Infinite-time blow-up for the 3-dimensional energy-critical heat equation

Abstract

We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension 3\n¶\n<math display="block">\n<mrow>\n<msub>\n<mrow>\n<mi>u</mi>\n</mrow>\n<mrow>\n<mi>t</mi>\n</mrow>\n</msub>\n<mo class="MathClass-rel">=</mo>\n<mi mathvariant="normal">Δ</mi>\n<mi>u</mi>\n<mo class="MathClass-bin">+</mo>\n<msup>\n<mrow>\n<mi>u</mi>\n</mrow>\n<mrow>\n<mn>5</mn>\n</mrow>\n</msup>\n<mspace class="quad" width="1em"/>\n<mtext> in </mtext>\n<msup>\n<mrow>\n<mi>ℝ</mi>\n</mrow>\n<mrow>\n<mn>3</mn>\n</mrow>\n</msup>\n<mo class="MathClass-bin">×</mo>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mn>0</mn>\n<mo class="MathClass-punc">,</mo>\n<mi>∞</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-punc">,</mo>\n<mspace class="qquad" width="2em"/>\n<mi>u</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>x</mi>\n<mo class="MathClass-punc">,</mo>\n<mn>0</mn>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">=</mo>\n<msub>\n<mrow>\n<mi>u</mi>\n</mrow>\n<mrow>\n<mn>0</mn>\n</mrow>\n</msub>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>x</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mspace class="quad" width="1em"/>\n<mtext> in </mtext>\n<msup>\n<mrow>\n<mi>ℝ</mi>\n</mrow>\n<mrow>\n<mn>3</mn>\n</mrow>\n</msup>\n<mo class="MathClass-punc">.</mo>\n</mrow>\n</math>\n¶ For each [math] we find initial data (not necessarily radially symmetric) with [math] such that as [math]\n¶\n<math display="block">\n<mrow>\n<mo class="MathClass-rel">∥</mo>\n<mi>u</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mo class="MathClass-bin">⋅</mo>\n<mo class="MathClass-punc">,</mo>\n<mi>t</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<msub>\n<mrow>\n<mo class="MathClass-rel">∥</mo>\n</mrow>\n<mrow>\n<mi>∞</mi>\n</mrow>\n</msub>\n<mo class="MathClass-rel">∼</mo>\n<msup>\n<mrow>\n<mi>t</mi>\n</mrow>\n<mrow>\n<mi>γ</mi>\n<mo class="MathClass-bin">−</mo>\n<mfrac>\n<mrow>\n<mn>1</mn>\n</mrow>\n<mrow>\n<mn>2</mn>\n</mrow>\n</mfrac>\n</mrow>\n</msup>\n<mspace class="quad" width="1em"/>\n<mtext> if </mtext>\n<mn>1</mn>\n<mo class="MathClass-rel"><</mo>\n<mi>γ</mi>\n<mo class="MathClass-rel"><</mo>\n<mn>2</mn>\n<mo class="MathClass-punc">,</mo>\n<mspace class="qquad" width="2em"/>\n<mo class="MathClass-rel">∥</mo>\n<mi>u</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mo class="MathClass-bin">⋅</mo>\n<mo class="MathClass-punc">,</mo>\n<mi>t</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<msub>\n<mrow>\n<mo class="MathClass-rel">∥</mo>\n</mrow>\n<mrow>\n<mi>∞</mi>\n</mrow>\n</msub>\n<mo class="MathClass-rel">∼</mo>\n<msqrt>\n<mrow>\n<mi>t</mi>\n</mrow>\n</msqrt>\n<mspace class="quad" width="1em"/>\n<mtext> if </mtext>\n<mi>γ</mi>\n<mo class="MathClass-rel">></mo>\n<mn>2</mn>\n<mo class="MathClass-punc">,</mo>\n<mspace class="qquad" width="2em"/>\n<mo class="MathClass-rel">∥</mo>\n<mi>u</mi>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mo class="MathClass-bin">⋅</mo>\n<mo class="MathClass-punc">,</mo>\n<mi>t</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<msub>\n<mrow>\n<mo class="MathClass-rel">∥</mo>\n</mrow>\n<mrow>\n<mi>∞</mi>\n</mrow>\n</msub>\n<mo class="MathClass-rel">∼</mo>\n<msqrt>\n<mrow>\n<mi>t</mi>\n</mrow>\n</msqrt>\n<msup>\n<mrow>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mo class="qopname">ln</mo>\n<mi>t</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n</mrow>\n<mrow>\n<mo class="MathClass-bin">−</mo>\n<mn>1</mn>\n</mrow>\n</msup>\n<mspace class="quad" width="1em"/>\n<mtext> if </mtext>\n<mi>γ</mi>\n<mo class="MathClass-rel">=</mo>\n<mn>2</mn>\n<mo class="MathClass-punc">.</mo>\n</mrow>\n</math>\n¶ Furthermore we show that this infinite-time blow-up is codimensional-1 stable. The existence of such solutions was conjectured by Fila and King (Netw. Heterog. Media 7:4 (2012), 661–671).

Topics & Concepts