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

Publicacions Matemátiques, Vol 36 (1992), 19-38 . Abstract UNIQUENESS OF VERY SINGULAR SELF-SIMILAR SOLUTION OF A QUASILINEAR DEGENERATE PARABOLIC EQUATION WITH ABSORPTION J .I . DIAZ * AND J .E . SAA * We show the uniqueness of the very singular self-similar solution of the equation 7tt - Op76 m -{- Y6 9 = 0 . The result is carried out by studying the stationary associate equation and by introducing a suitable chango of unknown . That allows to assume the zero-order perturbation term in the new equation to be monotone increasing . A careful study of the behaviour of solutions near the boundary of their support is also used in order to prove the main result . 1 . Introduction The main goal of this paper is to show the uniqueness of solutions of the following quasilinear elliptic problem 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 10) *Partially supported by the DGICYT project n° PB90/0620 . 20  J .I . DIAZ, J .E . San in which p > 1 and the functions h and g satisfy certain structural conditions which will be made explicit later . The main motivation for the consideration of such a problem comes from the study of very singular solutions of the quasilinear degenerate parabolic equation with absorption (4)  U t = A p U M -U q  in  Q = RN x (0, oo), where, as usual, O p u denotes the p-Laplacian operator N >_ 1 and m and q are nonnegative real numbers . Equation (4) contains, as special cases, the equations and O p v = div (1wIp -2 w),  1 <p< oo, Ut = DUm _ Uq (6)  ut = A P U -u q which have been intensively studied in the last years . For many different purposes it is interesting to study singular solutions of (4) Le . nonnegative functions u satisfying (4) in Q (in the sense of distributions) and such that u(x, 0) = 0 if x E RN- {0} . In many cases, the singularity at t = 0 of such a solution inust be as that of the fundamental solution Le . u(x,0) = cb(x) for some positive constant e, or, in other words, lim  u(x, t) dx = e t-0 fjxj<r for any r > 0 . Nevertheless, when the absorption is strong enough with respect to the diffusion, there exists another type of singular solution u called as very singular solution which has been discovered previously in the following cases : a) equation (5) with m = 1 and 1 < q < 1 + (2/N) : Brezis, Peletier and Terman [1] b) equation (5) with m > 1 and m < q < m + (2/N) : Peletier and Terman [7j c) equation (6) with p > qand p - 1 < q<p - 1 + (p/N) : Peletier and Wang [S] . UNIQUENESSOFVERY SINGULARSOLUTION  21 In all those cases this new singular solution satisfies (8)  lim  u(x, t) dx = +oo t--~o 1x1<r for any r > 0 and so it is more singular than the fundamental solution . As usual, the existente of a very singular solution is obtained in the caass of self-similar solutions (g)  W(x t) = t -1 /(q -1 ) f Ox1lt1/Q) where ,0 must be suitable chosen . For instante Q = 2(q - 1)/(qm) and / .i = p(q1)/(q+ 1 - p) in t he cases of equations (5) and (6) respectively (recall that q > p - 1) . More generally, we can consider self-similar solutions W of the equation (4) in which case the natural choice of ,0 is ( 10 )  Q= p(q - 1)1(q - m(p - 1)) A function W given by (8) is then a very singular solution if f satisfies (11) ~(r)'1 P-2 (r)')'+ (Nx 1)  P-2(fm)/+1x f'+ (q 1 1) ff q = (12)  f > 0 in (0, oo) ( 13 )  f'(0) = 0,  lira 71 1 /(q - m(p - 1))f(ri) = o The uniqueness of f solution of (11) (12) (13) was only given for the case m = 1 and p = 2, (see [1J) and was left open in [7] and [8J . The main goal of our work is to give an uniqueness result true for any value of m and p . Introducing v = f m , we remark that v satisfies an equation of the type (1) with 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 not monotone in u . Moreover the differential terms in equation (11) may Nave different homogeneity (m(p - 1) and 1 respectively) which leads te some special difficulties (solutions with compact .support if m(p - 1) > 1, etc) . 22  J .I . DIAZ, J .E . Sna 2 . The main results We shall prove the uniqueness of solutions of the problem ( 14 ) \I(Um)/IP-2(um)')'+ (Nx 1)I(um)~Ip_2(Um)'+ ~xu'+G(u) =0, (15)  u(x) > 0  (y0) (16)  (u'')'(0) = 0,  lim u(x) = 0 wherem>0,p>1,N>1  Q>0and G(u) =  1  uu9 . q-1 For some values of m and p problem (14) (15) (16) does not have any classical solution and it must be solved in a generalized way . This is the case when m(p - 1) > 1 because the solutions have as support a compact interval [0, xo] and u' may be discontinuous at x = xo (see part (v) of Lemma 1) . To define the notion of weak solution we .multiply the equation (14) by a smooth test function ~(x) with compact support in [0, oo) but not necessarily vanishing at x = 0 . By multiplying by x N -1 and integrating by parts we obtain (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 the other hand, by standard regularity results, it is clear that u E C ° ([0, oo)) and that in fact u E C 2 on the set where the equation is not degenerate Le . {x E (0, oo) : u(x) > 0 and (u m )'(x) 7~ 0} . We shall show that the closure of this set coincides with the support of u . We can assume that um E C l ([0,oo)), because taking a sequence ~ , such that lim ~ , (x) = 1 if x E [xo - E, xo] and lim ~  , (x) = 0 otherwise we have that N ~o N_l l( m ) ' I p_2 (U m )'] ao + x 41 . xo -E sp-E f  \ x N-1 (G(u) - N N u l dx . So °-  /E and so I (um)']p-2(um)'(xo) = 0 (the continuity at x = 0 is similarly j ustified) . In consequence, by a solution of (14), (15), (16) we shall mean a function u E C ° ([0, oo)) such that u,' E C l ([0, oo)), u >_ 0 ( :~- 0) and satisfies (16) and (17) for any smootll function ~ with compact support in [0, oo) . Now we are in a condition to state our uniqueness results : and (20) UNIQUI :NCSS OF VCRY SINGULAR SOLUTION  23 Theorem 1 . Assume that N > 1, m > 0, q > 0, p > 1, ( 18 )  m(p - 1) > 1 (19)  (p - 1)m Gq G (p - 1)m + Ñ Then there is at most one solution of problem (14), (15), (16) . Moreover, this solution has compact support . Theorem 2 . The conclusion of Theorem 1 holds if we replace the assumption (18) of Theorem 1 by In this case, the solution is positive in [0, oo) . Before giving the proofs we shall make some renrarks on the assunrptions of both results . First of all we notice that the reasonable assurnption on the parameters m and p is m(p - 1) _> 1, because otherwise the parabolic equation (4) corresponds to a fast diffusion and solutions vanish after a finite time . On the other hand, it is natural to expect a different behaviour of solutions of (14), (15), (16) according to whether m(p - 1) is greater or equal to one . Indeed, the first case corresponds to slow diffusion, and the solutions of (4) have corrlpact support for any value of t, although when m(p - 1) = 1 the solutions of (4) are strictly positive in RN x (0, oo) . Finally the assumption (19) include the assurriptions made in [1], [7] and [8] for the existence of very singular solutions . In that references it is also shown how boundary condition (16) implies the one given in (13) . 3 . Proofs and auxiliary results The following Lemma collects several properties of solutions of (14), (15), (16) . Lemma 1 . Assume m(p - 1) > 1 and condition (19) . Let, u be any solution of (14), (15), (16) . Then u E C o and um E C' . Moreover (21)  lim I(um)/(X)IP- (u - y(x) - - 1 C(u(0)), xlo x  N 2 4 (ü) (üi) (iv) (22) J .I . DIAZ, J .E . SAA u(x) < M for any x > 0 with M if xo E [0, oo) is such that u(xo) = 0 then u(x) = 0 for any x > xo, u(x) is monotone non-incresin,g in [0, oo) and u'(x) < 0 for any x > 0 such thatt u(x) > 0, exists a xo E (0, oo) such that supp u = [0, xo] then (v) if there lim I ( um ) , ( x )I P-  _ XIX 0 U(X) Remark . Condition (22) is equivalent to the differential equation of the interface of the solution of the parabolic equation (4) which comes from the Darcy law (see e .g . [7] for the case p = 2) . Proof of Lemma 1 : The regularity of u has already been proved in a previous remark, so we pass to consider the rest of the statement . Proof of (i) : We multiply equation (14) by a smoth sequences of text functions ~n(x) such that lim~ n (x) = 1 if x E [0, e] and lim~ n (x) = 0 otherwise, for some e > 0 . Integrating we have I(un°YWI`(1M),(E)+ f a N 1 'ex I(7l~n)'(X)IP-2(u')(x)dx= Dividing by a and making e -> 0 we obtain JoE xu'(x) dx - fE 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 therefore (i) . Proof of (ii) : Assume by contrary that 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) . If yo > 0, from the differential equation we deduce that If yo = 0, using (i) we find the same contradiction . Therefore u < M on [0, oo) . UNIQUEYESS OI' VERY SINGULAR SOLUTION  25 Proof of (iii) : Again we , shall argue by contradiction .  Assurne that (iii) is not true . Then it is easy to show that there exists e > 0 such that u(x) > 0 and u'(x) > 0 on (xo, xo + E) (otherwise we can found a sequence {x,,} of local minima of u such that x. - x0, which yields a contradiction with (14)) . Multiplying equation (14) by x N-1 and integrating over (x0,x) with xE (x0, x0 + E), we get x  -1 (U-y(x)IP-1 + fx SN 9/, ' ( s) ds+ (we recall that u is regular in (xo, x) and (u - )'(xo) = 0) .  Taking a sequence of smooth test functions ~  (s) in (14) such that lirn~,(s) = 1 if s E (xo, x) (where .co < .x, < .x0 + E) and lim~,(s) = 0 otherwise, we 1(rr -y(x)IP-I +  u(X) + (~ 1 1  N) Jxx s N-1 u(s) ds = 0 (notice that (u-)'(x0) = 0 and (u - )'(x) > 0) . Using that 1/(q - 1) > N/,Q we have or equivalently, x S N-1  x +  u(s) ds -  s N-1 u 9 (s) ds = 0 fx,, q - 1  f . o I xs N-1 uq(s) ds o xN ~  1 v, (x) < Jxo .s N-1 u`' (s) ds < Ñ (x N- .ró )u`'( :c), < N 1 U'-'(x) (1 -  0x  N) Making now x -+ xo we arrive to the inequality which is a contradiction . Proof of (iv) : Suppose that for some xo > 0  u'(x0) > 0, then by (i) there exists a x1 E (0, x0) such that 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 (ifi) that u(xl) > 0 . Arguing in the same way as in the proof of (i) we can show that lim NI (u-)'(x) Ip2(u-) , (x) _ _  1  (u(x1) - u 9(x1)) < 0 xlx l  x - x1  q _ 1 (since 0 < u(x1) < M) . Then we arrive to a contradiction with the fact that (hm)'I P -2 (u m )' l' (xl) > 0 . Thus u'(x) <_ 0 for all x >_ 0 . In fact the same argument shows that u'(x) < 0 for every x > 0 where 0 < u(x) < M . Thus it only remains to exclude the case u(x) = M  if x E [0, al u(x)<M if x E (a,, +oo) for sorrae a > 0 . Weassert tlaat there exists e > 0 sueh that for any x E (a, a + e) we have u(x)<0 x S N_1  x +  u(s) ds -  s N-1 ug(s) ds=0, fa ~- 1  a (the proof of these properties follow the same ideas used in the part (iii)) . By integrating by parts we obtain 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 decreasing on (a, a + e) sorne elementary manipulatign allows to obtain .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 If we take UNIQUENESS OFVERY SINGULAR SOLUTION  27 Dividing by x N (M-u(x)) and letting x - a we arrive to the contradicThus, we have excluded the possibility u = M on any interval [0, a], and the proof of (iv) is now complete . Proof of (v) : Choose E > 0 such that E < xo, u(x) > 0 and u'(x) < 0 in x E (xo - E, xo) . Then, as in part (iii), we obtain 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  fl )  xJ x with x E (xo - E, xo) . Since (um)'(x) < 0 and q 1 1 > Á we get ¡(u - )'IP  1 (x)  x  f . ° sN1ug(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) Letting x T xo in these two inequalities, we obtain at the limit lim I (u-)'I P-1 (x) _ XIX° U(X) Proof of Theorem 1 : The first step is to introduce a chango of unknown in such a way that the absorption term of the now equation be monotonically non-increasing . Let v(x) defined by ( 23 )  11 = (p - 1 )/(m(p - 1) - 1) it is easy to see that v satisfies (on the support of v) the equation (24) (Iv~IP-2v/) + N - 1 IVIIP-2vt + M I v ? + Pxv' +  1  - vi .(q-1) = 0 x  v  ,~av  (q - 1)a  a 34  J .I . Dmz, J .E . Snn Using part (v) of Lemma 1 we have in consequence _  x0 )  _1/(P-1)  1  1  N lizo V2 ( .r .)  - C&  1'm B( x) x x  a(lí+ 1 ) ( g - 1 On the other hand, if .x E (0, xo), by Lemma 2 we have and so lim B(x) > 0 . Then But (as ( 0 ) = 0) B(x)  av2(x) 1alv'(x)J 1-1 - Mx > 0 xjxp Fronl de notion of weak solution we have that .xp  +0 _~  xN-1 \l V1 ip_2V2) £~ -  x N_1 \lV1IP_2V2) ~1+ . 02  2 Ixp +xl -1 Chrrl lV2(x)l P-1 V2( . T , )) S(X0)+ xjxp + x N-1 ( 1 fo  (9 - 1)a 2 a_  x"'-1B(x) for any function ~ E WO'P(O,+oo) . Now we chose k > 0 such that v 1 (0) = v2 (0) < k < vi (xo) - v2(xo) = vi (xo) = h . We take agairl ~ = c puj _ 1  W = (Vi _ V2 - k)+ . xN-1 (¡V1 l P_ 2 v1 - lv2 ip-2v2)  ' _ J~riE01 +, Il(9 -1 ) W(9 -1 ) = xo -1 (lirrl lv2(x)lr' -1 )  (xo)+ f  xN-1 ( -y1  + v2  ) ~+ xTxp  Q,  CL +  +~ hXN-1 C lv l~ v1  + xv Qavl) ~ + f w x N-1 B(x)~ 0 ~(xo) =  ~'(s) ds < .0 l~ (s)l ds < ~£ X01 l~'l On the other hand, using Lemma 2 we have Noting that UNIQUCNCSS OI' VCRY SINGULAR SOLUTION  35 vill  + áávl = vi (- I v11P-1 +pa ) < 0 . Then arguing as in the step v2 (xo) > 0 and using that B E L°° (0, +00) we obtain that there exists C1, C2, C3 positive constants (not depending on k) such that C1 L  I w'I PePw dx < C2 L  ePw + C3 f  I (e')'Ie(P-1)w [w'~ol  [w~o1 [w'7Éol w1(x) _  vi (x) - v2 (x)  if x E (supp w) f1 (o,xo) vi (x)  if x E (supp w) fl (xo, yo), that vi (x) < 0 on (xo,yo) and that lim (vi (x) - v2 (x)) > 0, xjxp we can chose k, closed enough tó h, in order to have supe vi = suipp u) . Then, applying the Young inequality ab <_ eaP + C E bP/(P -1 ) for e small enough (e < CI) we get (Cl - e) J  I (e w ) ' I P ~ (C2 + C3CE) 11 .19É0] I ew1 P . [ .' 9,01 Now we are in the same situation than (31) . Thus inequality (32) holds and we obtain the contradiction by making k converging to h . In order to complete the proof we must prove the compactness of the support of any solution of problem (14), (15), (16) . For this surpose we sháll define a supersollltion of (14), (15), (16) with compact support . Let 0 be the function (37)  O(x) = ([C - x ° ]+)` ,  dx E [0, 00), where [v]+ = max{v, 0}, aE (1, p/(p - 1)) and C is a positive constant to be determinate . After some elementary manipulations one can verify that (-b n ')'(0) = 0 and that (I(~7n)/ip-2( 0nt),)' + N x 1 ROMA P-2(«n)' +  xo' +  1 1 OOq < ) 36  J .I . DIAZ, J.E . SAA assumed C largo errough . Hence 0 is a, supersolution of problem (14), (15), (16) . Arguing in the same way as in the proof of the uniqueness we can compare any solution of (14) with the supersolution 0 . Indeed : let u be a solution and apply the previous change of variables to the functions u and 0 . Then if we call v = u l /i~ and V) = 0 1 /x` = [C _ XQ]+, v is a solution of (24), (25), (26) and 0 verifies ,'-2  I É f + N  1 I, rl~-2zÚr+ V) x N ,0r  1  li(9-1) & V)  q-1  a Now, proving that sup (v - 0) < 0 consists in repeating the same arguments as in the uniqueness proof, where now v plays the role of vl and 0 the one of v2 . Hence 0 > u and since 0 lras a compact support, the same happens with u, and thc ; proof of Theorern 1 is complot . Proof of Theorern 2 : As in the previous theorem, wc ; introduce a change of unknown in order to arrive to a nc:w equation with a monotone perturbation term . More precisely, let v(x) defined by u (x) = e'(')  x > 0 (wc ; suppose here that u(x) > 0 as wc ; shall prove in thc ; last part of the Theorern) . It is easy to see that v satisfies (1v'j r-2 v ,)i + IV , I p-2 V ' + XVI +  1  - e (v-llv = 0 Q q - 1 Now the uniqueness reduces to repeat the same arguments as before (even in a easier way because the strict positivity of v and tire siniplicity of the absorption and transport terms) . In order to complete the proof of the Theorern 2 we just have to show that a solution u of the problem (14), (15), (16) with m,(p - 1) = 1 verifies u(x) > 0 in x[E [0, 00) . Let suppose that there exists some yo such that u(yo) = 0 . Then by Lernma 1 we ; know that supp u = [0, xo] for sorne .xo > 0 a,nd ( 3s)  limo I(u  ru(x x ) I r/~ G 0 in (0, C) . UNIQUGN13SS O1 , VBRY SINGULARSOLUTION  37 Let us define the function f (x) = In (u'(x» with .x E [0, xo), then we can write the previous lirnit as lim j '(x)  XO) Since lTm f (x) = -oo and fE C l ([0, xo)) we arrive to a contradiction with (38), and the proof is concluded . Rernark . The idea of obtaining a contradiction via Sobolev inequalities was already used in Uudinger [10] (see also [4, Theorem 10 .7]) to compare solutions of non-degenerate quasilinear elliptic problems . In that work the test function is defined as in the proof of Theorem 2 . Finally we point out that our arguments can be also applied in order to obtain comparison results for solutions of more general equations, as for instante -O r u - ~ ' Vul  + B(x, u, ¡Vul) +f (x, u) = 0 u where u ~-- f (x, u) and u , B(x, u, 77) are non-decreasing and 17 -> B(x, u, rt) is Lipschitz continuous . In particular, this allows to generalize the uniqueness result of [3] . Rernark . Sirnultaneously to the completion of our work (which irrlproves a previous version included in [9]) S . Kamin and L . Veron llave communicated to us their work [6] in which they give a new proof of the existente of the very singular solution of the equation (5) as lirnit of fundamental solutions satisfying (7) when c -, +oo . They also llave a proof of the uniqueness of the very singular solution (Le . a nonnegative not only self similar function satisfying (5)) and solutions of the parabolic equation (5) . In this way they are giving an indirect proof of the uniqueness of f for p = 2 and m > 1 arbitrary . It seems that their arguments, jointly with some ideas of Kamin-Vazquez [5], may allow to give the uniqueness of the very singular solution in the class of solutions of (6) or even (4) . In any case our arguments are of a different nature to those used in [6] and [5] and can be applied to other elliptic problems not necessarily related with the study of singular solutions of parabolic equations . Referentes 1 . H . BRrzls, L .A . PrLGTIGR AND D . TGRMAN, A very singular solution of the heat equation with absorption, Arch . Ratt . Mech . Anal . 9 6 (1986) . 38  J .I . DIAZ, J .E . SAA 2 .  J .I . DÍAZ, "Nonlinear partial differential equations and free boundaries . Vol 1 Elliptic equations," Pitman Research Notes in Math . 106, Longman, 1985 . 3 .  J .I . DíAZ AND J .E . SAA, Existence et unicité de solutions positivas pour certaines équations elliptiques quasilineaires, CRAS Acad . Se¡ . Paris 305 (1987), 521-524 . 4 .  D . GILBARG AND N .S . TRUDINGER, "Elliptic partial differential equations of second order," Springer-Verlag, 1983 . 5 .  S . KAMIN AND J .L . VÁzQUEz, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Revista Matemática Iberoamericana 4 (1988), 339-354 . 6 .  S . KAMIN AM) L . VERON, Existence and uniqueness of the very singular solutions of the porous media équations with absorption, Jourrzal d'Analyse Mathématique 51 (1988), 245-258 . 7 .  L .A . PELETII :1I AND D . TERMAN, A very singular solution of the porous media equation with absorption, J . Diff . Equ . 65 (1985), 396-410 . 8 .  L .A . PELETIER AND J . WANG, A very singular solution of a degenerate diffusion equation with absorption, Transactions of the AMS 307 (1988), 2813-2826 . 9 . J .E . SAA, Doctoral Thesis at the University Complutense of Madrid, Noverriber, 1988 . 10 . N .5 . TrtUD1NGEi2, On thc comparison principle for quasilinear divergc :nce structure equatións, Arch . Rational Mech . Anal . 57 (1973), 128--133 . Departamento de Matemática Aplicada Universidad Complutense de Madrid 28040 Madrid SPAIN Primera versió rabuda el 10 de Setembre de 1990, Barrera versió rebuáa el 23 d'Octubre de 1991