This paper investigates the asymptotic behavior of solutions to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u_t=\Delta u+|u|^{p-1}u$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>+</mo> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mi>u</mi> </mrow> </math></EquationSource> </InlineEquation> in the Sobolev critical case. Our main result is a classification of the dynamics near the ground states in the six dimensional case. It is shown that if the initial data <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u_0\in H^1({\mathbb R}^6)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <mo>∈</mo> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>6</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Vert u_0-\textsf{Q}\Vert _{\dot{H}^1({\mathbb R}^6)}\ll 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">‖</mo> </mrow> <msub> <mi>u</mi> <mn>0</mn> </msub> <msub> <mrow> <mo>-</mo> <mi mathvariant="sans-serif">Q</mi> <mo stretchy="false">‖</mo> </mrow> <mrow> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>6</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </msub> <mo>≪</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, then the solution falls into one of the following three scenarios: <OrderedList> <ListItem> <ItemNumber>1)</ItemNumber> <ItemContent> It is globally defined and converge to one of the ground states as <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. </ItemContent> </ListItem> <ListItem> <ItemNumber>2)</ItemNumber> <ItemContent> It is globally defined and converge to 0 in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\dot{H}^1({\mathbb R}^6)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mover accent="true"> <mi>H</mi> <mo>˙</mo> </mover> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>6</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$t\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. </ItemContent> </ListItem> <ListItem> <ItemNumber>3)</ItemNumber> <ItemContent> It exhibits finite time blowup with a type I rate. </ItemContent> </ListItem> </OrderedList> This paper extends the classification result in the case <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$n\ge 7$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>7</mn> </mrow> </math></EquationSource> </InlineEquation>, previously obtained by Collot-Merle-Raphaël, to the borderline case <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n=6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>.

Dynamics near the ground state for the Sobolev critical Fujita equation in 6D

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2026年4月29日

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Junichi Harada

摘要

This paper investigates the asymptotic behavior of solutions to $u_t=\Delta u+|u|^{p-1}u$ $u_{t} = Δ u + {| u |}^{p - 1} u$ in the Sobolev critical case. Our main result is a classification of the dynamics near the ground states in the six dimensional case. It is shown that if the initial data $u_0\in H^1({\mathbb R}^6)$ $u_{0} \in H^{1} (R^{6})$ satisfies $\Vert u_0-\textsf{Q}\Vert _{\dot{H}^1({\mathbb R}^6)}\ll 1$ $‖ u_{0} {- Q ‖}_{{\dot{H}}^{1} (R^{6})} ≪ 1$ , then the solution falls into one of the following three scenarios: 1)

It is globally defined and converge to one of the ground states as $t\rightarrow \infty $ $t \to \infty$ .

It is globally defined and converge to 0 in $\dot{H}^1({\mathbb R}^6)$ ${\dot{H}}^{1} (R^{6})$ as $t\rightarrow \infty $ $t \to \infty$ .

It exhibits finite time blowup with a type I rate.

This paper extends the classification result in the case $n\ge 7$

n \geq 7

, previously obtained by Collot-Merle-Raphaël, to the borderline case $n=6$

n = 6