Uniqueness of very singular self-similar solution of a quasilinear degenerate parabolic equation with absorption

Publicacions Ma emá iques, Vol 36 (1992), 19-38 . Abs ac UNIQUENESS OF VERY SINGULAR SELF-SIMILAR SOLUTION OF A QUASILINEAR DEGENERATE PARABOLIC EQUATION WITH ABSORPTION J .I . DIAZ * AND J .E . SAA * We show he uniqueness o he e y singula sel -simila solu ion o he equa ion 7 - Op76 m -{- Y6 9 = 0 . The esul is ca ied ou by s udying he s a iona y associa e equa- ion and by in oducing a sui able chango o unknown . Tha allows o assume he ze o-o de pe u ba ion e m in he new equa ion o be mono one inc easing . A ca e ul s udy o he beha iou o solu ions nea he bounda y o hei suppo is also used in o de o p o e he main esul . 1 . In oduc ion The main goal o his pape is o show he uniqueness o solu ions o he ollowing quasilinea ellip ic p oblem CU7 2 u') _  (1V - I) (1)  -  -  M  ju in-2u1 + h (x, u, u') = 9(x, u),  x > 0, (2)  ú (0) = 0,  lim u(x) = 0 x-oo U(X) > 0  (5 1- 0) *Pa ially suppo ed by he DGICYT p ojec n° PB90/0620 . 20  J .I . DIAZ, J .E . San in which p > 1 and he unc ions h and g sa is y ce ain s uc u al condi ions which will be made explici la e . The main mo i a ion o he conside a ion o such a p oblem comes om he s udy o e y singula solu ions o he quasilinea degene a e pa abolic equa ion wi h abso p ion (4)  U = A p U M -U q  in  Q = RN x (0, oo), whe e, as usual, O p u deno es he p-Laplacian ope a o N >_ 1 and m and q a e nonnega i e eal numbe s . Equa ion (4) con ains, as special cases, he equa ions and O p = di (1wIp -2 w),  1 <p< oo, U = DUm _ Uq (6)  u = A P U -u q which ha e been in ensi ely s udied in he las yea s . Fo many di e en pu poses i is in e es ing o s udy singula solu ions o (4) Le . nonneg- a i e unc ions u sa is ying (4) in Q (in he sense o dis ibu ions) and such ha u(x, 0) = 0 i x E RN- {0} . In many cases, he singula i y a = 0 o such a solu ion inus be as ha o he undamen al solu ion Le . u(x,0) = cb(x) o some posi i e cons an e, o , in o he wo ds, lim  u(x, ) dx = e -0 jxj< o any > 0 . Ne e heless, when he abso p ion is s ong enough wi h espec o he di usion, he e exis s ano he ype o singula solu ion u called as e y singula solu ion which has been disco e ed p e iously in he ollowing cases : a) equa ion (5) wi h m = 1 and 1 < q < 1 + (2/N) : B ezis, Pele ie and Te man [1] b) equa ion (5) wi h m > 1 and m < q < m + (2/N) : Pele ie and Te man [7j c) equa ion (6) wi h p > qand p - 1 < q<p - 1 + (p/N) : Pele ie and Wang [S] . UNIQUENESSOFVERY SINGULARSOLUTION  21 In all hose cases his new singula solu ion sa is ies (8)  lim  u(x, ) dx = +oo --~o 1x1< o any > 0 and so i is mo e singula han he undamen al solu ion . As usual, he exis en e o a e y singula solu ion is ob ained in he caass o sel -simila solu ions (g)  W(x ) = -1 /(q -1 ) Ox1l 1/Q) whe e ,0 mus be sui able chosen . Fo ins an e Q = 2(q - 1)/(q- m) and / .i = p(q- 1)/(q+ 1 - p) in he cases o equa ions (5) and (6) espec i ely ( ecall ha q > p - 1) . Mo e gene ally, we can conside sel -simila solu ions W o he equa ion (4) in which case he na u al choice o ,0 is ( 10 )  Q= p(q - 1)1(q - m(p - 1)) A unc ion W gi en by (8) is hen a e y singula solu ion i sa is ies (11) ~( )'1 P-2 ( )')'+ (Nx 1)  P-2( m)/+1x '+ (q 1 1) - q = (12)  > 0 in (0, oo) ( 13 )  '(0) = 0,  li a 71 1 /(q - m(p - 1)) ( i) = o The uniqueness o solu ion o (11) (12) (13) was only gi en o he case m = 1 and p = 2, (see [1J) and was le open in [7] and [8J . The main goal o ou wo k is o gi e an uniqueness esul ue o any alue o m and p . In oducing = m , we ema k ha sa is ies an equa ion o he ype (1) wi h and g(x~ u) = -uq/7n +  1  uI/m (q - 1) 1  _ mm>> h(x, u) _- Oxu  u' m = 0, in (0,oo) So g(x, u) is no mono one in u . Mo eo e he di e en ial e ms in equa- ion (11) may Na e di e en homogenei y (m(p - 1) and 1 espec i ely) which leads e some special di icul ies (solu ions wi h compac .suppo i m(p - 1) > 1, e c) . 22  J .I . DIAZ, J .E . Sna 2 . The main esul s We shall p o e he uniqueness o solu ions o he p oblem ( 14 ) I(Um)/IP-2(um)')'+ (Nx 1)I(um)~Ip_2(Um)'+ ~xu'+G(u) =0, (15)  u(x) > 0  (y- 0) (16)  (u'')'(0) = 0,  lim u(x) = 0 whe em>0,p>1,N>1  Q>0and G(u) =  1  u- u9 . q-1 Fo some alues o m and p p oblem (14) (15) (16) does no ha e any classical solu ion and i mus be sol ed in a gene alized way . This is he case when m(p - 1) > 1 because he solu ions ha e as suppo a compac in e al [0, xo] and u' may be discon inuous a x = xo (see pa ( ) o Lemma 1) . To de ine he no ion o weak solu ion we .mul iply he equa ion (14) by a smoo h es unc ion ~(x) wi h compac suppo in [0, oo) bu no necessa ily anishing a x = 0 . By mul iplying by x N -1 and in eg a ing by pa s we ob ain (17)  -  ~ x N 1I(um)'IP-2(Um)'~' d .T - 1 1 . XNu~' dx+ o  p o + ~~ x N-1 (G(U) - Nul  dx = 0 o  Q On he o he hand, by s anda d egula i y esul s, i is clea ha u E C ° ([0, oo)) and ha in ac u E C 2 on he se whe e he equa ion is no degene a e Le . {x E (0, oo) : u(x) > 0 and (u m )'(x) 7~ 0} . We shall show ha he closu e o his se coincides wi h he suppo o u . We can assume ha um E C l ([0,oo)), because aking a sequence ~ , such ha lim ~ , (x) = 1 i x E [xo - E, xo] and lim ~  , (x) = 0 o he wise we ha e ha N ~o N_l l( m ) ' I p_2 (U m )'] ao + x 41 . xo -E sp-E  x N-1 (G(u) - N N u l dx . So °-  /E and so I (um)']p-2(um)'(xo) = 0 ( he con inui y a x = 0 is simila ly j us i ied) . In consequence, by a solu ion o (14), (15), (16) we shall mean a unc- ion u E C ° ([0, oo)) such ha u,' E C l ([0, oo)), u >_ 0 ( :~- 0) and sa is ies (16) and (17) o any smoo ll unc ion ~ wi h compac suppo in [0, oo) . Now we a e in a condi ion o s a e ou uniqueness esul s : and (20) UNIQUI :NCSS OF VCRY SINGULAR SOLUTION  23 Theo em 1 . Assume ha N > 1, m > 0, q > 0, p > 1, ( 18 )  m(p - 1) > 1 (19)  (p - 1)m Gq G (p - 1)m + Ñ Then he e is a mos one solu ion o p oblem (14), (15), (16) . Mo eo e , his solu ion has compac suppo . Theo em 2 . The conclusion o Theo em 1 holds i we eplace he assump ion (18) o Theo em 1 by In his case, he solu ion is posi i e in [0, oo) . Be o e gi ing he p oo s we shall make some en a ks on he assun p- ions o bo h esul s . Fi s o all we no ice ha he easonable assu np- ion on he pa ame e s m and p is m(p - 1) _> 1, because o he wise he pa abolic equa ion (4) co esponds o a as di usion and solu ions anish a e a ini e ime . On he o he hand, i is na u al o expec a di e en beha iou o solu ions o (14), (15), (16) acco ding o whe he m(p - 1) is g ea e o equal o one . Indeed, he i s case co esponds o slow di usion, and he solu ions o (4) ha e co lpac suppo o any alue o , al hough when m(p - 1) = 1 he solu ions o (4) a e s ic ly posi i e in RN x (0, oo) . Finally he assump ion (19) include he assu ip- ions made in [1], [7] and [8] o he exis ence o e y singula solu ions . In ha e e ences i is also shown how bounda y condi ion (16) implies he one gi en in (13) . 3 . P oo s and auxilia y esul s The ollowing Lemma collec s se e al p ope ies o solu ions o (14), (15), (16) . Lemma 1 . Assume m(p - 1) > 1 and condi ion (19) . Le , u be any solu ion o (14), (15), (16) . Then u E C o and um E C' . Mo eo e (21)  lim I(um)/(X)IP- (u - y(x) - - 1 C(u(0)), xlo x  N 2 4 (ü) (üi) (i ) (22) J .I . DIAZ, J .E . SAA u(x) < M o any x > 0 wi h M i xo E [0, oo) is such ha u(xo) = 0 hen u(x) = 0 o any x > xo, u(x) is mono one non-inc esin,g in [0, oo) and u'(x) < 0 o any x > 0 such ha u(x) > 0, exis s a xo E (0, oo) such ha supp u = [0, xo] hen ( ) i he e lim I ( um ) , ( x )I P-  _ XIX 0 U(X) Rema k . Condi ion (22) is equi alen o he di e en ial equa ion o he in e ace o he solu ion o he pa abolic equa ion (4) which comes om he Da cy law (see e .g . [7] o he case p = 2) . P oo o Lemma 1 : The egula i y o u has al eady been p o ed in a p e ious ema k, so we pass o conside he es o he s a emen . P oo o (i) : We mul iply equa ion (14) by a smo h sequences o ex unc ions ~n(x) such ha lim~ n (x) = 1 i x E [0, e] and lim~ n (x) = 0 o he wise, o some e > 0 . In eg a ing we ha e I(un°YWI`(1M),(E)+ a N 1 'ex I(7l~n)'(X)IP-2(u')(x)dx= Di iding by a and making e -> 0 we ob ain JoE xu'(x) dx - E G(u(x)) dx in ¡I(7lna)i ( E )I E 2(Um ) /(E) + NE  1 1(u m ) , (E)I P-2(Um)'(E)] _ and he e o e (i) . P oo o (ii) : Assume by con a y ha u(yo) = sup{u(x) : x > 0} > M . (I(7 , ) I'' -2 (u na ) ,)' (?/o) = -G(u(yo)) >0 . _ -1ló G(u(e)), Then u'(yo) = 0 and (I(um)'IP-2(u,m)')1 (yo) <_ 0 (as um also has his maximum in ?Jo) . I yo > 0, om he di e en ial equa ion we deduce ha I yo = 0, using (i) we ind he same con adic ion . The e o e u < M on [0, oo) . UNIQUEYESS OI' VERY SINGULAR SOLUTION  25 P oo o (iii) : Again we , shall a gue by con adic ion .  Assu ne ha (iii) is no ue . Then i is easy o show ha he e exis s e > 0 such ha u(x) > 0 and u'(x) > 0 on (xo, xo + E) (o he wise we can ound a sequence {x,,} o local minima o u such ha x. - x0, which yields a con adic ion wi h (14)) . Mul iplying equa ion (14) by x N-1 and in eg a ing o e (x0,x) wi h xE (x0, x0 + E), we ge x  -1 (U-y(x)IP-1 + x SN 9/, ' ( s) ds+ (we ecall ha u is egula in (xo, x) and (u - )'(xo) = 0) .  Taking a sequence o smoo h es unc ions ~  (s) in (14) such ha li n~,(s) = 1 i s E (xo, x) (whe e .co < .x, < .x0 + E) and lim~,(s) = 0 o he wise, we 1( -y(x)IP-I +  u(X) + (~ 1 1  N) Jxx s N-1 u(s) ds = 0 (no ice ha (u-)'(x0) = 0 and (u - )'(x) > 0) . Using ha 1/(q - 1) > N/,Q we ha e o equi alen ly, x S N-1  x +  u(s) ds -  s N-1 u 9 (s) ds = 0 x,, q - 1  . o I xs N-1 uq(s) ds o xN ~  1 , (x) < Jxo .s N-1 u`' (s) ds < Ñ (x N- . ó )u`'( :c), < N 1 U'-'(x) (1 -  0x  N) Making now x -+ xo we a i e o he inequali y which is a con adic ion . P oo o (i ) : Suppose ha o some xo > 0  u'(x0) > 0, hen by (i) he e exis s a x1 E (0, x0) such ha u'(x1)=0 and (1(um)'IP-2(u-)')'(x1) 2 6 and - xN-'I(u')'(x)I''-, + J .I . DIAZ, J .E . SAA >_ 0 . Wc also know by (i i) ha u(xl) > 0 . A guing in he same way as in he p oo o (i) we can show ha lim NI (u-)'(x) Ip- 2(u-) , (x) _ _  1  (u(x1) - u 9(x1)) < 0 xlx l  x - x1  q _ 1 (since 0 < u(x1) < M) . Then we a i e o a con adic ion wi h he ac ha (hm)'I P -2 (u m )' l' (xl) > 0 . Thus u'(x) <_ 0 o all x >_ 0 . In ac he same a gumen shows ha u'(x) < 0 o e e y x > 0 whe e 0 < u(x) < M . Thus i only emains o exclude he case u(x) = M  i x E [0, al u(x)<M i x E (a,, +oo) o so ae a > 0 . We- asse laa he e exis s e > 0 sueh ha o any x E (a, a + e) we ha e u(x)<0 x S N_1  x +  u(s) ds -  s N-1 ug(s) ds=0, a ~- 1  a ( he p oo o hese p ope ies ollow he same ideas used in he pa (iii)) . By in eg a ing by pa s we ob ain xN  1 I(um) , ( x)Ip 1 - x N ~ _ a N M+M  ~ (X) xN = C  1  - N)  x SN-1u(s) ds -J xs N-1 u 9 (s) ds . i a  a As u is dec easing on (a, a + e) so ne elemen a y manipula ign allows o ob ain .c N  a N M + M Q _  ( x ) x N <  1  _ N  x N _  - a N M - x N - a N u q ( X ) , C q -1 Q/ N M _ U(X) xN < x N - a N ( M  _U, (x) Q N q-1 and I we ake UNIQUENESS OFVERY SINGULAR SOLUTION  27 Di iding by x N (M-u(x)) and le ing x - a we a i e o he con adic- Thus, we ha e excluded he possibili y u = M on any in e al [0, a], and he p oo o (i ) is now comple e . P oo o ( ) : Choose E > 0 such ha E < xo, u(x) > 0 and u'(x) < 0 in x E (xo - E, xo) . Then, as in pa (iii), we ob ain N - x"  I (U) (x) I P 2 (U) (x) - Q u(x)+ 1 N  ° +  -  ~° s N-1 u(s) ds -  px s N-1 u 9 (s) ds = 0 ( q - 1  l )  xJ x wi h x E (xo - E, xo) . Since (um)'(x) < 0 and q 1 1 > Á we ge ¡(u - )'IP  1 (x)  x  . ° sN- 1ug(s) ds u(x) < + xN-111,(x) I ( CL 7n ) , I P -1 (x) > x -  1  - 'N jT ° s N-1 u(s) ds u (x)  q - 1  ,0  xN-1u(x) Le ing x T xo in hese wo inequali ies, we ob ain a he limi lim I (u-)'I P-1 (x) _ XIX° U(X) P oo o Theo em 1 : The i s s ep is o in oduce a chango o un- known in such a way ha he abso p ion e m o he now equa ion be mono onically non-inc easing . Le (x) de ined by ( 23 )  11 = (p - 1 )/(m(p - 1) - 1) i is easy o see ha sa is ies (on he suppo o ) he equa ion (24) (I ~IP-2 /) + N - 1 IVIIP-2 + M I ? + Px ' +  1  - i .(q-1) = 0 x   ,~a  (q - 1)a  a 34  J .I . Dmz, J .E . Snn Using pa ( ) o Lemma 1 we ha e in consequence _  x0 )  _1/(P-1)  1  1  N lizo V2 ( . .)  - C&  1'm B( x) x x  a(lí+ 1 ) ( g - 1 On he o he hand, i .x E (0, xo), by Lemma 2 we ha e and so lim B(x) > 0 . Then Bu (as ( 0 ) = 0) B(x)  a 2(x) 1al '(x)J 1-1 - Mx > 0 xjxp F onl de no ion o weak solu ion we ha e ha .xp  +0 _~  xN-1 l V1 ip_2V2) £~ -  x N_1 lV1IP_2V2) ~1+ . 02  2 Ixp +xl -1 Ch l lV2(x)l P-1 V2( . T , )) S(X0)+ xjxp + x N-1 ( 1 o  (9 - 1)a 2 a_  x"'-1B(x) o any unc ion ~ E WO'P(O,+oo) . Now we chose k > 0 such ha 1 (0) = 2 (0) < k < i (xo) - 2(xo) = i (xo) = h . We ake agai l ~ = c puj _ 1  W = (Vi _ V2 - k)+ . xN-1 (¡V1 l P_ 2 1 - l 2 ip-2 2)  ' _ J~ iE01 +, Il(9 -1 ) W(9 -1 ) = xo -1 (li l l 2(x)l ' -1 )  (xo)+  xN-1 ( -y1  + 2  ) ~+ xTxp  Q,  CL +  +~ hXN-1 C l l~ 1  + x Qa l) ~ + w x N-1 B(x)~ 0 ~(xo) =  ~'(s) ds < .0 l~ (s)l ds < ~£ X01 l~'l On he o he hand, using Lemma 2 we ha e No ing ha UNIQUCNCSS OI' VCRY SINGULAR SOLUTION  35 ill  + áá l = i (- I 11P-1 +pa ) < 0 . Then a guing as in he s ep 2 (xo) > 0 and using ha B E L°° (0, +00) we ob ain ha he e exis s C1, C2, C3 posi i e cons an s (no depending on k) such ha C1 L  I w'I PePw dx < C2 L  ePw + C3  I (e')'Ie(P-1)w [w'~ol  [w~o1 [w'7Éol w1(x) _  i (x) - 2 (x)  i x E (supp w) 1 (o,xo) i (x)  i x E (supp w) l (xo, yo), ha i (x) < 0 on (xo,yo) and ha lim ( i (x) - 2 (x)) > 0, xjxp we can chose k, closed enough ó h, in o de o ha e supe i = suipp u) . Then, applying he Young inequali y ab <_ eaP + C E bP/(P -1 ) o e small enough (e < CI) we ge (Cl - e) J  I (e w ) ' I P ~ (C2 + C3CE) 11 .19É0] I ew1 P . [ .' 9,01 Now we a e in he same si ua ion han (31) . Thus inequali y (32) holds and we ob ain he con adic ion by making k con e ging o h . In o de o comple e he p oo we mus p o e he compac ness o he suppo o any solu ion o p oblem (14), (15), (16) . Fo his su pose we sháll de ine a supe solll ion o (14), (15), (16) wi h compac suppo . Le 0 be he unc ion (37)  O(x) = ([C - x ° ]+)` ,  dx E [0, 00), whe e [ ]+ = max{ , 0}, aE (1, p/(p - 1)) and C is a posi i e cons an o be de e mina e . A e some elemen a y manipula ions one can e i y ha (-b n ')'(0) = 0 and ha (I(~7n)/ip-2( 0n ),)' + N x 1 ROMA P-2(«n)' +  xo' +  1 1 O- Oq < ) 36  J .I . DIAZ, J.E . SAA assumed C la go e ough . Hence 0 is a, supe solu ion o p oblem (14), (15), (16) . A guing in he same way as in he p oo o he uniqueness we can compa e any solu ion o (14) wi h he supe solu ion 0 . Indeed : le u be a solu ion and apply he p e ious change o a iables o he unc ions u and 0 . Then i we call = u l /i~ and V) = 0 1 /x` = [C _ XQ]+, is a solu ion o (24), (25), (26) and 0 e i ies ,'-2  I É + N  1 I, l~-2zÚ + V) x N ,0  1  li(9-1) & V)  q-1  a Now, p o ing ha sup ( - 0) < 0 consis s in epea ing he same a gu- men s as in he uniqueness p oo , whe e now plays he ole o l and 0 he one o 2 . Hence 0 > u and since 0 l as a compac suppo , he same happens wi h u, and hc ; p oo o Theo e n 1 is complo . P oo o Theo e n 2 : As in he p e ious heo em, wc ; in oduce a change o unknown in o de o a i e o a nc:w equa ion wi h a mono one pe u ba ion e m . Mo e p ecisely, le (x) de ined by u (x) = e'(')  x > 0 (wc ; suppose he e ha u(x) > 0 as wc ; shall p o e in hc ; las pa o he Theo e n) . I is easy o see ha sa is ies (1 'j -2 ,)i + IV , I p-2 V ' + XVI +  1  - e ( -ll = 0 Q q - 1 Now he uniqueness educes o epea he same a gumen s as be o e (e en in a easie way because he s ic posi i i y o and i e siniplici y o he abso p ion and anspo e ms) . In o de o comple e he p oo o he Theo e n 2 we jus ha e o show ha a solu ion u o he p oblem (14), (15), (16) wi h m,(p - 1) = 1 e i ies u(x) > 0 in x[E [0, 00) . Le suppose ha he e exis s some yo such ha u(yo) = 0 . Then by Le nma 1 we ; know ha supp u = [0, xo] o so ne .xo > 0 a,nd ( 3s)  limo I(u  u(x x ) I /~ G 0 in (0, C) . UNIQUGN13SS O1 , VBRY SINGULARSOLUTION  37 Le us de ine he unc ion (x) = In (u'(x» wi h .x E [0, xo), hen we can w i e he p e ious li ni as lim j '(x)  XO) Since lTm (x) = -oo and E C l ([0, xo)) we a i e o a con adic ion wi h (38), and he p oo is concluded . Re na k . The idea o ob aining a con adic ion ia Sobole inequal- i ies was al eady used in Uudinge [10] (see also [4, Theo em 10 .7]) o compa e solu ions o non-degene a e quasilinea ellip ic p oblems . In ha wo k he es unc ion is de ined as in he p oo o Theo em 2 . Fi- nally we poin ou ha ou a gumen s can be also applied in o de o ob ain compa ison esul s o solu ions o mo e gene al equa ions, as o ins an e -O u - ~ ' Vul  + B(x, u, ¡Vul) + (x, u) = 0 u whe e u ~-- (x, u) and u , B(x, u, 77) a e non-dec easing and 17 -> B(x, u, ) is Lipschi z con inuous . In pa icula , his allows o gene alize he uniqueness esul o [3] . Re na k . Si nul aneously o he comple ion o ou wo k (which i l- p o es a p e ious e sion included in [9]) S . Kamin and L . Ve on lla e communica ed o us hei wo k [6] in which hey gi e a new p oo o he exis en e o he e y singula solu ion o he equa ion (5) as li ni o undamen al solu ions sa is ying (7) when c -, +oo . They also lla e a p oo o he uniqueness o he e y singula solu ion (Le . a nonneg- a i e no only sel simila unc ion sa is ying (5)) and solu ions o he pa abolic equa ion (5) . In his way hey a e gi ing an indi ec p oo o he uniqueness o o p = 2 and m > 1 a bi a y . I seems ha hei a gumen s, join ly wi h some ideas o Kamin-Vazquez [5], may allow o gi e he uniqueness o he e y singula solu ion in he class o solu ions o (6) o e en (4) . In any case ou a gumen s a e o a di e en na u e o hose used in [6] and [5] and can be applied o o he ellip ic p oblems no necessa ily ela ed wi h he s udy o singula solu ions o pa abolic equa ions . Re e en es 1 . H . BR zls, L .A . P LGTIGR AND D . TGRMAN, A e y singula solu ion o he hea equa ion wi h abso p ion, A ch . Ra . Mech . Anal . 9 6 (1986) . 38  J .I . DIAZ, J .E . SAA 2 .  J .I . DÍAZ, "Nonlinea pa ial di e en ial equa ions and ee bound- a ies . Vol 1 Ellip ic equa ions," Pi man Resea ch No es in Ma h . 106, Longman, 1985 . 3 .  J .I . DíAZ AND J .E . SAA, Exis ence e unici é de solu ions posi i as pou ce aines équa ions ellip iques quasilineai es, CRAS Acad . Se¡ . Pa is 305 (1987), 521-524 . 4 .  D . GILBARG AND N .S . TRUDINGER, "Ellip ic pa ial di e en ial equa ions o second o de ," Sp inge -Ve lag, 1983 . 5 .  S . KAMIN AND J .L . VÁzQUEz, Fundamen al solu ions and asymp- o ic beha iou o he p-Laplacian equa ion, Re is a Ma emá ica Ibe oame icana 4 (1988), 339-354 . 6 .  S . KAMIN AM) L . VERON, Exis ence and uniqueness o he e y singula solu ions o he po ous media équa ions wi h abso p ion, Jou zal d'Analyse Ma héma ique 51 (1988), 245-258 . 7 .  L .A . PELETII :1I AND D . TERMAN, A e y singula solu ion o he po ous media equa ion wi h abso p ion, J . Di . Equ . 65 (1985), 396-410 . 8 .  L .A . PELETIER AND J . WANG, A e y singula solu ion o a degen- e a e di usion equa ion wi h abso p ion, T ansac ions o he AMS 307 (1988), 2813-2826 . 9 . J .E . SAA, Doc o al Thesis a he Uni e si y Complu ense o Mad id, No e ibe , 1988 . 10 . N .5 . T UD1NGEi2, On hc compa ison p inciple o quasilinea di- e gc :nce s uc u e equa ións, A ch . Ra ional Mech . Anal . 57 (1973), 128--133 . Depa amen o de Ma emá ica Aplicada Uni e sidad Complu ense de Mad id 28040 Mad id SPAIN P ime a e sió abuda el 10 de Se emb e de 1990, Ba e a e sió ebuáa el 23 d'Oc ub e de 1991