Asymptotic profile of solutions of a singular diffusion equation as t→∞

Nonlinear Analysis 48 (2002) 781 – 790

www.elsevier.com/locate/na

Asymptotic prole of solutions of a singular di#usion equation as t→∞ Shu-Yu Hsu Department of Mathematics, The Chinese University of Hong Kong, Academia Sinica, Nankang, Taipei, 11529, Taiwan Received 1 December 1999; accepted 10 May 2000

Keywords: Singular di#usion equation; Dynamical system; Large time behaviour; Asymptotic prole

Recently, there is a lot of interest on the following equation [6,23,15]: ut = 8 log u; u ¿ 0

in R2 × (0; ∞);

u(x; 0) = u0 (x) ∀x ∈ R2

(0.1)

which arises naturally in many situations. Lions and Toscani [19] have shown that (0.1) arises as the singular limit for nite velocity Boltzmann kinetic models and Kurtz [17] have shown that it arises as the limiting density distribution of two gases moving against each other and obeying the Boltzmann equation. Recently, Hui [16] has shown that (0.1) also arises as the singular limit of the porous medium equation ut = 8(um =m) as m→0. We refer the reader to the survey paper of Aronson [1] and Peletier [21] for extensive references on the above type of equation. Eq. (0.1) and other similar type of equations arising from mean curvature Bow have been studied extensively recently by Giga, Evans, Soner, Souganidis, Spruck and others [7–11,22]. Existence of nite mass solutions of (0.1) which extinct in nite time as well as the existence of a global innite mass solution has been shown by Daskalopoulos and Del Pino [3]. Existence of multiple solutions of (0.1) which extinct in nite time for

E-mail address: [email protected] (S.-Y. Hsu). 0362-546X/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 0 ) 0 0 2 1 4 - 5

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S.-Y. Hsu / Nonlinear Analysis 48 (2002) 781 – 790

u0 ∈ L1 (R2 ) ∩ Lp (R2 ) for some constant p ¿ 1 was proved by Hui [15] and for radially symmetric u0 ∈ L1 (R2 ) by Esteban et al. [6]. Decay rate of solutions of (0.1) which extinct in a nite time and behaviour of solutions near the extinction time for solutions which extinct in a nite time and decay like |x|−4 as x→∞ was obtained by Hsu in papers [12,14]. Davis et al. have obtained some regularity properties and some other properties of solutions of (0.1) in their papers [5,4]. Existence of solutions for bounded domains in Rn with Neumann boundary condition was proved in [15]. Existence and uniqueness of global solution of (0.1) satisfying the conditions lim inf r→∞

log u(x; t) ≥ −2 uniformly in [t1 ; t2 ] log r

∀t2 ¿ t1 ¿ 0

(0.2)

and ut ≤

u t

(0.3)

in R2 × (0; ∞) under the very general condition u0 ∈ L1 (R2 );

u0 ∈ Lploc (R2 );

u0 ≥ 0

for some p ¿ 1

(0.4)

was recently proved by Hsu [12]. Existence of solution of (0.1) was also proved by Wu [24,25] for u0 ∈ L1 (R2 ) and satisfying some geometric (curvature) conditions. She also obtained a partial result on the large-time behaviour of solution of (0.1) under very strong assumptions on the initial data. She showed that under proper rescaling some rescaled function of the solution of (0.1) will have a subsequence that converges to a soliton solution of (0.1) of the form ; k (x; t)

=

(|x|2

2 + ke2t )

(0.5)

for some constants  ¿ 0; k ¿ 0, as t→∞. However, her result did not exclude the possibility of the existence of a subsequence of the rescaled solution which may diverge or converge to some other function. A natural question to ask is under what condition on the initial value will the solution converge to the soliton solution of (0.1) as time goes to ininity. This question was answered by Hsu [13]. In Hsu [13] used a dynamical system approach of Chayes et al. [2] to obtain the rst result on the large-time behaviour of solution of (0.1) which decay like |x|−2 as |x|→∞. Her method was based on the construction of a suitable Lyapunov functional for the equation (0.1). She proved that the rescaled function w(x; t) = e2t u(et x; t)

(0.6)

of the solution u of (0.1) satisfying (0.2) and (0.3) with initial value u0 satisfying ; k1 ≤ u0 ≤ ; k2

in R2

(0.7)

S.-Y. Hsu / Nonlinear Analysis 48 (2002) 781 – 790

783

for some constants p ¿ 1;  ¿ 0; k1 ¿ 0; k2 ¿ 0, will converge uniformly on R2 and also in L1 (R2 ) to some function ; k0 given by 2 ; k (x) = (0.8) 2 (|x| + k) as t→∞ where the constant k0 ¿ 0 is uniquely determined by the equation
(u0 − ; k0 ) d x = 0:

(0.9)

R2

Since the proof in [13] is not easy, in this paper we will give a conceptually simple proof of the above convergence result based on the idea that the positive part and negative part of a signed solution of a parabolic equation will cancel each other as time increases. We will use a modication of the arguments of [20,26] to prove the above convergence result. The plan of the paper is as follows. In Section 1, we will recall some results on the existence and properties of solutions of (0.1) [12,13]. In Section 2, we will use a modication of the arguments of [20,26] to give another proof of the convergence result. We rst start with some denition. We say that u is a solution of ut = 8 log u 2

(0.10) 2

2

in R × (0; T ) if u ¿ 0 in R × (0; T ) and is a classical solution of (0.10) in R × (0; T ). For any u0 ∈ L1loc (R2 ); u0 ≥ 0, we say that u is a solution of (0.1) if u ∈ C([0; ∞); L1loc (R2 )) and u is a solution of (0.10) in R2 × (0; ∞) with u(·; t) − u0 (·) L1 (K) →0 as t→0 for any compact set K ⊂ R2 . For any R ¿ 0; T ¿ 0; n ∈ Z+ , we let BR = BR (0) = {x ∈ Rn : |x| ¡ R}, QRT = BR × (0; T ), and QR = BR × (0; ∞). For any a ∈ R, we let a+ = max(a; 0), a− = −min(a; 0). We will assume that ; k , ; k , w, to be given by (0.5), (0.8) and (0.6) for the rest of the paper. 1. Some estimates In this section, we will recall some results on the existence and properties of solutions of (0.1): Theorem 1.1 (Theorem 1.5 of Hsu [12]). If u1 ; u2 ; are solutions of (0:1) satisfying (0:2) and (0:3) with initial values u0; 1 ≤ u0; 2 ; then u1 ≤ u2 . In particular; if u0; 1 = u0; 2 ; then u1 ≡ u2 . Theorem 1.2 (Theorem 1.2 of Hsu [13]). Let u1 ; u2 ; be solutions of (0:1) satisfying (0:2) and (0:3) with initial values u0; 1 and u0; 2 , respectively. If u0; 1 − u0; 2 ∈ L1 (R2 ); then u1 (· ; t) − u2 (·; t) L1 (R2 ) ≤ u0; 1 − u0; 2 L1 (R2 )

∀t ¿ 0:

Theorem 1.3 (Theorems 1.2 and 1.5 of Hsu [12]). If u0 satis5es (0:4); then there exists a unique solution of (0:1) satisfying (0:2); (0:3).

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S.-Y. Hsu / Nonlinear Analysis 48 (2002) 781 – 790

By Corollary 1.7 of [12] and the discussion at the beginning of Section 2 of [13] we have Theorem 1.4 (cf. Corollary 1.7 of Hsu [12]). Suppose u0 satis5es (0:7) for some constants k1 ¿ 0; k2 ¿ 0. If u is the unique solution of (0:1) satisfying (0:2); (0:3) given by Theorem 1:3; then u and w will satisfy ; k1

≤u≤

; k2

; k1 ≤ w ≤ ; k2

in R2 × (0; ∞)

(1.1)

and wt (x; t) = 8 log w(x; t) +  div (w(x; t)x)

in R2 × (0; ∞):

(1.2)

Lemma 1.5 (Lemma 2.1 of Hsu [13]). If u0 satis5es (0:7); then there exists a unique constant k0 ¿ 0 satisfying (0:9). 2. Large time behaviour of solutions In this section, we will use a modication of the technique of [20,26] to give another proof of the large-time behaviour of solution of (0.1) with initial value satisfying (0.7). We rst start with a lemma. Lemma 2.1. Let u1 ; u2 ; be the solutions of (0:1) given by Theorem 1:3 satisfying (0:2); (0:3) with initial values u0; 1 ; u0; 2 ; satisfying (0:7) and let w1 ; w2 ; be given by (0:6) with u = u1 ; u2 ; respectively. If min( (u0; 1 − u0; 2 )+ L∞ ; (u0; 1 − u0; 2 )− L∞ ) ¿ 0;

(2.1)

(w1 − w2 )(·; t) L1 (R2 ) ¡ u0; 1 − u0; 2 L1 (R2 )

(2.2)

then ∀t ¿ 0:

Proof. We will use a modication of the argument in [20,26] to prove the lemma. By Theorem 1.4 both w1 ; w2 , satisfy (1.1) and (1.2). Let q = w1 − w2 . Then, q satises the equation qt = 8(a1 (x; t)q) +  div(q(x; t) x) in R2 × (0; ∞); where


a1 (x; t) =

0

1

(2.3)

1 d: w1 + (1 − )w2

Since both w1 ; w2 , satisfy (1.1), a1 (x; t) ∈ C ∞ satises 1 ; k2

≤ a1 (x; t) ≤

1 ; k1

:

(2.4)

S.-Y. Hsu / Nonlinear Analysis 48 (2002) 781 – 790

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Hence, (2.3) is uniformly parabolic on any compact subset of R2 × (0; ∞). By standard parabolic theory for any R ¿ 0 there exist solutions q1; R ; q2; R , of (2.3) in QR with initial values (u0; 1 − u0; 2 )+ ; (u0; 1 − u0; 2 )− , and boundary values (w1 − w2 )+ ; (w1 − w2 )− on @BR ×(0; ∞), respectively. Similarly, there exist solutions qK1; R ; qK2; R of (2.3) in QR with initial values (u0; 1 − u0; 2 )+ ; (u0; 1 − u0; 2 )− , and zero boundary value on @BR × (0; ∞). By the maximum principle, we have 0 ≤ qK1; R ≤ q1; R 0 ≤ qK2; R ≤ q2; R

in QR

(2.5)

and qK1; R ≤ qK1; R

in QR

∀R ≥ R ¿ 0;

qK2; R ≤ qK2; R

in QR

∀R ≥ R ¿ 0:

(2.6)

Since w1 ; w2 , satises (1.1), we have in R2 × (0; ∞):

(w1 − w2 )+ ; (w1 − w2 )− ≤ ; k2 − ; k1

Let q˜R be the solution of (2.3) in QR with initial value and lateral boundary value ; k2 − ; k1 . Then by the maximum principle, 0 ≤ q1; R ; q2; R ≤ q˜R :

(2.7)

As q1; R − q2; R is the solution of (2.3) in QR with initial values u0; 1 − u0; 2 = q and boundary values w1 −w2 =q on @BR ×(0; ∞). By the maximum principle, q ≡ q1; R −q2; R on QR . Let  ∈ C0∞ (R2 ); 0 ≤  ≤ 1, be such that (x) = 1 for all |x| ≤ 12 and (x) = 0 for all |x| ≥ 1 and let R (x) = (x=R). Then |8R | ≤ C=R2 , |∇R | ≤ C=R. Hence,

|u0; 1 − u0; 2 |R d x |q(x; t)|R (x) d x − R2

R2


=

R2




|(q1; R − q2; R )(x; t)|R (x) d x −


=

R2

R2

|u0; 1 − u0; 2 |R d x

(max(q1; R (x; t); q2; R (x; t)) − min(q1; R (x; t); q2; R (x; t)))R (x) d x




−

R2


=

R2

|u0; 1 − u0; 2 |R d x

(max(q1; R (x; t); q2; R (x; t)) + min(q1; R (x; t); q2; R (x; t)))R (x) d x




−

R2


|u0; 1 − u0; 2 |R d x − 2

R2

min(q1; R (x; t); q2; R (x; t))R d x

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S.-Y. Hsu / Nonlinear Analysis 48 (2002) 781 – 790


=




R2

q1; R (x; t)R (x) d x −




− =

R2

R2


t
0

+

R2

R2

R2

q2; R (x; t)R (x) d x

min(q1; R (x; t); q2; R (x; t))R d x


t
R2

0

(q2; R )t (x; t)R (x) d x dt

(a1 (x; t)q1; R (x; t)8R (x) − q1; R (x; t)x · ∇R (x)) d x dt

R2

0


R2

R2

min(q1; R (x; t); q2; R (x; t))R d x


t


−2




(q1; R )t (x; t)R (x) d x dt +




−2 =

(u0; 1 − u0; 2 )+ R d x +

(u0; 1 − u0; 2 )− R d x − 2


t
0

R2




(a1 (x; t)q2; R (x; t)8R (x) − q2; R (x; t)x · ∇R (x)) d x dt

min(q1; R (x; t); q2; R (x; t))R d x:

Hence by (2.7),

|q(x; t)|R (x) d x − |u0; 1 − u0; 2 |R d x R2

C ≤ 2 R


−2

R2


t
0

BR0

R=2≤|x|≤R

a1 (x; t)q˜R (x; t) d x dt + C


t
0

R=2≤|x|≤R

q˜R (x; t) d x dt

min(qK1; R (x; t); qK2; R (x; t))R d x

= I1 + I2 + I3

∀R ≥ R0 ¿ 0:

(2.8)

Now, observe that q˜R = q˜1; R + q˜2; R ; where q˜1; R is the solution of (2.3) in QR with initial value q˜1; R (x; 0) = 0 for all |x| ≤ R=4, q˜1; R (x; 0) = ; k2 (x) − ; k1 (x) for all R=4 ¡ |x| ≤ R and lateral boundary value ; k2 − ; k1 on @BR × (0; ∞) and q˜2; R is the solution of (2.3) in QR with initial value q˜2; R (x; 0) = ; k2 − ; k1 for all |x| ≤ R=4, q˜1; R (x; 0) = 0 for all R=4 ¡ |x| ≤ R and lateral boundary value equal to zero on @BR × (0; ∞). By the maximum principle 0 ≤ q˜1; R (x; t) ≤ q˜2; R ≥ q˜2; R ≥ 0

max

BR ×{0}∪@BR ×(0;∞)

in QR

q˜1; R ≤

∀R ¿ R ¿ 0:

C R4

∀(x; t) ∈ QR ;

S.-Y. Hsu / Nonlinear Analysis 48 (2002) 781 – 790

787

Hence, q˜2 = limR→∞ q˜2; R exists. Now,



@ 8(a1 q˜2; R ) d x +  div(q˜2; R x) d x q˜2; R d x = (q˜2; R )t d x = @t BR BR BR BR


@ @ = (a1 q˜2; R ) d +  q˜2; R x · n d = (a1 q˜2; R ) d ≤ 0; @n @n @BR @BR @BR where @=@n is the exterior normal derivative on @BR . Hence,


(; k2 − ; k1 ) d x (; k2 − ; k1 ) d x ≤ q˜2; R (x; t) d x ≤ BR

R2

|x|≤R=4

Letting R→∞ we get by the Fatou’s lemma,

q˜2 (x; t) d x ≤ (; k2 − ; k1 ) d x = C ¡ ∞ R2

⇒

R2


t
0

R2

q˜2 (x; t) d x dt ≤ Ct

∀t ¿ 0:

∀t ¿ 0

∀t ¿ 0:

Now, q˜R = q˜1; R + q˜2; R ≤

C + q˜2 R4

∀(x; t) ∈ QR :

Hence by the Lebesgue Dominated Convergence Theorem we have  

1 1 C t d x ds + q ˜ I1 ≤ 2 2 R 0 R=2≤|x|≤R ; k1 R4 ≤

Ct +C R2


t
0

R=2≤|x|≤R

q˜2 d x ds→0

Similarly, for any t ¿ 0 we have
t
Ct I2 ≤ 2 + C q˜2 d x ds→0 R 0 R=2≤|x|≤R

as R→∞

∀t ¿ 0:

as R→∞:

By (2.5) – (2.7) both qK1; R and qK2; R are monotone increasing as R increases and are uniformly bounded above. Hence, qK1 = lim qK1; R ; R→∞

qK2 = lim qK2; R R→∞

exist and are both solutions of (2.3) in R2 × (0; ∞). Thus, letting R→∞ in (2.8) we get


|q(x; t)| d x− |u0; 1 −u0; 2 | d x ≤ −2 min(qK1 (x; t); qK2 (x; t)) d x ∀R0 ; t¿0: R2

R2

BR0

(2.9)

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S.-Y. Hsu / Nonlinear Analysis 48 (2002) 781 – 790

By the denition of qK1 and qK2 , we have qK1 ≥ qK1; 2R0 ; qK2 ≥ qK2; 2R0 : Hence by (2.9) we have


|q(x; t)| d x − |u0; 1 − u0; 2 | d x ≤ −2 R2

R2

BR0

min(qK1; 2R0 (x; t); qK2; 2R0 (x; t)) d x: (2.10)

Since qK1; 2R0 and qK2; 2R0 are the solutions of (2.3) in Q2R0 with zero boundary value and initial values (u0; 1 − u0; 2 )+ , (u0; 1 − u0; 2 )− ; respectively, by the Green function representation of the solutions, for any t ¿ 0 there exists a constant ct ¿ 0 such that min qKi; 2R0 (x; t) ≥ ct ¿ 0

x ∈ BR0

∀i = 1; 2:

(2.11)

By (2.10) and (2.11), we get (2.2) and the lemma follows. We next recall a technical lemma of [20]: Lemma 2.2 (cf. Lemma 1 of Osher and Ralston [20]). Suppose w(·; ti )→wK in L1 (R2 ) as i→∞ for some sequence ti →∞. If w˜ is the solution of (1:2) in R2 × (0; ∞) with initial value w(x; ˜ 0) = w(x); K then w(·; ˜ t) − ; k L1 (R2 ) = wK − ; k L1 (R2 )

∀t ¿ 0; k ¿ 0:

We are now ready to give another proof of the convergence result: Theorem 2.3. If u0 satis5es (0:7) for some constant  ¿ 0; k1 ¿ 0; k2 ¿ 0; then the rescaled function w of the solution u of (0:1) satisfying (0:2) and (0:3) given by (0:6) will converge to ; k0 uniformly on R2 and also in L1 (R2 ) as t→∞; where ; k0 is given by (0:8) and k0 ¿ 0 is chosen such that (0:9) holds. Proof. As in the proof of Theorem 2.2 of [13], we rst observe that since w satises (1.1), Eq. (1.2) is uniformly parabolic on BK 2R ×( 12 ; ∞) for any R ¿ 0. By the Schauder’s estimates [18] w is equi-Holder continuous on BK R ×[1; ∞) for any R ¿ 0. Hence, by the Ascoli’s Theorem and a diagonalization argument any sequence {w(· ; ti )}, ti →∞ as i→∞, of {w(· ; t)} will have a convergent subsequence {w(· ; ti )} converging uniformly on every compact subset of R2 as i→∞. Without loss of generality, we may assume that {w(· ; ti )} converges uniformly on every compact subset of R2 as i→∞ and ti ≥ 1 for all i = 1; 2; : : : : Let w(x) K = limi→∞ w(x; ti ) and let k0 be given by (0.9). By (1:1), 0 ≤ w(· ; ti ) − ; k1 ≤ ; k2 − ; k1 ∈ L1 (R2 ); ; k1 ≤ wK ≤ ; k2

in R2 × (0; ∈ fty):

S.-Y. Hsu / Nonlinear Analysis 48 (2002) 781 – 790

789

Hence, (w(· ; ti ) − ; k1 ) will also converge to wK − ; k1 in L1 (R2 ) as i→∞. Also K ≤ (w(· ; ti ) − ; k1 ) + (wK − ; k1 ) ≤ 2(; k2 − ; k1 )→0 |w(· ; ti ) − w| uniformly in ti as |x|→∞. Hence, {w(· ; ti )} converges uniformly on R2 to wK as i→∞. By the
same argument as the
proof of (2:26) in [13] we have R2

(wK − ; k0 ) d x =

R2

(u0 − ; k0 ) d x = 0:

(2.12)

Now by Theorem 1.3 there exists a unique solution u˜ of (0:1) with initial value wK satisfying (0:2); (0:3). Let w˜ be given by (0:6) with u replaced by u˜ there. Then w˜ will satisfy (1.2). If min( (wK − ; k0 )+ L∞ ; (wK − ; k0 )− L∞ ) ¿ 0:

(2.13)

Since ; k0 is also a solution of (0.1), by Lemma 2.1 we have w(· ˜ ; t) − ; k0 L1 (R2 ) ¡ wK − ; k0 L1 (R2 )

∀t ¿ 0:

This contradicts Lemma 2.2. Hence (2.13) is false and we have (wK − ; k0 )+ L∞ = 0 or

(wK − ; k0 )− L∞ = 0

⇒ w(x) K ≤ ; k0 (x) ∀x ∈ R2

or

⇒ w(x) K ≡ ; k0 (x) ∀x ∈ R2

by (2:12):

w(x) K ≥ ; k0 (x)

∀x ∈ R2

Since by Lemma 1.5 k0 is uniquely given by (0:9), w(· ; t) will converge uniformly on R2 and also in L1 (R2 ) to ; k0 as t→∞ and the theorem follows. References [1] D.G. Aronson, The porous medium equation, CIME Lectures, in Some problems in Nonlinear Di#usion, Lecture Notes in Mathematics, vol. 1224, Springer, New York, 1986. [2] J.T. Chayes, S.J. Osher, J.V. Ralston, On singular di#usion equations with applications to self-organised criticality, Comm. Pure Appl. Math. XLVI (1993) 1363–1377. [3] P. Daskalopoulos, M.A. Del Pino, On a singular di#usion equation, Comm. Anal. Geom. 3 (3) (1995) 523–542. [4] S.H. Davis, E. Dibenedetto, D.J. Diller, Some a priori estimates for a singular evolution equation arising in thin-lm dynamics, SIAM J. Math. Anal. 27 (3) (1996) 638–660. [5] E. Dibenedetto, D.J. Diller, About a singular parabolic equation arising in thin lm dynamics and in the Ricci Bow for complete R2 , in: P. Marcellini, G.G. Talenti, E. Vesentini (Eds.), Partial Di#erential Equations and Applications, Lecture Notes in Pure and Appl. Math., Vol. 177, Dekker, New York, 1996, pp. 103–119. [6] J.R. Esteban, A. Rodriguez, J.L. Vazquez, The fast di#usion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane, Adv. Di#eretial Equations 1 (1) (1996) 21–50. [7] L.C. Evans, H.M. Soner, P.E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (9) (1992) 1097–1123. [8] L.C. Evans, J. Spruck, Motion of level sets by mean curvature I, J. Di#erential Geom. 33 (3) (1991) 635–681. [9] L.C. Evans, J. Spruck, Motion of level sets by mean curvature II, Trans. Amer. Math. Soc. 330 (1) (1992) 321–332. [10] Y. Giga, S. Goto, Motion of hypersurfaces and geometric equations, J. Math. Soc. Japan 44 (1992) 99–111.

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