BLOWUP PROPERTIES FOR A CLASS OF NONLINEAR DEGENERATE DIFFUSION EQUATION WITH NONLOCAL SOURCE*

Transcription

1 Applied Mamatics Mechanics (English Edition, Vol 24, No 11, Nov. 2003) Published by Shanghai University, Shanghai, China Article ID: (2003) BLOWUP PROPERTIES FOR A CLASS OF NONLINEAR DEGENERATE DIFFUSION EQUATION WITH NONLOCAL SOURCE* DENG Wei-bing (~q-j~), LIU Qi-lin (~]~1~;~), XIE Chun-hong (i~(]~) (Department Mamatics, Nanjing University, Nanjing , P.R. China) (Communicated by JIANG Fu-ru, Original Abstract Member Editorial Committee, AMM) The Abstract: one-dimensional A nonlinear problem degenerate parabolic motion equation a rigid flying nonlocal plate source under was explosive considered. attack has an analytic It was solution shown only under when certain polytropic assumption.q index solution detonation equation products blows equals up in to finite three. In general, time a numerical set analysis blowup is required. points is In this whole paper, region. however, The integral by utilizing method is "weak" used to shock behavior investigate reflection blowup shock properties in explosive solution. products, applying small parameter purterbation method, an analytic, first-order approximate solution is obtained problem flying plate driven Key by words: various degenerate high explosives equation; noniocal polytropic source; indices blowup or in finite than time; but nearly global equal blowup to three. Final velocities Chinese Library flying plate Classification obtained number: agree very O well numerical Document results code: by computers. A Thus an analytic 2000 mula Mamatics two Subject parameters Classification: high explosive 35K55; (i.e. 35K65 detonation velocity polytropic index) estimation velocity flying plate is established. Introduction Main Results Let 2 C R N be a bounded domain 1. Introduction smooth boundary 012. We set QT = t2 (0, TI, Sr = 00 x (0,T], Qt,r = 12 x (t,t], 0 < t < T < ao. Consider following Explosive driven flying-plate technique ffmds its important use in study behavior materials nonlinear under parabolic intense initial-boundary impulsive loading, value shock problem: synsis diamonds, explosive welding cladding metals. t The method estimation flyor velocity way raising it are questions., =.r( u + (x 12,t > 0), common interest. Under (1) u(x,t) assumptions = 0 one-dimensional (x plane 6 detonation o12,t > 0), rigid flying plate, normal approach solving u(~,0) problem u0(x) motion flyor (~ 6 is 12), to solve following system equations governing flow field detonation products behind flyor (Fig. I): where 0 < r < 1. Such problems model a variety physical phenomena, which arise, example, in study flow a fluid through a homogeneous isotropic rigid porous medium or in studies population dynamics ap (see +u_~_xp Refs. + [ 1,2 au ] ). Moreover, it has also been suggested nonlocal growth terms present a au more realistic au 1 model a population ( see Refs. [ 3,4 ] ). y =0, Over last few years, much eft has been devoted to study blow up properties nonlocal sernilinear parabolic equations as (see as Refs. [ 5,6,7,8 ] or ir references). In Ref. [ 5 ], Chadam et al. discussed case r = 0 ( 1). They proved solution blows up in finite time if where p, p, f(s) S, u are I> pressure, 0, f'(s) density, i> 0, specific f( s entropy ) is convex, particle I = f--~ velocity ds < ~, detonation products respectively, u0(x) is sufficiently trajectory large. R However, reflected shock blowup detonation properties wave are D as not a boundary well studied trajectory F flyor as anor boundary. Both are unknown; position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage motion flyor also; position F state parameters products Received date: ; Revised date: Biography: DENG Wei-bing (1971 ~ ), Doctor ( wbdeng@nju.edu.cn) 1362

2 Blowup Properties Nonlinear Degenerate Diffusion Equation 1363 degenerate nonlocal problem (see Ref. [ 31 ). In this paper, we inlxoduce nonlinear term u r, which corresponds to degenerate (slow)diffusion equation (see Ref. [31). By making some similar assumptions as Ref. [ 5 ], we get same blowup result. Furrmore, we prove solution ( 1 ) has global blowup ( i. e., blowup in whole domain), prile is unim in all compact subsets domain. These results give better improvements Ref. [5]. The method to discuss prile we utilize in this paper is different to Ref. [5] gives a development method Ref. [ 61 Bee stating our main results, we make some assumptions initial data u0 (x) f( s ) as follows : (H1) uo(x) E Ca(O), ~*0(x) I.~e~ > O, inuo(x) I.~6~a = OSuo/3v < O, x E 8,0, here ~ is outward normal vector on 3 O Abstract ; (H2) Au o + ~/(uo)dx >t O, x e O; The one-dimensional problem motion a rigid flying plate under explosive attack has an (H3) analytic f(s) solution 6 C([0, only w)) when N C'((0, polytropic w)) such index f detonation I> 0f' products I> 0on (0, equals w); to three. In general, a numerical analysis is required. In this paper, however, by utilizing "weak" shock behavior (tt4) f( s ) is convex, reflection shock in ds < explosive ~ products, applying small parameter purterbation method, an analytic, first-order approximate solution is obtained problem flying plate driven By (H3) by various (H4), high explosives we assertlimf(s) polytropic -- ~ indices or than but nearly equal to three. Final velocities flying plate obtained agree very well numerical results by computers. Thus lim f(s) - ~. (2) an analytic mula two parameters high explosive (i.e. detonation velocity polytropic index) The first estimation assertion is trivial. velocity To see flying second, plate is denote established. k -- limf ' ( s. By L' Hospital' s rule, it comes 1. Introduction lira f(s) - limf'(s) = k. Explosive driven flying-plate technique ffmds its important use in study behavior materials Now, if under k < intense, n impulsive re exists loading, an s0 shock > 0such synsis f(s) diamonds, ~< 2ks, s explosive I> So. welding Thus cladding metals. The method estimation flyor velocity way raising it are questions common interest. 'o f(s) >~ "0 2ks - ~' Under assumptions one-dimensional plane detonation rigid flying plate, normal which is a contradiction. approach solving problem motion flyor is to solve following system equations governing In our considerations flow field detonation a crucial products role is played behind by flyor f'~rst (Fig. eigenvalue I): Dirichlet problem - A~(x) = 2~(x) in ~, ~(x) = 0 ~n 90. ap Denote by,~1 first eigenvalue +u_~_xp + au by q~(x) corresponding eigenfunction normalization ~(x) > 0, in O x)dx = y 1. =0, Then, let us state our main results. as Theorem 1,1 Suppose Uo(X) f(s) satisfy (H1) ~ (H4). Then solution (1) blows up in finite time if u 0 (x) is sufficiently large. as where Theorem p, p, S, u are 1.2 pressure, Under density, assumptions specific entropy Theorem particle 1.1 velocity blowup set detonation (1) is products whole respectively, domain. trajectory R reflected shock detonation wave D as a boundary trajectory Theorem F flyor 1. 1 as shows anor boundary. if f( s)/s Both is slightly are unknown; superlinear position when s is R large, state example parameters on it are governed by flow field I central rarefaction wave behind detonation wave D f(s)/s by ~ initial e(lns) stage v, motion some s > flyor 0, p also; > 1, n position (1) has F solutions state parameters blow up in finite products time some initial data.

3 1364 DENG Wei-bing, LIU Qi-lin XIE Chun-hong 2 Local Existence Since u = 0, on boundary, equation (1) is not strictly parabolic type. The stard parabolic ory cannot be used directly to prove local existence solution to (1). To overcome this difficulty we will use stard regularization method ( see Ref. [ 9 t ) We first give a maximum principle, which is a conclusion Lemma 2.1 Ref. [ 8 ~. Lernma 2.1 Suppose w(x, t) E C 2'1 (Qr) ~ C( Qr) satisfies W t -- d(x,t)aw >I cl(x,t)w + c3(x't)fnc'(x't)w(x't)dx ((x,t) E QT), w(x,t) >. 0 ((x,t) E St); w(x,0) I>0 (xe ~), where cl ( x,t ),c2 ( x,t ),c3( x,t ) are bounded Abstract functions c~ (x, t ~ 0, c3 (x, t) >~ 0, d(x,t) > 0in QT. Thenw(x,t) The one-dimensional problem ~> 0on motion (Qr). a rigid flying plate under explosive attack has an analytic To show solution existence only when polytropic solution index Eq. (1), detonation consider products following equals regularization to three. In general, problems a : numerical [ analysis is required. In this paper, however, by utilizing "weak" shock behavior reflection shock in explosive products, applying small parameter puru. = (u~)~(~u~ terbation method, an analytic, first-order + ff(~, approximate )ax) solution (xe is obtained O,t >0), problem flying plate driven by various high explosives polytropic indices or than but nearly equal to three. u~(~,t) = e (x E ao,t >~ O), (3) Final velocities flying plate obtained agree very well numerical results by computers. Thus an analytic mula u~(x,o) two = parameters u o + e high explosive (i.e. (~ E detonation ~), velocity polytropic index) whereo < estimation e ~< 1. velocity flying plate is established. The following lemma gives local existence-uniqueness classical solution (3). Lemma 2.2 Suppose (H1) 1. Introduction (H3) hold. Then re exists a unique maximal in time functions u~ (x, t) ~> e, defined on ~ x [0, T~ ), some T~ E (0, ~ ], such all0 Explosive < r < T driven < T[, flying-plate u~ E C(QT) technique N C 2+~(Q~,T), ffmds its important somea use E'(0,1) in study u~ behavior is a classical materials under intense impulsive loading, shock synsis diamonds, explosive welding solution (3). cladding metals. The method estimation flyor velocity way raising it are questions common It can interest. be proved in much same way as pro Theorem A. 4 in Ref. [ 7 ]. We omit pro. Under assumptions one-dimensional plane detonation rigid flying plate, normal approach To problem solving (3), problem we give a comparison motion flyor principle, is to solve which can following be proved system by Lemma equations 2.1 governing easily, flow field detonation products behind flyor (Fig. I): Lemma2.3 Assume u~(x,t) E C-'l(Qr) N C(Qr) is a classical solution (3), nonnegative function w(x, t) E C2'1(Qr) N C( ap +u_~_xp + au QT), satisfies W t ~ (~)wr(~w+ fj(w)d~) y ((x,t) =0, E QT),~ w(~,t) >I (<<)e as as ((~,t) E s~), w(x,0) I> (~<)%(~) + e (~ E ~). Then w(x,t) >i (<~)u~(x,t) on p Qr- =p(p, s), Using Lemma 2.3, we can show u~ has following monotonicity respect to e. where p, p, S, u are pressure, density, specific entropy particle velocity detonation products respectively, Lemma 2.4 Suppose trajectory R (H1) reflected (H3) shock hold. detonation Let i ~> wave el ~ D e~ as > a boundary 0, u~, u~,. are trajectory solutions F (3) flyor as E anor 1 e~_, boundary. respectively. Both are Then unknown; u~ I> u~ on position (0, T "~, ) R T ~ ~< state T "~ para-. meters on it are governed by flow field I central rarefaction wave behind detonation wave D By by Lemma initial stage 2.4, it motion comes flyor u~ are also; monotone position respect F to e state parameters bounded from products below. So limit

4 Blowup Properties Nonlinear Degenerate Diffusion Equation exists, as well point-wise limit exists (x,t) E Qz, T < T* T~ _---- hmt~. u(x,t) - limu~(x,t), (4) To prove u ( x, t ) defined by (4) is a classical solution ( 1 ), we still need following lemma, which gives a unim lower bound to u~. Lemma2.5 Suppose (H1) (H3) hold. Let (x, t) = ke, t~(x), where k > 0 is small enough such k~(x) ~< uo(x), p =,~l(k + 1)". Then all0 < e ~< 1, solution (3) satisfies ue ( x, t) >I ( x, t) on (0, T~ ). Substituting ( x, t ) + e into (3), it comes result by Lemma 2.3 easily, we omit Abstract pro. The Then, one-dimensional by stard arguments problem (see Refs. motion [9,10]), a rigid it flying comes plate under u~ --~ explosive u unimly attack has an second analytic derivatives solution in only compact when subsets polytropic O index u ( x, t) detonation is continuous products at any equals point to ( y, three. t ), y In E general, 30 or t a = numerical 0, which analysis means is u required. is a classical In this solution paper, however, (1) on O by x utilizing [ 0, T ~ ). "weak" shock behavior reflection shock in explosive products, applying small parameter pur- The uniqueness can be proved by stard method ( see Ref. [ 11 ] ). terbation method, an analytic, first-order approximate solution is obtained problem flying plate driven Thus, by we various bave high explosives polytropic indices or than but nearly equal to three. Final velocities Theorem 2.1 flying Suppose plate obtained (H1) agree (H3) very hold. well Then numerical ( 1 ) has results a unique by computers. classical solution Thus an u analytic EC~"I(~ mula x (0,T ~)) two N parameters C(~ x [0, T*)). high explosive Moreover, (i.e. if detonation T* < ~ velocity nlimsupmaxu(x, polytropic index) estimation velocity flying plate is established. ~r" ~E-q t) Pro Main Results 1. Introduction Explosive By Lemma driven 2.1 flying-plate stard technique upper-lower ffmds its solution important method use It2] in, it comes study behavior materials Lemnm under 3.1 intense Suppose impulsive loading, (H1) ~ shock (H3) synsis hold, ue ( x, diamonds, t) is a classical explosive solution welding (3) on cladding x [0, T~* metals. ). Then The u~, method I> 0 in ~ estimation x (0, T~ ). flyor velocity way raising it are questions common interest. Under According assumptions to Theorem 2.1 one-dimensional Lemma 3.1 plane we detonation have following rigid conclusion. flying plate, normal approach Lemma solving 3.2 Suppose problem (H1) motion ~ (H3)hold. flyor is Then to solve classical following solution system u, (x, equations t ) (1) governing satisfies u~ I> flow 0 in field any compact detonation subsets products ~ behind x [0, T" flyor ). (Fig. I): Pro Theorem 1.1 Set H(t) = faul-'q~dx, n ap +u_~_xp + au y =0, 1 -H'(t) = Auq~dx + (u)dx = 1-r a as as --'I;/ u~gd-+ ~(f u~gd.), (5) where p, p, S, u are pressure, density, specific entropy particle velocity detonation products where respectively, C = max{ ~ (x)} trajectory we R have reflected used shock Jensen' detonation s inequality. wave By D (2) as a boundary it comes ~ED trajectory F flyor as anor boundary. Both are unknown; position R state parasomea > 2C~, re exists ans o > 0suchf(s) I> As, s ~ s o. Now, ifu oislarge meters on it are governed by flow field I central rarefaction wave behind detonation wave D enough by to initial satisfy stage fo u ~dx motion I> s o, it flyor comes also; fa uq~dx position >I s o F by u, ~> state 0. Then, parameters by (5), products it comes

5 1366 DENG Wei-bing, LIU Qi-lin XIE Chun-hong 1 -H'(t) + id(foo. ) 1 - r That is, Letv = ul-" in (6), it comes ftv,(x,t)q~dx >I 12-~rf(fj'(1-')~vdx). 1 Since ( 1 - r) > 1, by Jessen inequality, yields Abstract r \ 1/(a-r) 3 ~t~ v~dx 9 The one-dimensional problem motion a rigid flying plate under explosive attack has an analytic solution only when polytropic index detonation products equals to three. In if u0 is large enough to satisfy[ U a -- general, a numerical analysis required. roq~dx In this > paper, 1. Then, however, by (6), by utilizing yields "weak" shock 3 n behavior reflection shock in explosive products, applying small parameter pur- 1- r terbation method, an analytic, first-order approximate solution is obtained problem flying plate driven by various high explosives polytropic indices or than but nearly equal to three. Final which velocities comes, by flying integrating plate (7) obtained from agree 0 to T, very well numerical results by computers. Thus an analytic mula two parameters fh(r) high ds explosive k-s (i.e. detonation velocity polytropic index) estimation velocity flying H(O) f(s) plate is ~ established. zt~ T" That is, H'(t) >I :-~-c-f(h(t)), (7) 1. 2C Introduction 0 ds T ~ 1--Z~rJ.(o) f(~) < zo, (8) which Explosive means u driven blows up flying-plate finite time. technique ffmds its important use in study behavior materials In under next intense part, we impulsive denote loading, shock synsis diamonds, explosive welding cladding metals. The method estimation flyor velocity way raising it are questions common interest. g(t) = ft~f(u)dx' G(t)= flg(s)ds Under assumptions one-dimensional plane detonation rigid flying plate, normal approach Co, Ca, solving C',"" will problem denote various motion positive flyor constants. is to solve Moreover, following we assume system equations solution governing u (x, t ) ( flow 1 ) blows field up detonation in finite time products T" < behind w. The global flyor (Fig. blowup I): result can be derived by following orem. Theorem 3.1 Suppose Uo(X) f(s) satisfy (H1) ~ (H4). Then ap +u_~_xp + au ua-r(x,t)/(1 - r) I u(',t) I~-'/(1 - r) y =0, lim G ( t ) =,-r lim G(t) = 1 (9) t~ T' " converge unimly on compact subsets 2. as as In order to prove Theorem 3.1, we first derive a number preliminary facts on solution problem ( 1 ). The following lernma can be proved by sarr~ method Ref. [ 6, Lemma 4.1 ]. Lemma 3.3 Under assumptions Theorem 1.1, u~ ( x, t) is solution (3) on where x[0,t~'). p, p, S, u ThenAu~ are pressure, ~< density, C'in~ specific x [T[12, entropy T;). particle velocity detonation products respectively, It can be proved trajectory in much R same reflected way as shock pro detonation Lemma wave 4. D I in as a Ref. boundary [ 61. We omit trajectory F flyor as anor boundary. Both are unknown; position R state para pro. meters on it are governed by flow field I central rarefaction wave behind detonation wave D According by initial to stage Theorem motion 2.1 flyor Lemma also; 3.3, position we have F state parameters products Lemrna 3.4 Under assumptions Theorem 1.1, u ( x, t) is solution ( 1 ) on (6)

6 Blowup Properties Nonlinear Degenerate Diffusion Equation 1367 x[0,t "). ThenAu ~< Co in any compact subsets 2 x [T0,T ") some Co > 0 To E(0, T "). Lemma 3.5 Under assumptions Theorem 1.1, we have G ( t ) --~ w as t ~ T ~. Pro According to equation (1), we have u-% = A,, + g(t). (lo) Integrating (10) between To t E ( To, T ~ ) yields u~-r(x,t)/(1 - r) <~ ul-r(x,to)/(1 - r) + Au(x,s)ds + G(t). S' ro Hence, bylemma3.4, allx E O,T0 < t < T",wehave ul-r(x,t)/(1 -- r) <~ G(t) + C1, (11) where C1 = supul-~(x, T0)/(1 - r) + Abstract Co T ~ is bounded, which comes conclusion. [] Po Theorem 3.1 The one-dimensional problem motion a rigid flying plate under explosive attack has Define an analytic solution only when polytropic index detonation products equals to three. In general, a numerical analysis is required. In this paper, z(x,t) = G(t) however, r ul_r(x,t)" by utilizing "weak" shock behavior reflection shock in explosive products, applying small parameter purterbation method, an analytic, first-order approximate solution is obtained problem flying plate driven by various high explosives polytropic indices or than but nearly equal to three. Final velocities flying plate obtained fl(t) agree = faz(y't)q~(y)dy" very well numerical results by computers. Thus an analytic mula two parameters high explosive (i.e. detonation velocity polytropic By Green's mula, we bave index) estimation velocity flying plate is established. ~'(t) = (g(t)- u-rut)~(y)dy = I Au(y,t)q)(y)dy = 3 1. Introduction f u(y,t)qz(y)dy. (12) Explosive driven flying-plate technique ffmds its important use in study behavior Integrating (12) between 0 t yields materials under intense impulsive loading, shock synsis diamonds, explosive welding cladding metals. The method _< estimation c=<,+ 'flfau(y's)q)(y)dyds') flyor velocity way raising it are questions (13) common interest. By (11), Under we have assumptions one-dimensional plane detonation rigid flying plate, normal approach solving problem infz(x,t) motion >I- CI, flyor is to t solve E (To,T following ~). system equations (14) o governing flow field detonation products behind flyor (Fig. I): Denote ~5(t) 2 u(y,s)~(y)dyds. By (13) (14) it comes 0 0 ap +u_~_xp + au f Iz(y,t) lso(y)dy<~c3(l+@(t)), te(to,tx). (15) 0 Let Kp = {y ~ ~Q ;dist(y,3t-2) I> p}. We assert y - Az =0, <~ C4p -r. Indeed, by factinfq~ ~> Csp u o >I C(Kp)~ in Kp, we as have Uo as ~> 6'6to. Thus, u ~ C6p by u t I> 0. Hence, - Az =- ru -r-i I V u 12 + u-'au < C4p -r, where C4 = Cg r C0. Now, using I_emma 4.5 in Ref. [ 6, p.387 ], we get where p, p, S, u are pressure, density, C7 specific entropy particle velocity detonation products su~z(x,t) <. pn+l r(l + ~P(t)), t E (To,T~ ). respectively, trajectory R reflected shock detonation wave D as a boundary trajectory Combined F flyor ( 11 as ), anor it follows boundary. Both are unknown; position R state parameters on it are governed by flow field I central rarefaction wave behind detonation wave D by initial stage C1 motion 1/(1 flyor r)u also; 1-r position C7 (1 + F ff)(t)) state parameters products G(t) <~ 1 - t,k-~---et) <~ pn+1+~ G(t) ' (16)

7 1368 DENG Wei-bing, LIU Qi-lin XIE Chun-hong tbrxinkp t ~ (T0,T *). We assert ( 1 + ( t ) ) / G ( t ) --~ 0 as t --~ T". Indeed, by convex nondecreasing assumptions f, we have 1 ' 1 ' f(t f u( y,s)9( y)dyds) <~ t f J(u(y,s ) )9( y)dy ds ~ CG(t), where C = max { 9 ( x ) } ( t ) <~ t2 if- 1 ( CG ( t ) / t ), which comes q~(t) f-x(cg(t)/t) G(t) ~ C/t1 CG(t)/t --~0 as G (t) --~ ~ (namely, t --~ T" ). Indeed, by (H3) (H4), non-decreasing also lirnf -1 ( s ) = w. Hence f-l(s) (Yy) lim Abstract - hm = O. s~ S f it is obvious f-l( s ) is The So, one-dimensional (16) implies (9). problem Then we complete motion a pro. rigid flying plate under explosive attack has [] an analytic The results solution Theorem only when 3.1 show polytropic asymptotic index detonation behavior is products mally equals given to be three. balancing In general, u-ru, a numerical nonlocal analysis source is required. term g(t), In this resulting paper, in however, a flat blowup by utilizing prile, which "weak" has shock been behavior reflection shock in explosive products, applying small parameter purasserted in Ref. [ 6 ] semilinear parabolic equations. terbation method, an analytic, first-order approximate solution is obtained problem flying By Theorem 3.1 Lemma 3.5, it comes Theorem 1.2 easily. plate driven by various high explosives polytropic indices or than but nearly equal to three. Final velocities flying plate obtained agree very well numerical results by computers. Thus References: an analytic mula two parameters high explosive (i.e. detonation velocity polytropic index) [ 1 ] CantreU estimation R S, Consner velocity C. Diffusive flying logistic plate equations is established. indef'mite weights:population models in disrupted environments II [J]. SlAM J Math Anal, 1991,22(4) : [2 ] Diaz J I, Kersner R. On a nonlinear 1. Introduction degenerate parabolic equation in infiltration or evaporation through a porous medium[ J]. J Differential Equations, 1987,69 (3) : [ 3 ] Explosive Anderson driven J R, Deng flying-plate K. Global technique existence ffmds degenerate its important parabolic use equations in study a nonlocal behavior cing materials [J]. under Math intense Mech impulsive Appl Sci, 1997,20(13) loading, shock : 1069 synsis diamonds, explosive welding cladding [ 4 ] Furter metals. J, Grinfeld The method M. Local vs. estimation non-local interactions flyor velocity in population way dynamics[ raising J]. it J are Math questions Biology, common 1989,27(1) interest. : [ 5 ] Under Chadam J assumptions M, Peirce A, Yin one-dimensional H M. The blow-up plane property detonation solutions rigid to flying some plate, diffusion normal equations approach loca~zed solving nonlinear problem reactions motion [ J ]. J Math flyor Anal is Appl, to solve 1992,169 following (2) : 313 system equations governing [ 6 ] Souplet flow Ph. field Unim detonation blow-up priles products behind boundary flyor behavior (Fig. I): diffusion equations nonlocal nonlinear source [ J ]. J Differential Equations, 1999,153 (2) : [ 7 ] Souplet Ph. Blow up in nonlocal reaction-diffusion equations[ J]. SIAM J Math Anal, 1998,29(6) : ap +u_~_xp + au [ 8 ] WANG Ming-xin, WANG Yuan-ming. Properties y positive =0, solutions non-local reaction-diffusion problems[j]. Math Mech Appl Sci,1996,19(4) : [ 9 ] WANG Shu, WANG Ming-xin, as XIE Chun-hong. as Nonlinear degenerate diffusion equation not in di vergence m[ J]. Z Angew Math Phys, 2000,51( 1 ) : [ 10] Friedman A, McLeod B. Blow-up p =p(p, solutions s), nonlinear degenerate parabolic equations [ J]. Arch Rational Mech Anal, 1987,96(1) : where p, p, S, u are pressure, density, specific entropy particle velocity detonation products [ 11 ] Anderson J R. Local existence uniqueness solutions degenerate parabolic equations[ J] respectively, trajectory R reflected shock detonation wave D as a boundary Commun Partial Differential Equations, 1991,16( 1 ) : trajectory F flyor as anor boundary. Both are unknown; position R state parameters [ 12] on Pao it C are V. governed Nonlinear by Parabolic flow field Elliptic I central Equations[ rarefaction M]. New wave York:Plenum behind detonation Press, wave D by initial stage motion flyor also; position F state parameters products