Abstract UNIQUENESS OF VERY SINGULAR SELF-SIMILAR SOLUTION OF A QUASILINEAR DEGENERATE PARABOLIC EQUATION WITH ABSORPTION

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 Op76m -{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 (510) *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) Ut = ApUM -Uq in Q = RN x (0, oo), where, as usual, Opu denotes the p-Laplacian operator N >_ 1 and m and q are nonnegative real numbers . Equation (4) contains, as special cases, the equations and Op v = div (1wIp-2 w), 1 <p< oo,

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) Ut = ApUM -Uq in Q = RN x (0, oo), where, as usual, Opu denotes the p-Laplacian operator N >_ 1 and m and q are nonnegative real numbers .Equation (4) contains, as special cases, the equations and Op 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 > q and p -1 < q < p -1 + (p/N): Peletier and Wang [S] .
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)/(q-m) and / .i= p(q-1)/(q+ 1 -p) in the 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 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) 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 .supportif m(p -1) > 1, etc) .

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, 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 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 C2 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 Cl ([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 and so I (um)']p-2(um)'(xo) = 0 (the continuity at x = 0 is similarly j ustified) .
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) .

. Proofs and auxiliary results
The following Lemma collects several properties of solutions of ( 14), (15), (16).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 Dividing by a and making e -> 0 we obtain and therefore (i) .
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)) .
By integrating by parts we obtain As u is decreasing on (a, a + e) sorne elementary manipulatign allows to obtain and If we take Dividing by xN (M-u(x)) and letting x -a we arrive to the contradic-Thus, 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 with x E (xo -E, xo) .Since (um)'(x) < 0 and q 1 1 > Á we get I ( CL 7n ) , I P -1 (x) > x - 1 -'N \ jT °s N-1 u(s) ds 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 By comparing the value of for i = 1, 2, it is not difficult to see that xo > 0 .Indeed, if xo = 0 we deduce, in the same way that in part (iii) of Lcrnma 1, that vl(x) < v2 (x) in (0,6) and therefore From this inequality and the property of vi (see part (i) of Lemma 1) we obtain v, (0) < vz(0) and so the maxirnum of VI -vz can not be attaint at .xo = 0. Assurric now that xo > 0 and V2(X0) > 0. Obviously we also have that vr (xo) > 0 because h > 0. Then there exists a constant L > 1 such that IaI I' -Ia 2 I' 5 CIa l -021 Val, a2 E R, (where C denotes again a generie constant and so it will denotes in the following), using that vi (x) > h/2 for any x such that w(x) > 0, and integrating by parts in the lasa integral we deduce that But [w' :~0] C_ [w =~0] and from the choice of w we deduce that these exists these positive constants 61, 52, 63 such that if x E [0, oo) satisfies that w(x) > 0 then x < di (because supp vi and supe v2 are bounded), x > 52 and v2 (x) > 53 (as consequence of ( 28 In the case p > N conclusion (32) is obtained from the Sobolev inequality by replacing p* by any number greater than p* .Since these inequalitíes are independent of k they must hold as k tends to h.That is, the function Vi -v2 attain its supremum on a set of positive measure, where at the same tirne (VI -v2)' = 0, which is a contradiction with the inequality .(32) .
In the case 1 < p < 2 inequality (30) must be replaced by (instead of (31)) and so the conclusion follows .Now we consider the last case: it is when the point .co is such that v2(x0) = '0.We shall need a qualitative information which gives an additional information to part (v) of Lemma 1 .Lemma 2. Assume m(p -1) > 1 and (19) .Let u be _any solution of ( 14), ( 15  1 -N l u_u`r~>O. q 1 Q Integrating the equation ( 14) on (0, x1) we also llave Thus (35) follows from this identity, (36), and the monotonicity of u (part (iv) of Lemma 1).
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 is a solution of (24), ( 25), (26) and 0 verifies ,'-2 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 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 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 ru -~' 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 .
the function f (x) = In (u' (x» with .xE [0, xo), then we can write the previous lirnit as lim j'(x)XO)    Since lTm f(x) = -oo and f E Cl ([0, xo)) we arrive to a contradiction with (38), and the proof is concluded .