On the asymptotics of global solutions of higher-order semilinear parabolic equations in the supercritical range

Abstract

We study the asymptotic behaviour of global bounded solutions of the Cauchy problem for the semilinear 2 m th order parabolic equation u t =−(− Δ ) m u +| u | p in R N × R + , where m &gt;1, p &gt;1, with bounded integrable initial data u 0 . We prove that in the supercritical Fujita range p &gt; p F =1+2 m / N any small global solution with nonnegative initial mass, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∫</mml:mo> <mml:msub> <mml:mi mathvariant="normal">u</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mspace width="1.69998pt"/> <mml:mi>dx</mml:mi> <mml:mi>⩾</mml:mi> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , exhibits as t →∞ the asymptotic behaviour given by the fundamental solution of the linear parabolic operator (unlike the case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">p</mml:mi> <mml:mo>∈</mml:mo> <mml:mspace width="1.69998pt"/> <mml:mo>]</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> <mml:msub> <mml:mi mathvariant="normal">p</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> where solutions can blow-up for any arbitrarily small initial data). A discrete spectrum of other possible asymptotic patterns and the corresponding monotone sequence of critical exponents <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> <mml:mo>/</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>l</mml:mi> <mml:mo>+</mml:mo> <mml:mi>N</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width="3.30002pt"/> <mml:mi>l</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>...</mml:mo> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , where p 0 = p F , are discussed.