Let f∈L1(RN), N≥1, f≥0, and consider the Cauchy problem ut=Δum on ]0,∞[×RN, u(0,⋅)=f on RN. The authors prove that as m→∞, the corresponding solutions um(t)→u_=f+Δw in L1(RN), uniformly for t in a compact set in ]0,∞[, where 0≤w_∈L1(Rn) is the solution of the variational inequality Δw_∈L1(RN), 0≤f+Δw_≤1, w_(f+Δw_ −1)=0 a.e. The authors also show similar results for the same equation on a bounded open set Ω in RN with Dirichlet or Neumann boundary condition

Boccardo, L.

Bénilan, Philippe

Herrero, Miguel A.

English

Proceedings of the conference held in Turin, October 2–6, 1989Let f∈L1(RN), N≥1, f≥0, and consider the Cauchy problem ut=Δum on ]0,∞[×RN, u(0,⋅)=f on RN. The authors prove that as m→∞, the corresponding solutions um(t)→u_=f+Δw in L1(RN), uniformly for t in a compact set in ]0,∞[, where 0≤w_∈L1(Rn) is the solution of the variational inequality Δw_∈L1(RN), 0≤f+Δw_≤1, w_(f+Δw_ −1)=0 a.e. The authors also show similar results for the same equation on a bounded open set Ω in RN with Dirichlet or Neumann boundary conditionsDepto. de Análisis Matemático y Matemática AplicadaFac. de Ciencias MatemáticasTRUEpu

Docta Complutense

Rend. Sem. Mat. Univ. Poi. Torino Fascicolo Speciale 1989 Nonlinear PDE's Ph. Bénilan - L. Boccardo - M.A. Herrero ON THE LIMIT OF SOLUTIONS OF ut = Aum AS m -> oo A b s t r a c t . We consider the Cauchy problem (1) ii« = A t i m , w.(0) = / . wehere / G Ll(VL\ ), N > 1 and / > 0 a.e., and prove tliat as in —• oo, the corresponding solutions w m ( 0 converge in L , uniformly for t in a compact set in ] 0 , o o [, to the solution of a suitable limit problem. We alsoshowsiiuilar results for the Catichy-Dirichlet and Cauchy-Neumanii boundary value problems for (1) in bounded domains. Key words and phrases. Porous medium equation, singular limit, mesa problem, asymptotic behavior. Let / e Ll(\RN), / > 0 be given and consider the problem (1) ut = Aum on]0, oolxlil^, w(0,.) = / on IR^ . It is vvell known (see for instance [1]) tliat for any m > 1, there exists a unique "strong solution" of (1), tliat is a function u(t) (x) = u(t} x) satisfying ti e CT([0, oolLl(W.N))nC{]Qt oo [xlRN), ti > 0 on ]0, oo[xJRNf ti(0,.) = / on Ili/ for any r > 0, u e L°°(] r, oo [ xlR^), utt Aum <E L°°i]rt oo[, Ll(VXN)) and ut = A«m a.e. on ] 0, oo [ xIRN . We note um the solution of (1) and prove the following 2 THEOREM 1. As m -+ oo, um(t)-+u=: f + Awin Ll(KlN) uniformly for t in a compact set in ]0, oo[, where m is the solution of the variationai inequality (2) weLl(M,N)y Aui€Ll(Utn), 0 < / + A M < 1, i*i > 0, i£(/ + Aw-1) = 0 a.e. . Existence and uniqueness of a solution w of (2) follows by the results in [3]: indeed the problem rnay be rewritten under the forni (3) IL, we Ll{lllN)+, u- Aw = f in V(\\lN),ueP(w) a.e. . where j3 is the sigli graph. li me H^cClll"), which is tiie case if N = 1 or / e L ^ I R * ) with some e > 0, then Aw = 0 a.e. on {w = 0} so that (4) a = XE + / \ r ^ \ s w i t h S = ^N\{w > 0} . The fact that for 7?? large the solution of the porous medium equation develops "mesas" on the set of noncoincidence of the solution of the variationai inequality (2), and tends to / on the complementary set, has been noticed in [8]. In [7], it has been proved that for / bounded and satisfying strong geometrie assumptions, um(t) —• u given by (4) in the weak-* topology of L°°(\[\N) as 7?i —• oo, uniformly for t in a compact set in ]0, oo[. In [9], Theorem 1 has been proved in the cases N = 1 and N > 2 with / radialiy symmetric. We also consider equation (1) on a bounded open set Q in IR^ with Dirichlet or Neumann boundary conditions, and prove the results correspon-ding to Theorem 1; for the Cauchy-Dirichlet boundary value problem, sudi result has been shown in [9] in the case N = 1. The paper is organized as follows: 1. Proof of Theorem 1. 2. The Cauchy-Dirichlet boundary value problem. 3. The Cauchy-Neumann boundary value problem. 3 S E C T I O N 1. Proof of Theorem 1 We first recali that the map / —• um(t) is a contraction in Ll(MN) for any m > 1 and t > 0; a similar result holds for the map / —• M. Therefore, as it was noticed in [9], it is enough to prove the Theorem assuming that / is bounded and compactly supported. Namely, we will assume throughout this Section that (5) 0 < / < M a.e. on {\x\ < R0}, / = 0 a.e. on {\x\ > RQ} . By the maximum principle we have (6) 0 < um(t) < M a.e. for any t > 0 and m > 1 . Fix now T > 0 and m0 > 1. It follows from Lemma 2.1 in [9] that there exists /2, depending on N, A/, R0, T and wQì sudi that (7) um(t) = 0 on {\x\ > 11} for any t e [0, 7'] and m > m0 . By the translation in vari ance and the L^contractivity of the maps / —• um(t)j we have that for any t > 0 and in > 1 (8) / |tim(<, x + y) - um(tt x)\ dx < J \f(x + y) - f(x)\ dx for any y € 111" . Therefore, as in [9], it follows from (6)-(8) that (9) {um(i)i < £ [U>?]> m > mo} i« precompact in Ll(\SXN) . We now recali the following oneside estimate (see [1]) for the solution u = um o f ( l ) (10) ut = Aum > -w./(7;i - 1 + 2/N)t a.e. . Since Au(t)m e Ll{\\lN) and / Au(t)m = 0 a.e. t> 0, one tlien has (11) IMOIIL« = il Any rii£.« = 2ii( At to rn i t i < 2||u(0IU»/ (m - 1 + j ^ t < 2 | | / |U. / (m - 1 + J : ) t a.e. t > 0. From (6), (7) and (10), it follows that (12) (um (t,x))m < ME(x)/ (m - 1 + -^ j t on ]0, T[ xlllN for m > 7?i0 . 4 where EeC(lRN) is the solution of £ = 0 on {\x\ > R], -AE = 1 in V'({\x\ < R}) . In particular for 0 < r < T, we have (13) («m)m -»• 0 uniformly on [r, T] x IR^ as m -+ oo . Thanks to (6) we have (um)m G C([0, oo[, Ll(ÌRN)) and we may defme, for t > 0 and m > ì (14) wm(t) = f (um(s))mds . Jo which satisfies (15) um(t) - Awm(t) = f in D'(1RN) . lf for a subsequence m* —• oo we have u„,fc(l)—+u in Ll(V& ), then b y ( H ) (16) timfc (*) -* a in Ll(\ìiN) uniformly for * G [r, T] . whereas, by (7) and (15) (17) tvmk{l)-+w\n Ll{\\lN) . with (18) w - Au; = / in £>'(lllN) , w > 0 a.e. on IR* . and using (13) (19) 0 < n < 1 a.e. on IR* . We claim that (20) w = 0 a.e. on {u< 1} . This will end the proof the Theorem 1. In order to prove (20), we first reniark that according to (10), the map t —• t(n+l+N) u(t) is nondecreasing so that (21) um(t) < t~ 1 / (m-1+(^) )w.m(l) for any 0 < t < 1 . 5 By (6) (22) tifn(Om < Mum(t)m-1 . so that by the definition (14) of wm(t) and (21) (23) wm(ì) < Mum(ì)m-l(ì + N(m - 1)/2) . Property (20) is novv clear: we may assume wnlfc(l) —+ u a.e., sudi that , a.e. x € {u < 1} we willl have for k large, tinlfc(l)(x) < 6 < 1 and then, using (23), wmk(l)(x)-*0 as k —• oo. REMARK 1. Under assumptìon (5), we have justified the definition (14) of wm(i), and actually proved that (24) wm(t) —• w in W2,P(ÌR.N) for any 1 < p < oo, as m —• oo . uniformly for t in a compact set in ]0, oo[. Actually, according to (15), one lias that for any / 6 Ll(ÌR.N)1 wm(t) is well defìned and converges to w in W^(ÌRN) for any 1 < p < N/(N - 1). S E C T I O N 2. T h e Cauchy-Dirichlet boundary value problem In this section Q will be an open set in 1RN and / € Ll(£l), / > 0. We consider now the problem (25) ut = Awm on ] 0, oo[xfì, ti(0,.) = / on Q, w = 0 o n ] 0 , o o [ x a « . For simplicity we will assume Q bounded with smootli boundary ^fì, although the results which follow can be easily extended to a general open set 12. Using for instance the results of [2], it follows that for m > 1 there exists a unique "strong solution" of (25) satisfying u 6 C([0, oo[,Ll(Q))nC(]0y oo[xS), u > 0 on ]0, oo[ xQ, u = 0 on ]0, oo[ xdtì, M(0,.) = / on Q\ for any r > 0, utì Aum G L°°(] r, o o [ , ^ ( 0 ) ) and ut = Aum a.e. on ]0, o o [ x ^ . 6 We shall denote by um the strong solution of (25). On the other band there is existence and uniqueness of a solution of the variational inequality (26) we WyiQ), 0 < / + Aui< lin'P'(ft), rr> > 0, i ( / + A w - l ) = 0a.e.onfì . This follovvs by the result of [6]. We have the follovving THEOItEM 2. With the notations of this section, as m —• oo «m(0 —• li = / + AMI in Ll(Q) uniformiy for t in a compact set in ]0,oo[. To show this we will adapt the proof of Section 1. Using the contraction property in L1^) of the maps / —• um(t) and / —• w, we rnay assume / bounded, namely 0 < / < A/, such tliat (G) stili holds. The oneside estimate (10), with the Constant (m- 1-f — j there is no more true, but as it is proved in [4] for general homogeneous evolution equation, we have for u = um (27) ut = Atim > -ti/(m - 1)/. a.e. . and also (28) IMOIIn = IIMOmIU« < 2| | / |Ui/(m- ì)t a.e. t > 0 . From (27) and (6), we deduce (29) («,„(*, x))m < 'ME(x)/(m - ì)l on ]0, T] x Q for m > 1 . where E is now the solution of the Dirichlet problem on Q - A £ = 1 on il, E = 0 on diì and it follows that for 0 < r < T (30) («m)m —• 0 unifornily on [r, T] x Q as m—> oo . 7 We may defme again wm(t) by (14), and vve have (31) um(t) - Awm(t) = / on Q} wm(t) = 0 on dSl . If for a subsequence m* —• oo we have timk(l) —• IL in £*(ft), then we will have by (30), (29) and (31) 0 < i£ < 1 a.e. on U umk(t) —• IL in £*(fì) uniformly for / € [r, T] tvmk(ì) —• w in L!(Q) where u; > 0 is the solution of u — Aw - / on i2, w = 0 on dQ . The proof of (20) can be done as in Section 1, with slight modifications: according to (27), the map t —• tl^m~l^u(l) is nondecreasing, so that replacing (22) by um(t)m < M2um(t)m-2, we will have in place of (23) (32) wm(l)<M2um(l)m-\m-l). which gives also (20) exactly in the sanie way. In other words, according to these remarks, the proof of Theorem 2 reduces to showing that (33) {«m(0i * £ [0, T], m > ino] is precompact in L 1 ^) . According to (6), it is actually enough to prove that {um(t)\ t e [0, T], m > m0} is precompact in Lloc(tì) . To prove this, fix p € V(Q), p > 0. Let u = um, and for y e lllN with supp(/?-fy) contained in Q, let v(tt x) = p(x)\u(t,x + y) - u(t,x)\. By Kato's inequality, we have ut < pAu; in Z>'(]0, oo[xQ) with u;(i, x) = |«(<, a; + y)m - u(t, x)m\ and then integrating I p(x)\u(t, l' + i / ) - «(<, x)\dx < Ip(x)\f(x + y) - /(ar)|<fa? + i2|y| with R=\y\~l I I Ap(x)w(s} x)dxds <\\Ap\\Looin)\\ grsid um\\Lii]0,t[xiì) • Therefore (33) will follow from LEMMA 1. For T>0 and m0 > 1, there exists C sudi that ||grad (um)m|U»(]o,r[xn) < C for m > m0 . Proof of lemmn 1. Set u = um. For any t > 0, let v(t) be the solution of -Av(t) = u(t) on Q, v(t) = 0 on dtt . We have (34) v£ClQ0too[xQ)t vt = -um . sudi that, taking integrating over ]0, T[ xQ, vve obtain jj^nm+l = ^ 2«,Ai; = -JJ(\gr*à v\2)t < f |grad t;(0)|2 and then (35) ffum+l<C. Using now (27) we have that for any t > 0 //. / r \ m/m+l Igrad u"'(OI2 < ^ u"*+1(«) < ||«(0IU-+> f y « m + 1 (0 j Then using Holder and the fact that ||w(/)||Lm+i < ||/||L»»+I, we deduce that (771-1) ( ^ Igrad um | ) < < l«lll/llx»+« Uj um+l) il<///<(m+1)/(m+2) m/m+l / r \ (m+2)/(m+l) <i«iii/iii-+^yy«m+ij and ( Il |grad tim |) < M|fì|(m+2)^m+1)7 ,1/m+1Cm/m+1(m + 2)<m+2>'<m+1>(m - l )" 1 , whence the Lemma follows. 9 SECTION 3. The Cauchy-Neumann boundary value problem In this section we consider the problem (36) ut = Aum on ] 0, oo [ xU, u(0,.) = / on Q, dum/dn = 0 on ] 0, oo [ xdQ . where Q is a bounded connected open set with smooth boundary dQ, and / € L1^), / > 0. Using again the results of [2], for any m > 1 there exists a unique "strong solution" of (3G) satisfying u GC([0, oo [ ,^(^))0^(10, oo[x£J), u> 0 on ]0, oo[xft, u(0,.) = / on ft, ut e L°°(] T, oo [, L\Q)) for any r > 0, um G ££.(] 0, oo [, W2ì(ty) and ut = Aum a.e. on ]0, oo[ x£2, dum/dn = 0 a.e. on ]0, oo[ xdQ . We denote now by um this solution of (36). On the other hand, consider the variational inequality (37) w e WM(n) , 0 < / + Aw < 1 in Z>'(Q), w > 0, w(f + A i - 1) = 0 a.e. on Q and / pAw = — grad p grad w for any p e C1^) . According to the results in [5], (37) has a solution if and only if (38) ff=\n\-lJf<Ì; Moreover, if ff< 1, then the solution w of (37) is unique if - / / = 1 , for any solution w of (37), / + Aw = 1 a.e. on Q. We have THEOREM 3. With the notations of this section, i) if / / > ! , then um(t)^J-fin I ' (0 ) as ,n-* oo, « / a n n / y / o r , ina compact set in ] 0, oo [. ll) if / / < 1J then um(t)-+u = f + Aw in LJ(f2) as m —* oo, uniformly for t in a compact set in ] 0, oo [. 10 To prove this Theorem we ma,y assume again that / is bounded and then that (6) holds. According to the results in [4], (27) and (28), stili are true in this case. In partìcular, it is enough to prove that the conclusion is satisfied at t = 1: it will then hold uniformly for / in a compact set in ]0,oo[. We denote by G the Green operator in Ll(Q) associated to the Neumann problem for the Laplaeian: for w £ ^(Q), v = Gw is the unique solution of the problem (39) v G W 1 , 1 ^ ) , Y V = 0, p(w- jw)= /grad/> gradi; for any p£Cl(p). It is clear that G is a bounded (actually compact) linear operator from Ll(SÌ) into Wl'l(Q) (see [6]). Finaly, we set / = / / . We then have (40) jum(t) = I f o r a n y m > l , t>0. Proof of part i): case / > 1. We note for simplicity um = wm(l); using (40), we have / |wm - / | = 2 I(I — um)+, where r+ = sup(r,0), and then it is enough to prove that (41) / (/ — wrn)+ -+ 0 as m -> oo . Using (28), since (um)m - f(um)m = G(-A(t /m )m ) , we see that (42) em = |(«m)m - j(um)m\ - 0 in W^1-1^) a s m - o o . Now, by convexity, we have so that (um)m>(l-em)+ 11 and £m > Im ~ ( « m ) m > m(um)m"l(I - t l m ) . Then I-um<em(ì-em)\r1,mm'1 and, tlianks to (42), (41) holds by Lebesgue dominated convergence theorem. Proof of part ii): case I < 1. We will prove LEMMA 2. With the notations of this Section 3, if I < ì then for T > 0 there exists C sudi that ll(tim)m+1|U«(]olT[xn)<C f o r m > l . Using Lemma 2, and repeating the proof of Lemma 1, one sees that for mo > 1, there exists C such that ||grad (wm)m|U»(]o,T[xn) < C for m > m0 and then (33) holds also in this case. Another consequence of Lemma 2 is that liminf(tim)m+l < oo a.e. on ]0, T\xQ m-+oo which we will use instead of (30) to obtain, tlianks to (28), that if for a subsequence m* -+ oo we have umk(ì) -+« in ^(Q), then we will have 0 < u < 1 a.e. on Q. The proof of Theorem 3 in this case will follow then exactly as that of Theorem 2. To end up, we give the Proof of lemma 2. Let w = um, v(t) vt(t) = Gu{t). We have that = fum(t) - um(t)t u(t) = / - Av(t) 12 and then y«m+i(o=jumMju(t) - fu(t)vt(t)= = / fum(t) - I fvt(t) - / grad vt(t) grad v(t) . Using the convexity inequality (m + l)wm < mum+l + 1, we obtain (m + 1 — m/) / / « m + 1 < / r + (m + 1){I jv(Q) + 1 / 2 ^ |grad t;(0)|2} whence the result. REFERENCES [1] D.G. Aronson, Ph. Bénilan. Régularité des solutions de l'équations des milieux poreux dans UlN. C.R.Ac. Se. Paris 288 (1979), pp. 103-105. [2] Ph. Bénilan, A strong regularity Lp for solutions of the porous medium equation. Contribution nonlinear pde. Res. Notes in Math. 89. Pitman (1983), pp. 39-58. [3] Ph. Bénilan, H. Brézis, M.G. Crandall. A semilinear elliptic equation in L\MN). Ann. Se. Norm. Sup. Pisa. V,2. (1975), pp. 523-555. [4] Ph. Bénilan, M.G. Crandall. RegularizingefTect of homogeneousevolutions equations. Contributions to Analysis and Geometry, D.N. Clarke & al eds, John Hopkins Un. Press, Baltimore (1981), pp. 23-30. [5] Ph. Bénilan, M.G. Crandall, P.E. Sacks. Some L1 existence and depen-dance results for semilinear elliptic equations under nonlinear boundary conditions. Appi. Math. Optim. 17 (1988), pp. 203-224. [6] H. Brézis, W. Strauss. Semilinear elliptic equations in L1. J. Math. Soc. Japan, 25 (1973), pp. 565-590. [7] L.A. CafTarelli, A. Friedman. Asymptotic behavior of solutions of ut = Awm as m -+ oo. Indiana Un. Math. J. 36,4 (1987), pp. 711-718. [8] C.M. Elliot, M.A. llerrero, J.R. King. J.R. Ockendon. The mesa problemi di/rusion pattems for ut = V(wmVw) as m —• oo. IMA J. Appi. Maths 37 (1986), pp. 147-154. [9] P.E. Sacks. A gingillar limit problem for the porous medium equation. J.Math. Anal. Appi. 140,2 (1989), pp. 456-466. PHILIPPE BENILAN - Equipe Mathématiques de Besangon U.A. CNRS 741 - Université de Franche-Comté 25030 Besancpn Cedex - FRANCE LUCIO BOCCARDO - Dipartimento di Matematica Università di ROMA 1 - Piazzale Aldo Moro 2 00185 Roma - ITALIA MIGUEL A. HERRERO - Departamento de Matematica Aplicada Facultad de Matematica^ - Universidad Complutense 28040 Madrid - ESPANA 

On the limit of solutions of ut=Δum as m→∞

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On the limit of solutions of ut=Δum as m→∞

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