Bebernes, Jerrold W.

English

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 6Jerrold W. BebernesA description of blow-up for the solid fuel ignition modelIn: Jaromír Vosmanský and Miloš Zlámal (eds.): Equadiff 6, Proceedings of the InternationalConference on Differential Equations and Their Applications held in Brno, Czechoslovakia,Aug. 26 - 30, 1985. J. E. Purkyně University, Department of Mathematics, Brno, 1986.pp. [193]--196.Persistent URL: http://dml.cz/dmlcz/700189Terms of use:© Masaryk University, 1986Institute of Mathematics of the Academy of Sciences of the Czech Republic provides accessto digitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czA DESCRIPTION OF BLOW-UP FOR THE SOLID FUEL IGNITION MODEL J. W. BEBERNES Department of Mathematics, University Colorado Boulder, CO 80309, U.S.A. The nondimensional ignition model for a supercritical high acti-vation energy thermal explosion of a solid fuel in a bounded container ft can be described by (1) u - Au = e (2) u(x,0) = <J)(x) __ 0, x £ ft, u(x,t) = 0, x € 9ft, t > 0 where ft = (x £ HR : |x| <_ R> and <j> is radially decreasing, i.e., <}>(x) :L <i> (y) > 0 whenever |x| <_ |y| <_ R and A<|) + e^ __ 0 on ft. Assume R > 0 is such that the radially symmetric solution u(x,t) blows up in finite time T > 0. Then by the maximum principle u(«,t) is radially decreasing for t £ [0,T) and u (x,t) __ 0 for all (x,t) € r = ft x [0,T) . Friedman and McLeod [4] recently proved that blow-up occurs only at the origin x = 0 and in addition that u(x,t) satisfies the fol-2 lowing estimates: I) u(x,t) <_ - — in | x | + c for all a < 1 and (x,t) 6 r; II) there exists t < T such that |Vu(x,t)| < 2 e u ( 0 ' t ) / / 2 , t 6 [t,T), |x| <_ R; III) there exists 6 > 0 such that u (x,t) __ 6 eu(x,t)^ t ^ [| , T ) , x € [- | , | ] ; and iv) -£n(T-t) < u(0,t) < -£n(T-t) - £nS, t € [| , T ) , 6 > 0. Since u(x,t) is radially symmetric, the initial boundary value problem (l)-(2) can be reduced to a problem in one spatial dimension. Let D = { ( r , t ) : 0 < _ t < _ T , 0 < _ r < _ R } . Then if r = | x | , v ( r , t ) = u(x,t) satisfies: /^ , n-1 , v (3) v. = v + v + e t rr r r (4) v(r,0) = (t)(r), vr(0) = 0, v(R,t) = 0. To study the asymptotic behavior of v as t -*• T, consider the fol--1/2 lowing change of variables: T = -£n(T-t), n = r(T-t) ' , 6 = v + -T -T/2 £n(T-t) = v - T whose inverse is t = T - e , r = ne , v = 8 -T /? Jln(T-T). The domain D transforms to D' = {(TI,T): 0<_r) <_ Re ' , T > -£nT} and 0 ( n , T ) = v - T solves 194 (5) 9 = 9 + ?2li - B e + e 6 - 1 v ' t nn [n 2} n (6) 9(n,-£nT) = <|)(nT1//2) + £nT 9 (0,T) = 0 , 9 (ReT//2,T) = -T The following theorem is similar to a result proven by Giga-Kohn [ 5 l. Theorem 1. As T ->• + °°, the solution 9(n,T) tends uniformly to a function y(n) on compact subsets of JR where y(n) is a solution of the problem: (7) y" + [ ^ - ajy + ey - 1 = 0 (8) y'(0) = 0, y(0) = a > 0 which is globally Lipschitz continuous and nonincreasing in n• Thus, to describe how the blow-up occurs at (T,0) for (l)-(2), we need to analyze the solutions of the steady-state equation (7)-(8) which are globally Lipschitz and are nonincreasing on [O,00). Theorem 2. For n = 1 or 2, the only solution of (7)-(8) which is globally Lipschitz continuous and nonincreasing in r\ is y(n) = 0. Proof. For n = 1, this result was first proven by Bebernes-Troy [2l . The following proof is essentially due to D.Eberly. For n > 2, the proof fails. Let g(n) = ^ y ' ( n ) + 1 and h ( n ) = y " ( n ) + — y ' ( n ) n where y ( n ) is a solution of ( 7 ) - ( 8 ) . Then g ( n ) satisfies , g " + {%-^-± - | ) g ' + (ey - l ) g = 0 (9 ) I [ g ( 0 ) = 1, g ' ( 0 ) = 0 and h ( n ) satisfies f h " + (̂ ----i - ^ ) h ' + (ey - l ) h < 0 n 2 (10 ) [ h ( o ) = 1 - e a , h ' ( o ) I t i s c l e a r t h a t g ( n ) > 0 on I = [ 0 , x j where x e (0,°°] S e t W(n) = g h ' - g ' h , t h e n W(n) s a t i s f i e s rW ' + {2—-± - | ) w = - ey ( y ' ) 2 g ( n ) < 0 ( l l ) l,W(0) = 0 195 on I. This implies W(n) < 0 on I and hence h(n)/g(n) < h(0)/g(0) = = 1 - ea on I. Thus, we have (12) h(n) < (1 - ea)g(n) on I. We now must consider two cases. We assume now that n = 1 or 2. a) If XQ < °°, then g(xQ) = 0 and g'- -?-g = - §ey < 0 implies g(n) < 0 for all n > nQ. Thus (ny')' = ng(n) - ney < 0 and y(n) is not globally Lipschitz on [ 0 ,°°) . b) If xQ = + °° and g(n) > e > 0 for all n > 0, then (ny' )' < 0 by (12) and again y" (n) < 0. If lim inf g(n) = 0 as n - °° with g(n) > 0, we observe that (11) can be solved for h(n) to give (13) h ( n ) = ( l - e a ) g ( n ) -2 ( x r11 1 e s / 4 , fs - u 2 / 4 y . , N 2 . . , , . - g ( n ) / ~5 ( / ue ' e J (y ) g ( u ) d u ) d s 0 gZ(s) s 0 By analyzing (13), we can show that h ( n ) - " - 0 0 a s n - + 00. Once again we have that y''(n) < 0 for n large and y(n) cannot be globally Lipschitz on [ 0 , ° ° ) . This completes the proof in dimensions 1 and 2. As an immediate consequence of theorems 1 and 2, we have Theorem 3. Let n = 1 or 2. As t - T~, v(r,t) - ln(T - t)" - 0 l j 2 uniformly on 0 < r < c(T - t) These results will appear in [ 3l . Several open questions remain. What can be said for n > 3? What 1/2 -happens outside the parabolic domain r < c(T-t) as t - T ? References [ ll J . B e b e r n e s and D.Kassoy, A mathe.matlcat anaty6l6 o£ btowup ion. the,n.mat n.e.actlon6 - thz tpatiatty monhoge.ne.ou6 cai>z, SIAM J . A p p l . Math. 40 ( 1 9 8 1 ) , 476 -484 . [ 2l J . B e b e r n e s and W.Troy, Mone.xl6te.ncz ion. the. Ka66oy pn.obte.rn, SIAM J . Math. A n a l y s i s , s u b m i t t e d . [ 3l J . B e b e r n e s , A . B r e s s a n and D . E b e r l y , A de.6cn,kptlon o& btoiA)-up ion. the. 6ottd iue.t Ignition mode.t, s u b m i t t e d . « [41 A.Fr iedman and B.McLeod, Btow-up oi po6ltlve. 6otutlon6 oi 6e.mltlne.an. he.at e.quatlon6, I n d i a n a Univ . Math. J . 34 (19 8 5 ) , 4 2 5 - 4 4 7 . 196 [5] Y.Giga and R.Kohn, Asymptotically szlí-similan blo^-up o<J s dmilinean. kdat equation*, Comm. Pure Appl. Math. 38 (19 85), 297-320. [6] D. Kassoy and J.Poland, Tkd tkdnmal explosion confined by a constant temp 2, statute, boundary:I. Tka induction period solution, SIAM J. Appl. Math. 39 (1980), 412-430. 

A description of blow-up for the solid fuel ignition model

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A description of blow-up for the solid fuel ignition model

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Institute of Mathematics AS CR, v. v. i.