s work [<CitationRef CitationID="CR38">38</CitationRef>, <CitationRef CitationID="CR39">39</CitationRef>] on quantized slow blow-up harmonic maps and Schweyer’s work [<CitationRef CitationID="CR40">40</CitationRef>] on heat equations.</p>

A slow blow-up solution for the four dimensional energy critical semilinear heat equation

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2026年4月1日

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Tongtong Li,
Liming Sun,
Shumao Wang

摘要

We consider the energy critical four dimensional semi-linear heat equation $ \partial _{t}v-\Delta v-v^{3}=0, \quad (t,x)\in \mathbb {R}\times \mathbb {R}^4. $ $\partial_{t} v - Δ v - v^{3} = 0, (t, x) \in R \times R^{4} .$ Formal computation of Filippas et al. (R. Soc. Lond. Proc. 2000) conjectures the existence of a sequence of type II blow-up solutions with various blow-up rates $ \Vert v(t)\Vert _{L^\infty (\mathbb {R}^4)}\approx \frac{|\log (T-t)|^{\frac{2L}{2L-1}}}{(T-t)^L} ,\quad L=1,2,\cdots .$ ${‖ v (t) ‖}_{L^{\infty} (R^{4})} \approx \frac{{| log (T - t) |}^{\frac{2 L}{2 L - 1}}}{{(T - t)}^{L}}, L = 1, 2, \dots .$ Schweyer (J. Funct. Anal. 2012) rigorously constructs a type II blow-up solution for the case $L=1$ $L = 1$ . In this paper, we show the existence of type II blow-up solution for $L=2$ $L = 2$ . It is inspired by Raphaël and Schweyer’s work [38, 39] on quantized slow blow-up harmonic maps and Schweyer’s work [40] on heat equations.