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" Blowup in nonlinear heat equations "
Shuangcai Wang
I. M. Sigal
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Latin Dissertation
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| Language of Document
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English
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| Record Number
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55502
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| Doc. No
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TL25456
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| Call number
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3265647
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| Main Entry
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Shuangcai Wang
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| Title & Author
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Blowup in nonlinear heat equations\ Shuangcai Wang
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| College
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University of Notre Dame
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| Date
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2006
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| Degree
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Ph.D.
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2006
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| Page No
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70
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| Abstract
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In this paper we study the blowup problem for nonlinear heat equation [Special characters omitted.] <display-math> <fd> <fl>u<inf>t</inf>=<g>6</g><sup>2</sup><inf>u</inf>u+u<sup> p-1</sup>u</fl> <fl>u<fen lp="par">x,0<rp post="par"></fen>=u<inf>0</inf><fen lp="par"> x<rp post="par"></fen><hsp sp="0.212"><hsp sp="2.000"><hsp sp="0.265"> <hsp sp="0.265"><hsp sp="0.265"><hsp sp="2.000"><fen lp="par"> 1<rp post="par"></fen></fl> </fd> </display-math> where u (t ) : x ∈ [Special characters omitted.] <math> <f> <sc><blkbd>R</blkbd></sc></f> </math> [arrow right] u (x, t ) ∈ [Special characters omitted.] <math> <f> <sc><blkbd>R</blkbd></sc></f> </math> and p > 1. We show that if the initial data is close enough to a 2-dimensional manifold of approximately homogenous solutions, the solution blows up in a finite time and the asymptotical profile is an approximate solution with parameters evolving according to a certain dynamical system plus a small fluctuation in L∞ . Main Theorem . Assume the initial data u 0 ∈ L∞ ([Special characters omitted.] <math> <f> <sc><blkbd>R</blkbd></sc></f> </math> ) is even and satisfies [Special characters omitted.] <display-math> <fd> <fl><fen lp="vb" style="d">u<inf>0</inf><fen lp="par">x<rp post="par"></fen> -<fen lp="par"><fr><nu>2c<inf>0</inf></nu><de>p-1+b<inf>0</inf> x<sup>2</sup></de></fr><rp post="par"></fen><sup><fr><nu>1</nu> <de>p-1</de></fr></sup><rp post="vb" style="d"></fen><inf> </inf><g>d</g><inf>0</inf>c<sup><fr><nu>1</nu><de>p-1</de></fr> </sup><inf>0</inf></fl> <fl><fen lp="vb" style="d"><fen lp="ang">x<rp post="ang"></fen><sup> -3</sup><fen lp="sqb">u<inf>0</inf><fen lp="par">x<rp post="par"></fen> -<fen lp="sqb"><fr><nu>2c<inf>0</inf></nu><de>p-1+b<inf>0</inf> x<sup>2</sup></de></fr><rp post="sqb"></fen><sup><fr><nu>1</nu> <de>p-1</de></fr></sup><rp post="sqb"></fen><rp post="vb" style="d"></fen><inf> </inf>b<sup>2</sup><inf>0</inf>c<sup>-<fr><nu>3</nu> <de>2</de></fr>+<fr><nu>1</nu><de>p-1</de></fr></sup><inf>0</inf> </fl> </fd> </display-math> where [left angle bracket]x [right angle bracket] = [Special characters omitted.] <math> <f> <rad><rcd>1+x<sup>2</sup></rcd></rad></f> </math> with b0 ≥ 0 and c 0 > 0 constant. If 0 ≤ [Special characters omitted.] <math> <f> <fr><nu>b<inf>0</inf></nu><de>c<inf>0</inf></de></fr></f> </math> « 1 and 0 ≤ δ0 « 1, then (1) There exists a finite time t * ∈ (0, ∞) such that the solution u (x, t ) blows up at t = t *. (2) When t < t *, there exist unique positive, C 1 functions λ(t), b ( t ) and c (t ) with b ( t ) [Special characters omitted.] <math> <f> </f> </math> b (0) such that λ(0) = [Special characters omitted.] <math> <f> <rad><rcd>2c<inf>0</inf>+<fr><nu>2</nu><de>p-1</de></fr>b<inf> 0</inf></rcd></rad></f> </math> and u (x, t ) can be decomposed as [Special characters omitted.] <display-math> <fd> <fl>u<fen lp="par">x,t<rp post="par"></fen>=<g>l</g><sup><fr><nu> 2</nu><de>p-1</de></fr></sup><fen lp="par">t<rp post="par"></fen> <fen lp="sqb"><fen lp="par"><fr><nu>2c<fen lp="par">t<rp post="par"></fen> </nu><de>p-1+b<fen lp="par">t<rp post="par"></fen><g>l</g><sup> 2</sup><fen lp="par">t<rp post="par"></fen>x<sup>2</sup></de></fr> <rp post="par"></fen><sup><fr><nu>1</nu><de>p-1</de></fr></sup> +<g>h</g><fen lp="par">x,t<rp post="par"></fen><rp post="sqb"></fen> </fl> </fd> </display-math> with the fluctuation part , η, admitting the estimate ||[left angle bracket]λ(t ) x [right angle bracket]-3 η(x, t )|| ∞ ≤ Cb2 (t ). (3) As t [arrow right] t *, λ( t ), a (t ), b ( t ) and c (t ) have the following approximations , [Special characters omitted.] <display-math> <fd> <fl><g>l</g><fen lp="par">t<rp post="par"></fen>=<g>l</g><sup> -1</sup><inf>0</inf><fen lp="par">t<sup>*</sup>-t<rp post="par"></fen><sup> -<fr><nu>1</nu><de>2</de></fr></sup><fen lp="par">1+o<fen lp="par"> 1<rp post="par"></fen><rp post="par"></fen></fl> <fl>b<fen lp="par">t<rp post="par"></fen>=-<fr><nu><fen lp="par"> p-1<rp post="par"></fen><sup>2</sup></nu><de>4p<rm>ln<fen lp="par"> <it>t<sup>*</sup>-t</it><rp post="par"></fen></rm></de></fr><fen lp="sqb"> 1+O<fen lp="par"><fen lp="vb"><fr><nu>1</nu><de><rm>ln<fen lp="par"> <it>t<sup>*</sup>-t</it><rp post="par"></fen></rm></de></fr><rp post="vb"></fen><sup> 1/2</sup><rp post="par"></fen><rp post="sqb"></fen></fl> <fl>c<fen lp="par">t<rp post="par"></fen>=<fr><nu>1</nu><de>2 </de></fr>-<fr><nu>p-1</nu><de>4p<fen lp="vb"><rm>ln<fen lp="par"> <it>t<sup>*</sup>-t</it><rp post="par"></fen></rm><rp post="vb"></fen> </de></fr><fen lp="sqb">1+O<fen lp="par"><fr><nu>1</nu><de><rm> ln<fen lp="par"><it>t<sup>*</sup>-t</it><rp post="par"></fen> </rm></de></fr><rp post="par"></fen><rp post="sqb"></fen>.<hsp sp="0.212"> <hsp sp="0.265"><hsp sp="0.265"><hsp sp="0.265"><hsp sp="0.265"> <fen lp="par">2<rp post="par"></fen></fl> </fd> </display-math> with λ0 = [Special characters omitted.] <math> <f> <rad><rcd>2c<inf>0</inf>+<fr><nu>2</nu><de>p-1</de></fr>b<inf> 0</inf></rcd></rad></f> </math> . Furthermore, we have not only the asymptotic expressions for the parameters b and c determining the leading term and the size of the reminder, but also the dynamical equations for the parameters. Finally, we point out that we can easily construct an initial data with more than one local maximum while it still satisfies our constraints. So we have a completely different result from that by Herrero and Velazquez. We also demonstrated that our result is not same as the theorem proved by Bricmont and Kupiainen.
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Pure sciences; Blowup; Nonlinear heat equations; Partial differential equations; Asymptotic stability; Self-similar; Mathematics; 0405:Mathematics
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I. M. Sigal
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University of Notre Dame
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