Pacific Journal of Mathematics

Transcription

1 Pacific Journal of Matheatics SINGULAR LIMIT OF SOLUTIONS ( ) OF THE EQUATION u t = u AS 0 Kin Ming Hui Volue 187 No. 2 February 1999

2 PACIFIC JOURNAL OF MATHEMATICS Vol. 187, No. 2, 1999 SINGULAR LIMIT OF SOLUTIONS ( ) OF THE EQUATION u t = u AS 0 Kin Ming Hui We will show that for n = 1, 2, as 0 the solution u () of the fast diffusion equation u/ t = (u /), u > 0, in R n (0, ), u(x, 0) = u 0 (x) 0 in R n, where u 0 L 1 (R n ) L (R n ) will converge uniforly on every copact subset of R n (0, T ) to the axial solution of the equation v t = log v, v(x, 0) = u 0 (x), where T = for n = 1 and T = R 2 u 0 dx/4π for n = 2. The degenerate parabolic equation ( ) u u t =, u > 0, in R n (0, ) (0.1) u(x, 0) = u 0 (x) 0 x R n where > 0 arises in the odelling of any physical phenoenon such as the flow of gases through a porous ediu or the flow of viscous fliud on a surface. When > 1, the above equation is called the porous ediu equation. When 0 < < 1, it is called the fast diffusion equation and when = 1, it becoes the faous heat equation. We refer the reader to the papers by Aronson [A] and Peletier [P] for extensive reference on the above equation. In this paper we will investigate the convergence of solution u () of the fast diffusion equation (0.1) as 0. We will show that for n = 1, 2, if u 0 L 1 (R n ) L (R n ), then as 0 the solution u () of the above fast diffusion equation will converge uniforly on every copact subset of R n (0, T ) to the axial solution of v t = log v, v > 0, in R n (0, T ) (0.2) v(x, 0) = u 0 (x) x R n where (0.3) T = if n = 1 1 4π R u 2 0 dx if n = 2. As proved in [ERV] for u 0 L 1 (R) when n = 1 and in [DP], [H] for u 0 L 1 (R 2 ) L p (R 2 ) for soe p > 1 when n = 2, there exists infinitely 297

3 298 KIN MING HUI any solutions of (0.2). However by the results of [ERV] for n = 1 and [DD] for n = 2 there exists only a single axial solution of (0.2) for n = 1, 2. So the liit li 0 u () is unique. When n = 2 L.F.Wu [W1], [W2] showed that (0.2) is related to the study of Ricci flow and when n = 1 K.G. Kurtz [K] showed that (0.2) is related to the liiting density of two type of particles oving against each other obeying the Boltzann equation. Siilar type of singular liits for solutions of (0.1) as and 1 are obtained by L.A. Caffarelli and A. Friedan [CF2], P.L. Lions, P.E. Souganidis and J.L. Vazquez [LSV] and for the p-laplacian equation which are related to the sandpile odel by L.C. Evans, M. Feldan, etc. [AEW], [EFG], [EG]. For any doain Ω R n, T > 0, φ(u) = u / for soe 0 < < 1 or φ(u) =log u, we say that u is a solution (respectively subsolution, supersolution) of (0.4) u t = φ(u) in Ω (0, T ) if u C(Ω (0, T )) C (Ω (0, T )), u > 0 on Ω (0, T ), and satisfies (0.4) (respectively, ) in Ω (0, T ) in the classical sense and we say that u has initial value u 0 if the following holds li t 0 Ω u(x, t)η(x)dx = Ω u 0 (x)η(x)dx η C 0 (Ω). We say that u is a solution (respectively subsolution, supersolution) of u t = φ(u), u > 0, in R n (0, ) u(x, 0) = u 0 (x) 0 x R n if u is a solution (respectively subsolution, supersolution) of (0.4) in R n (0, T ) with initial value u 0. We say that ṽ is a axial solution of (0.2) if ṽ v for any solution v of (0.2). For any x 0 R n, R > 0, we let B R (x 0 ) = x R n : x x 0 < R} and B R = B R (0). The plan of the paper is as follows. In Section 1 we will recall soe existence results and construct soe subsolutions of (0.1). In Section 2 we will use the subsolutions obtained in Section 1 to obtain lower bound estiates for the solution of (0.1). We will also prove a Harnack type inequality for solution of (0.1) and a convergence result for the case n = 1, 2, under the assuption that u 0 L 1 (R n ) L (R n ) and ε(1 + x 2 ) α u 0 (x) M for soe constants ε > 0, M 1, and 1 < α < 2 for all x R n. The general convergence result for u 0 L 1 (R n ) L (R n ) then follows by an approxiation arguent. Since we are concerned with the liit 0, we will assue 0 < < 1 and let u () be the solution of (0.1) for the rest of the paper.

4 SINGULAR LIMIT OF SOLUTIONS 299 Section 1. In this section we will recall soe existence results and coparsion principles for solutions of (0.1) in [AB], [BBC], [DK]. We will also construct soe subsolutions of (0.1). We first start with an existence result of [AB], [DK]: Theore 1.1 ([AB], [DK]). If u 0 L 1 (R n ) and (n 2) + /n < < 1, then (0.1) will have a unique solution u () L ((0, ); L 1 (R n )) C (R n (0, )), u () > 0 on R n (0, ) and satisfying u () t u() (1 )t in R n (0, ). Proof. Existence of solution to (0.1) is proved in [DK], [AB]. Let p = u () t /u (). Then as in [AB], [CF1], [ERV] p will satisfy the equation p t = u () 1 p + 2u () 2 u () p (1 )p 2. Since 1/(1 )t also satisfies the above equation but with initial value. By the axiu principle and the theore follows. u () t u() (1 )t Theore 1.2 ([BBC],[E]). If (n 2) + /n < < 1 and u () 1, u () 2, are subsolution and supersolutions of (0.1) with initial values u 0,1, u 0,2 L 1 (R n ) respectively, then ( ) u () 1 (x, t) u () 2 (x, t) dx (u 0,1 u 0,2 ) + dx t > 0 R n + R n and if u () 1 and u () 2 are two solutions of (0.1) with initial values u 0,1, u 0,2 L 1 (R n ) respectively, then u () 1 (x, t) u () 2 (x, t) dx u 0,1 u 0,2 dx t > 0. R n R n In particular if u 0,1 u 0,2, then u 2 u 2. Lea 1.3. If T 1 > 0, 0 < < 1, α > 1, 0 < k < 1 are constants satisfying the condition (1 ) 1 < α < 2(1 ) 1 and 0 < k < (4α) 1/(α(1 ) 1), then the function (1.1) w(x, t) = [(1 )(T 1 t) + ] 1/1 (k + x 2 ) α is a subsolution of (1.2) u t = (u /)

5 300 KIN MING HUI in R n (0, T 1 ), n = 1, 2, with initial value [(1 )T 1 ] 1/1 (k + x 2 ) α. Proof. Write w(x, t) = f(t)g(x) where f(t) = [(1 )(T 1 t) + ] 1/1 and g(x) = g(r) = (k + x 2 ) α where r = x. For n = 1 by direct coputation, (g ) (r) = 2αr(k + r 2 ) α 1 Hence (g ) (r) = 4α(α + 1)r 2 (k + r 2 ) α 2 2α(k + r 2 ) α 1. (g ) + g = (k + x 2 ) α (1 + 4α(α + 1)(k + x 2 ) α(1 ) 2 x 2 2α(k + x 2 ) α(1 ) 1 ) = (k + x 2 ) α (1 + (2α + 4α 2 )(k + x 2 ) α(1 ) 1 4α(α + 1)(k + x 2 ) α(1 ) 2 k) (k + x 2 ) α (1 + (2α + 4α 2 )k α(1 ) 1 4α(α + 1)k α(1 ) 1 ) (k + x 2 ) α (1 2αk α(1 ) 1 ) 0. Siilarly for n = 2, we have (g ) + g = (k + x 2 ) α (1 + 4α(α + 1)(k + x 2 ) α(1 ) 2 x 2 4α(k + x 2 ) α(1 ) 1 ) (k + x 2 ) α (1 4αk α(1 ) 1 ) 0. Hence ( ) w w t = f ( g ) f t g = f ( (g ) + g) 0 Thus w is a subsolution of (0.1) with initial value [(1 )T 1 ] 1/1 (k+ x 2 ) α and the lea follows. As a consequence of Theore 1.2 and Lea 1.3 we have the the following corollary. Corollary 1.4. If u 0 L 1 (R n ) and u 0 (x) ε(k + x 2 ) α for soe constants ε > 0, 0 < k < 1, 0 < < 1, satisfying (1 ) 1 < α < 2(1 ) 1 and 0 < k < (4α) 1/(α(1 ) 1), then the solution u () of (0.1) is bounded below by (ε 1 (1 )t) 1/1 + (k + x 2 ) α. Theore 1.5. If u 0 L 1 (R 1 ) for n = 1 and u 0 L 1 (R 2 ) L p (R 2 ) for soe p > 1 for n = 2, then there exists a unique axial solution v of (0.2) in R n (0, T ) where T is given by (0.3).

6 SINGULAR LIMIT OF SOLUTIONS 301 Proof. For n = 1 the theore is proved in [ERV]. For n = 2 by Theore 4.3 of [H] (cf. [DP] Theore 1.2) there exists a solution v of (0.2) in R 2 (0, T ) satisfying the condition (1.3) v(x, t)dx = u 0 dx 4πt 0 < t T R 2 R 2 where T is given by (0.3). By Prop 3.1 and Prop 4.1 of [DD], there exists a axial solution v 1 of (0.2) in R 2 (0, T ). By Theore 1.3 of [DP], v 1 (x, t)dx u 0 dx 4πt 0 < t T R 2 R 2 v 1 (x, t)dx v(x, t)dx 0 < t T by (1.3) R 2 R 2 v(x, t) = v 1 (x, t) x R 2, 0 < t T since v v 1. Hence v is the axial solution solution of (0.2). Lea 1.6. There exists a constant 0 < 1 1/2 such that the function u 1 (x, t) = A 2 t 1/1 x 2/(1 ) ( x 1) 2 is a subsolution of (1.2) in R 2 \ B 2 (0, ) for all 0 < 1 where 0 < A (2/3) 4 is a constant. Proof. Let b = 2/(1 ) and g(x) = g(r) = r b (r 1) 2 where r = x. Then (1.4) g = g + 1 r g = (b) 2 r b 2 (r 1) br b+ 2 (r 1) r b+ 2 (r 1) (2 + 1)r b+ 2 (r 1) r b+ 2 (r 1) 2 2 r 2 ( ) u 1 u 1,t = A 2 1 t /1 g A 2 (1 ) 1 t /1 r b (r 1) 2 2A 2+2 t /1 r b+ 2 (r 1) 2 2 A 2 (1 ) 1 t /1 r b (r 1) 2 = 2 A t /1 r b (r 1) 2 2 (2 2 r (1 ) 1 A 1 (r 1) 2 ).

7 302 KIN MING HUI Since li 0 = 1, there exists 0 < 1 1/2 such that > 2/3 for all 0 < 1. Thus 2 2 r (1 ) 1 A 1 (r 1) 2 2(2/3) 2 r 2A 1/2 r 22 2(2/3) 2 r 22 (r (1 2) 1) 0 0 < A (2/3) 4, 0 < 1, r 2. Hence the right hand side of (1.4) is positive for all 0 < 1, r 2, and the lea follows. By direct coputation we also have the follow result: Lea 1.7. For any 0 < A < 1, 0 < < 1, the function ( ) t 1/(1 ) u 2 (x, t) = A x 2 is a subsolution of (1.2) in R \ B 2 (0, ). Theore 1.8. condition If v is a solution of (0.2) in R 2 (0, T ) and satisfies the v(x, t) Ct ( x log x ) 2 x R, 0 < t T for soe constants C > 0 and R > 0, then v is the unique axial solution of (0.2) in R 2 (0, T ). Proof. For any 0 < t 1 < T, since v(x, t) Ct 1 ( x log x ) 2 x R, t 1 < t T by Prop 2.1 of [DD] v is the unique axial solution of w t = log w, w > 0, in R n (t 1, T ) (1.5) w(x, t 1 ) = v(x, t 1 ) x R n with n = 2. By Theore 4.3 of [H] or Theore 1.2 of [DP] there exist a solution v of (1.5) satisfying v(x, t)dx = R 2 v(x, t 1 )dx 4π(t t 1 ) R 2 0 < t 1 t T. Since v is the unique axial solution of (1.5), v v. Thus (1.6) v(x, t)dx R 2 v(x, t)dx = R 2 v(x, t 1 )dx 4π(t t 1 ) R 2 0 < t 1 t T.

8 SINGULAR LIMIT OF SOLUTIONS 303 We next let R > 0 and let η R C0 (R2 ) be such that 0 η R 1, η R (x) 1 for x R, η R (x) 0 for x 2R. Then v(x, t 1 )dx v(x, t 1 )η R (x)dx R 2 R 2 li inf v(x, t 1 )dx li inf v(x, t 1 )η R (x)dx t 1 0 R 2 t 1 0 R 2 = u 0 (x)η R (x)dx R 2 li inf v(x, t 1 )dx u 0 (x)dx as R. t 1 0 R 2 R 2 Hence letting t 1 0 in (1.6) we have (1.7) v(x, t)dx R 2 u 0 (x)dx 4πt R 2 0 < t T. By Theore 4.3 of [H] and the proof of Theore 1.5 there exists a unique axial solution ṽ of (0.2) which satisfies (1.8) ṽ(x, t)dx = R 2 u 0 (x)dx 4πt R 2 0 < t T. Since ṽ v, by (1.7) and (1.8) we get v(x, t)dx = R 2 ṽ(x, t)dx = R 2 u 0 (x)dx 4πt R 2 0 < t T v ṽ on R 2 (0, T ). Hence v is the unique axial solution of (0.2). Theore 1.9. condition If v is a solution of (0.2) in R (0, T ) and satisfies the (1.9) v(x, t) Ct x 2 x R, 0 < t T for soe constants C > 0 and R > 0, then v is the unique axial solution of (0.2) in R (0, T ). Proof. For any 0 < t 1 < T, since v satisfies (1.9) log (1/v) log Ct + 2log x o( x ) x R, t 1 < t T. By the result in Section 3 of [ERV] v is the unique axial solution of (1.5) with n = 1. By an arguent siilar to the proof of Theore 1.8 we get that v is the unique axial solution of (0.2) and the theore follows.

9 304 KIN MING HUI Section 2. In this section we will first prove the convergence of solution u () of (0.1) as 0 for n = 1, 2, under the assuption that u 0 L 1 (R n ) L (R n ) and ε(1 + x 2 ) α u 0 (x) M for soe constants ε > 0, M 1, and 1 < α < 2 for all x R n. The general convergence theore then follows by an approxiation arguent. We will first start with a Harnack type inequality. Lea 2.1. If u () L ((0, ); L 1 (R n )) L (R n (0, )) and 0 u () M for soe constant M, then for any (n 2) + /n < < 1, 0 < t 1 < t 2, t > 0, R > 0, and x 0 R n, the following inequalities hold, (i) (ii) 1 B R 1 B R B R (x 0 ) t2 t 1 (iii) I 1 1 B R u () (x, t) dx u() (x 0, t) + C MR2 (1 )t B R (x 0 ) t2 t 1 1 u () dxdt I 2 + CMR 2 B R (x 0 ) 1 u () dxdt + CMR 2 where C > 0 is a constant independent of and (2.1) I i = (t 2 t 1 ) ( 1 (1 )u () (x 0, t i )t /1 1/1 t i 2 t 1/1 )} 1, t 2 t 1 i = 1, 2. Proof. We will use a odification of the arguent of Lea 6 of [V] and Lea 4 of [GH] to prove the lea. Let (cf. [V] p. 509, [DD] p. 653) x x 0 2 n R 2 n + n 2 2 R n ( x x 0 2 R 2 ) if n 3 G R (x) = log (R/ x x 0 ) R 2 ( x x 0 2 R 2 ) if n = 2 R x x 0 + R 1 2 ( x x 0 2 R 2 ) if n = 1 be the Green s function for B R (x 0 ). Then G R 0, G R (R) = G R (R) = 0, and n(n 2)R n (n 2) B 1 δ x0 if n 3 G R = 2R 2 2πδ x0 if n = 2 R 1 2δ x0 if n = 1

10 where δ x0 (2.2) SINGULAR LIMIT OF SOLUTIONS 305 is the delta ass at x 0. Then by Theore 1.1 we have ( 1 u () ) (x, t) a n dx u() (x 0, t) B R B R (x 0 ) ( ) u () = G R (x) (x, t)dx B R (x 0 ) = G R (x)u t (x, t)dx B R (x 0 ) 1 (1 )t B R (x 0 ) G R (x)u(x, t)dx C MR2 (1 )t where a 1 = 2, a 2 = 2π, and a n = (n 2) B 1 for n 3. Thus (i) follows. Integrating (2.2) over (t 1, t 2 ) we get 1 t2 1 u () dxdt B R t 1 B R (x 0 ) t2 1 u () = dt 1 t 2 G R (x)u(x, t)dx a n t 1 By Theore 1.1, u(x, t 2 ) t 1/1 2 Hence t2 1 u () dt t 1 B R (x 0 ) u(x, t) t 1/1 u(x, t 1) t 1/1 1 t2 t 1 t 1 t t 2. t 1. 1 u () (x 0, t 2 )(t/t 2 ) /1 dt I 2 where I 2 is given by (2.1). Since G R 0 and G R (x)u(x, t)dx CMR 2 t > 0 B R (x 0 ) (ii) follows. Siilarly t2 t 1 1 u () dt I 1 where I 1 is given by (2.1) and (iii) follows. Lea 2.2. Let u () L ((0, ); L 1 (R n )) L (R n (0, )) be the solution of (0.1). If there exists a constant M 1 such that 0 u () M for all 0 < < 1 and (2.3) li inf 0 u() (x 0, t 0 ) > 0

11 306 KIN MING HUI for soe x 0 R n, n = 1, 2, 0 < t 0 < T, T > 0. Then for any R > 0, 0 < t 1 < t 2 < t 0, there exists C > 0 and 0 < 0 < 1 depending on t 1, t 2, t 0 t 2, and R > 0 such that (2.4) u () (x, t) C 0 < 0, x B R, t 1 t t 2. Proof. Without loss of generality we ay assue that R > x 0. By (2.3) there exists 0 < 1 < 1 and δ > 0 such that u () (x 0, t 0 ) δ 0 < 1. Since B 1 (y) B 2R+1 (x 0 ) for any y B R, by Lea 2.1 we have for any y B R, 0 < t 1 t t 2 < t 0, ( 1/1 (t 0 t) t 1 (1 )u () (y, t)t /1 0 t 1/1 )} (2.5) t 0 t 1 B 1 1 B 1 t0 t t0 t (2R + 1)n B 2R+1 (2R + 1)n B 2R+1 + t0 t (2R + 1)n B 2R+1 B 1 (y) 1 u () dxds + CMR 2 B 1 (y) u () 1} t0 t 1 u () dxds + CMR 2 B 2R+1 (x 0 ) u () 1} B 2R+1 (x 0 ) t0 (2R + 1) n (t 0 t) + (2R + 1)n B 2R+1 t t0 t 1 u () dxds B 2R+1 (x 0 ) u () >1} 1 u () dxds + CMR 2 u () 1 dxds + CMR 2 1 (1 )u () (x 0, t 0 )t /1 0 ( 1/1 t 0 t 1/1 )} t 0 t B 2R+1 (x 0 ) u () >1} u () 1 dxds + CMR 2 where C > 0 is a constant. Since z 1 z log z Mlog M 1 z M. The second ter on the right hand side above is bounded above by (2.6) (t 0 t)(2r + 1) n Mlog M.

12 SINGULAR LIMIT OF SOLUTIONS 307 By the ean value theore there exists t 3 (t, t 0 ) and 0 < θ 1 < 1 such that ( (t 0 t) 1 (1 )u () (x 0, t 0 )t /1 1/1 t 0 t 1/1 )} (2.7) 0 t 0 t = (t } 0 t) 1 u () (x 0, t 0 )(t 3 /t 0 ) /1 = (t 0 t)(u () (x 0, t 0 )(t 3 /t 0 ) 1/1 ) θ 1 log (t 0/t 3 ) 1/1 T Mlog (T/t 1) 1/1 1 δ 0 < 1. Siilarly there exists t 4 (t, t 0 ) such that (t 0 t) (2.8) 1 (1 )u () (y, t)t /1 0 = (t 0 t) (t 0 t 2 ) 1 u () (y, t)(t 4 /t) /1 } 1 u () (y, t)(t/t 1 ) /1 }. u () (x 0, t 0 ) ( t 1/1 0 t 1/1 t 0 t By (2.5), (2.6), (2.7), (2.8), there exists a constant c 1 > 0 such that for all 0 < 1 we have } (t 0 t 2 ) (2.9) 1 u () (y, t)(t/t 1 ) /1 c 1 ( u () (y, t) 1 c ) 1/ 1 (t 1 /T ) 1/1 0 < 1. t 0 t 2 Since the right hand side of (2.9) tends to soe positive constant independent of y B R, t 1 t t 2, there exists constants 0 < 0 1 and C > 0 such that u () (y, t) C > 0 y B R, t 1 t t 2, 0 < 0 )} and the lea follows. Theore 2.3. For n = 1, 2, if u 0 L 1 (R n ) L (R n ) and ε(k+ x 2 ) α u 0 (x) M for soe constants ε > 0, 0 < k < 1, 1 < α < 2, M 1, satisfying α > (1 2 ) 1 and 0 < k < (4α) 1/(α(1 2) 1) for soe constant 0 < 2 < 1/2, then as 0 the solution u () of (0.1) will converge uniforly on every copact subset of R n (0, T ) to the unique axial solution v of (0.2) where T is given by (0.3).

13 308 KIN MING HUI Proof. We first consider the case n = 2. For any 0 < 2, we have (1 ) 1 < α < 2(1 ) 1 and α(1 ) 1 α(1 2 ) 1. Thus 0 < k < (4α) 1/(α(1 2) 1) (4α) 1/(α(1 ) 1) for all 0 < 2. By Theore 1.2 and Corollary 1.4, (ε 1 (1 )t) 1/1 + (k + x 2 ) α u () (x, t) M x R 2, t > 0, 0 < 2. Since (ε 1 (1 )t) 1/1 + ε/2 for all 0 < t ε/2 and 0 < 1/2, M u () (x, t) C K > 0 (x, t) K for any copact subset K of R 2 (0, ε/2). Let T 0 = ax s > 0 : li inf 0 u() (x 0, s) > 0 for soe x 0 R n}. Then T 0 ε/2 > 0. For any 0 < t 1 < t 2 < T 0, let t 0 = (t 2 + T 0 )/2. Then there exists x 0 R n such that (2.10) li inf 0 u() (x 0, t 0 ) > 0. By Lea 2.2 for any R > 0 there exists 0 < 0 < 2 such that (2.4) holds. Then there exists constants C 2 > 0, C 3 > 0 independent of 0 < < 0 such that C 2 u () 1 (x, t) C 3 x B R, t 1 t t 2. Hence (0.1) is uniforly parabolic for u () } 0<<0 on any copact subset of R 2 (0, T 0 ). By standard parabolic theory [LSU], u () } 0< 0 is uniforly equi-holder continuous on any copact subset of R 2 (0, T 0 ). Let u ( i) }, i 0 as i 0, be a sequence of u (). By the Ascoli Theore and a diagonalization arguent u ( i) will have a subsequence u ( i ) converging uniforly on every copact subset K of R 2 (0, T 0 ) to a continuous function v satisfying M v C K > 0 on K for soe constant C K > 0. Without loss of generality we ay assue u ( i) converges uniforly on every copact subset K of R 2 (0, T 0 ) to v as i. Since u ( i) satisfies (0.1), for any η C 0 (R2 (0, T 0 )) we have (2.11) t 2 t 1 R 2 ( u (i) η t + u( i) i ) 1 η dxds = u (i) ηdx R 2 t 2 t 1 i 0 < t 1 < t 2 < T 0.

14 SINGULAR LIMIT OF SOLUTIONS 309 Now by the ean value theore, for each (x, t) K, there exists θ = θ(x, t) such that u ( i) i (x, t) 1 log v i u ( i) i (x, t) 1 log u ( i) log i + u ( i ) log v e ilog u ( i ) 1 log u ( i) log i + u ( i ) log v = u( i)θ i log u (i) log u ( i) log + u ( i ) log v u ( i)θ i 1 log u ( i) + log u (i) log v ax(m i 1, C i K 1 ) ax(log M, log C K ) + log u (i) log v 0 as i uniforly on K. Hence letting i in (2.11), for any η C 0 (R2 (0, T 0 )) we have t 2 t 1 R 2 ( ) vη t + log v η dxds = vηdx R 2 t 2 t 1 0 < t 1 < t 2 < T 0. By standard parabolic theory [LSU] v C (R 2 (0, T 0 )). Thus v is a solution of (0.3) in R 2 (0, T 0 ) with φ(v) = log v. We next clai that v has initial value u 0. To prove the clai we let η C0 (R2 ) be such that supp η B R for soe constant R > 2 and fix a 0 < t 2 < T 0. Then by the definition of T 0, if t 0 = (t 2 + T 0 )/2, then there exists x 0 R 2 such that (2.10) holds. By Lea 2.2 there exists δ > 0 and 0 < M 0 < in( 1, 2 ) such that u () (x, t 2 ) δ x R, 0 < < M 0 where 1 is as in Lea 1.6. By Theore 1.1, we have for any 0 < < M 0, (2.12) u () (x, t) u() (x, t 2 ) t 1/1 t 1/1 2 δ t 1/1 2 t 1/1 c 1 t 1/1 x R, 0 < t t 2

15 310 KIN MING HUI where c 1 = δ/(1 + T 0 ) 2. Hence u () (x, t)η(x)dx u 0 η(x)dx R 2 R 2 t = u () t ηdxds 0 R 2 t ( = u () ) 1 ηdxds 0 R 2 t ( = u () ) 1 ηdxds 0 B R ( t u () 1 η L dxds 0 B R u () 1} t 1 u () ) + dxds 0 B R u () <1} ( t M 1 η L 0 B R u () 1} dxds t 1 (c 1 s 1/1 ) ) + dxds 0 B R u () <1} ( t ) η L B R tmlog M (c 1 s 1/1 ) θ log (c 1 s 1/1 )ds C(t + tlog t) 0 < t < t 2 0 for soe constants C > 0 and 0 < θ < 1 by the ean value theore. Letting = i 0, we get v(x, t)η(x)dx u 0 η(x)dx C(t + tlog t) 0 as t 0. R 2 R 2 Hence v have initial value u 0 and is thus a solution of (0.2) in R 2 (0, T 0 ). We next clai that v is the unique axial solution of (0.2). To prove the clai we observe that by (2.12) ( u () (x, t) 22/1 δ 2 ) t 1/1 (1 + T 0 ) 2 x 2/(1 ) ( x 1) 2 (2.13) where A 2 t 1/1 x 2/(1 ) ( x 1) 2 x = 2, 0 < t t 2, 0 < M 0 ( A = in (2/3) 4 (log 2) 2 ) δ, 16 8 (1 + T 0 ) 2

16 SINGULAR LIMIT OF SOLUTIONS 311 since (2 1)/ = 2 θ log 2 log 2 for soe constant 0 < θ < 1. By Lea 1.6, (2.12), and the axiu principle (Lea 3.4 of [HP]), for any 0 < M 0 we have (2.14) (2.15) u () (x, t) A 2 t 1/1 x 2/(1 ) ( x 1) 2 0 < t t 2, x 2 v(x, t) = li i u ( i) (x, t) At ( x log x ) 2 0 < t t 2, x 2. Then by Theore 1.8 v is the unique axial solution of (0.2) in R 2 (0, t 2 ) for all 0 < t 2 < T 0. Hence v is the unique axial solution of (0.2) in R 2 (0, T 0 ) and u () converges uniforly to v on any copact subset of R 2 (0, T 0 ) as 0. Suppose T 0 < T where T = R u 2 0 dx/4π. By Theore 1.5 v can be extended to the unique axial solution of (0.2) in R 2 (0, T ). Since v > 0 in R 2 (0, T ) and v C (R 2 (0, T )), there exists a constant δ 1 > 0 such that Let v(x, t) δ 1 > 0 x 2, T 0 /2 t (T 0 + T )/2. ( A 1 = in (2/3) 4, (log ) 2)2 (δ 1 /2) 16 8 (1 + T 0 ) 2 and choose t 3 > 0 such that T 0 /(1 + A 1 ) < t 3 < T 0. Since u () v as 0 uniforly on B 2 t 3 }. There exists 0 < M 1 1 such that u () (x, t 3 ) δ 1 /2 x 2, 0 < M 1. By the sae arguent as the proof of (2.14) we get u () (x, t) A 1 2 t 1/1 x 2/(1 ) ( x 1) 2 x 2, 0 < t t 3, 0 < M 1. Let α = (1 ) , k 1 = (2A 1 (1 + T 0 ) 2 /δ 1 ) 1/α, and let w be as in Lea 1.3 with T 1 = (1 + A 1 )t 3 and k = k 1. Then 1/(1 ) < α < 2/(1 ) < 4 for 0 < < 1/2 and α (1 ) 1 = (1 + )(1 ) = 1 2 > α (1 ) 1 < 2 < 8 α 0 < < 1/2 0 < k 1 = (2A 1 (1 + T 0 ) 2 /δ 1 ) 1/α 16 8/α (4α ) 1/α(1 ) 1. Since by the ean value theore there exists 0 < θ < 1 such that x 1 = x θ log x x log x x 1+ x 2

17 312 KIN MING HUI and (2.16) T 1 = (1 + A 1 )t 3 ((1 )(T 1 t 3 ) + ) 1/1 (A 1 t 3 ) 1/1 A 1 t 1/1 3 hence for all x 2, 0 < < M 1, we have A 1 2 t 1/1 3 x 2/(1 ) ( x 1) 2 A 1 t 1/1 3 x 2+2+2/(1 ) ((1 )(T 1 t 3 ) + ) 1/1 (k 1 + x 2 ) α u () (x, t 3 ) ((1 )(T 1 t 3 ) + ) 1/1 (k 1 + x 2 ) α = w(x, t 3 ) x 2, 0 < M 1. Since k α 1 = 2A 1 (1 + T 0 ) 2 /δ 1, by (2.16) for any x 2, 0 < < M 1, we have Hence u () (x, t 3 ) δ 1 2 = A 1(1 + T 0 ) 2 k1 α By the axiu principle, A 1t 1/1 3 (k 1 + x 2 ) α ((1 )(T 1 t 3 ) + ) 1/1 (k 1 + x 2 ) α = w(x, t 3 ). u () (x, t 3 ) w(x, t 3 ) x R 2. u () (x, t) w(x, t) x R 2, t 3 t < T 1, 0 < M 1. Since T 1 = (1 + A 1 )t 3 > T 0, we have (T 0 + T 1 )/2 > T 0 and u () (x, (T 0 + T 1 )/2) w(x, (T 0 + T 1 )/2) > 0 x R 2. This contradicts the axiality of T 0. Hence T 0 = T where T is given by (0.3) and the theore for the case n = 2 follows. For n = 1 by an arguent siilar to the case n = 2 but with the subsolution u 2 of Lea 1.7 in place of u 1 of Lea 1.6 in the arguent we get that for n = 1 as 0 the solution u () also converges uniforly on every copact subset of R (0, ) to the axial solution v of (0.2) in R (0, ) and the theore follows. Theore 2.4. If n = 1, 2, and u 0 L 1 (R n ) L (R n ), then as 0, the solution u () of (0.1) will converge uniforly on every copact subset of R n (0, T ) to the unique axial solution v of (0.2) where T is given by (0.3). Proof. Choose 1 < α < 2, 0 < 2 < 1/2, 0 < k < 1, satisfying the conditions α > (1 2 ) 1 and 0 < k < (4α) 1/(α(1 2) 1) and let u () j be

18 SINGULAR LIMIT OF SOLUTIONS 313 the solution of (0.1) with initial value u 0j (x) = u 0 (x) + 1 j (k + x 2 ) α j = 1, 2,... Then by Theore 2.3 for each j = 1, 2,, u () j will converge uniforly on every copact subset of R n (0, T j ) to the unique axial solution v j of (0.2) with initial value u 0j as 0 where if n = 1 T j = 1 4π R 2 u 0,j (x)dx if n = 2. Suppose the theore is false. Then there exists ε > 0, R > 0, 0 < t 1 < t 2 < T and a sequence u ( i), i 0 as i, of u (), such that (2.17) u ( i) v L (B R (t 1,t 2 )) ε i = 1, 2,... where v is the unique axial solution of (0.2) with initial value u 0. Since u () L u 0 L, u (i) will have a subsequence u ( i ) converging weakly in L (R n (0, T )) and a.e. (x, t) R n (0, T ) to soe function ṽ as i. By the uniqueness of axial solution and coparsion principle for axial solution of (0.2) (Lea 4.2 of [H] or Lea 4.1 of [DP] for n = 2 and [ERV] for n = 1) we have v 1 v 2 v > 0 in R n (0, T ) Hence v j are uniforly bounded below by soe positive constant on any copact subset of R n (0, T ). Thus (0.4) with φ(u) = log u is uniforly parabolic for v j on any copact subset of R n (0, T ). Hence v j are equi- Holder continuous on any copact subset of R n (0, T ) and v j will converge uniforly on any copact subset of R n (0, T ) to soe function v v as j. By Theore 1.2 and Fatou s lea, for any 0 < t < T we have u () j (x, t) u () (x, t) dx u 0j u 0 dx R n R n 1 (k + x 2 ) α dx C j R n j T u () j (x, t) u () (x, t) dxdt CT 0 R n j T v j (x, t) ṽ(x, t) dxdt CT as = i 0 0 R n j T v(x, t) ṽ(x, t) dxdt = 0 as j 0 R n v(x, t) = ṽ(x, t) a.e. (x, t) R n (0, T ).

19 314 KIN MING HUI Since v v > 0 in R n (0, T ), ṽ(x, t) > 0 for a.e. (x, t) R n (0, T ). Thus there exists a set E R n (0, T ) of easure zero such that li i u( i ) (x, t) = ṽ(x, t) = v(x, t) v(x, t) > 0 (x, t) (R n (0, T ))\E. Hence for any R > 0, 0 < t 1 < t 2 < T, there exist x 0 R n, t 2 < t 0 < T, such that li i u( i) (x 0, t 0 ) > 0. By Lea 2.2 there exists a constant C > 0 such that u 0 L u ( i ) (x, t) C > 0 x B R, t 1 t t 2, i = 1, 2,.... Then there exists constants C 1 > 0, C 2 > 0, independent of i such that C 1 u ( i ) i 1 (x, t) C 2 x B R, t 1 t t 2, i = 1, 2,.... Hence (1.2) is uniforly parabolic for u ( i ). By standard parabolic theory [LSU], u ( i ) } is uniforly equi-holder continuous on any copact subset of R 2 (0, T ). By the Ascoli Theore and a diagonalization arguent siilar to the proof of Theore 2.3 u ( i ) will have a subsequence converging uniforly on every copact subset of R 2 (0, T ) to the axial solution v of (0.2). This contradicts (2.17). Hence the theore ust be true and u () converges uniforly on every copact subset of R 2 (0, T ) to the axial solution v of (0.2) as 0 and we are done. By the proof of Theores 2.3 and 2.4 we have the following corollary: Corollary 2.5. If v is the axial solution of (0.2) in R n (0, T ), n = 1, 2, where T is given by (0.3), then for any 0 < T 0 < T there exists a constant C > 0 such that Ct x 2, 0 < t T x v(x, t) 2 0 if n = 1 Ct x 2, 0 < t T ( x log x ) 2 0 if n = 2. [A] [AB] References D.G. Aronson, The porous ediu equation, CIME Lectures, in Soe probles in Nonlinear Diffusion, Lecture Notes in Matheatics, 1224, Springer-Verlag, New York, D.G. Aronson and P. Benilan, Régularité des solutions de l equation des ilieux porous dans R n, C.R. Acad. Sci., Paris, 288 (1979), [AEW] G. Aronsson, L.C. Evans and Y. Wu, Fast/slow duffusion and growing sandpiles, J. Diff. Equations, to appear. [BBC] P. Benilan, H. Brezis and M.G. Crandall, A seilinear elliptic equation in L 1 (R n ), Ann. Scuola Nor. Sup. Pisa, Ser. IV, 2 (1975),

20 SINGULAR LIMIT OF SOLUTIONS 315 [CF1] L.A. Caffarelli and A. Friedan, Continuity of the density of a gas flow in a porous ediu, Trans. A.M.S., 252 (1979), [CF2], Asyptotic behaviour of solutions of u t = u as, Indiana Univ. Math. J., (1987), [DK] [DP] [DD] [ERV] [E] [EFG] [EG] [GH] B.E.J. Dahlberg and C. Kenig, Non-negative solutions to fast diffusions, Revista Mateática Iberoaericana, 4 (1988), P. Daskalopoulos and M.A. Del Pino, On a singular diffusion equation, Co. in Analysis and Geoetry, 3(3) (1995), E. Dibenedetto and D.J. Diller, About a singular parabolic equation arising in thin fil dynaics and in the Ricci flow for coplete R 2, Partial differential equations and applications, , Lecture Notes in Pure and Applied atheatics, Vol. 177, edited by P. Marcellini, Giorgio G. Talenti and E. Vesentini, Dekker, New York, J.R. Esteban, A. Rodriguez and J.L. Vazquez, A nonlinear heat equation with singular diffusivity, Co. in P.D.E., 13 (1988), L.C. Evans, Applications of nonlinear seigroup theory to certain partial differential equations, in Nonlinear Evolution Equations (M.G. Crandall, Ed.), Acadeic Press, New York/London, L.C. Evans, M. Feldan and R.F. Gariepy, Fast/slow diffusion and collapsing sandpiles, 137(1) (1997), L.C. Evans and W. Gangbo, Differential equations ethods for the Monge- Kantorovich ass transfer proble, Me. Aer. Math. Soc., to appear. B.H. Gilding and L.A. Peletier, Continuity of solutions of the porous edia equation, Ann. Scuola Nor. Sup. Pisa, 8 (1981), [HP] M.A. Herrero and M. Pierre, The Cauchy proble for u t = u when 0 < < 1, Trans. A.M.S., 291(1) (1985), [H] Kin Ming Hui, Existence of solutions of the equation u t = log u, Nonlinear analysis TMA, to appear. [K] [LSU] [LSV] [P] [V] T.G. Kurtz, Convergence of sequences of seigroups of nonlinear operators with an application to gas kinetics, Trans. A.M.S., 186 (1973), O.A. Ladyzenskaya, V.A. Solonnikov and N.N. Uraltceva, Linear and quasilinear equations of parabolic type, Transl. Math. Mono., 23, Aer. Math. Soc., Providence, R.I., P.L. Lions, P.E. Souganidis and J.L. Vazquez, The relation between the porous ediu and the eikonal equations in several space diensions, Revista Mat. Iberoaericana, 3(3) (1987), L.A. Peletier, The porous ediu equation in Applications of Nonlinear Analysis in the Physical Sciences, H. Aann, N. Bazley, K. Kirchgassner editors, Pitna, Boston, J.L. Vazquez, Nonexistence of solutions for nonlinear heat equations of fastdiffusion type, J. Math. Pures Appl., 71 (1992), [W1] L.F. Wu, The Ricci flow on coplete R 2, Co. in Analysis and Geoetry, 1 (1993),

21 316 KIN MING HUI [W2], A new result for the porous ediu equation, Bull. Aer. Math. Soc., 28 (1993), Received Septeber 4, Acadeia Sinica Nankang, Taipei, Taiwan, R. O. C. E-ail address: akhui@ccvax.sinica.edu.tw