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Existence theorems for fast diffusion equations
Nonlinear Analysis 43 (2001) 575 – 590
www.elsevier.nl/locate/na
Existence theorems for fast di usion equations Guy Bernard ∗ Department of Mathematics, Eberly College of Science, The Pennsylvania State University, University Park, PA 16802, USA Received 9 November 1998; accepted 6 April 1999
Keywords: Nonlinear di erential equations; Fast di usion equations; Parabolic di erential equations
1. Introduction Nonlinear di usion equations of the form @u = ∇ · (u m−1 ∇u); @t where ∇ is the gradient operator and m ¡ 1 are often referred to as fast di usion equations. They arise in many applications in physics, chemistry and engineering sciences. Perhaps the most important of these occur in gas kinetics theory and in plasma physics. In gas kinetics, the above equation (with m = 0) results from the uid dynamical limit of Carleman’s equations [12,13,16]. In plasma physics [4,5], this equation (with 0 ¡ m ¡ 1) models the geometry-free case of plasma di using across a magnetic eld. Central to the understanding of this equation is its corresponding Cauchy Problem. Hence, this paper focusses on positive solutions to the Cauchy Problem for the fast di usion equation, that is (E1 )
@u = ∇ · (u m−1 ∇u) in Rn × [0; T ]; @t u(x; 0) = u0 (x) for all x ∈ Rn
∗ Tel.: +902-585-1382. E-mail address: [email protected] (G. Bernard).
0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 2 2 0 - 5
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G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
or equivalently (E)
@u = u m−1 u + (m − 1)u m−2 |∇u|2 @t u(x; 0) = u0 (x) for all x ∈ Rn
in Rn × [0; T ];
where u0 ≥ 0 is the initial value, 0 ¡ T ¡ ∞ and m ¡ 1. The question of existence to the fast di usion equation in Rn has been treated by only a few writers. Global solutions in time have been demonstrated by Herrero and Pierre [11] for the restricted case of 0 ¡ m ¡ 1. Distributional solutions are shown to exist by a semigroup method for any intial value u0 ∈ L1loc (Rn ). For such nonnegative initial values, the positive solutions obtained are classical solutions if (n − 2)+ =n ¡ m ¡ 1: For the case of m = 0, few existence theorems are known for the Cauchy Problem. In the one-dimensional case i.e. n = 1, Takac [18,19] demonstrated the existence of solutions but with a Neumann condition at −∞ and a Dirichlet condition at ∞. Here the Cauchy Problem is transformed into a Boundary Value Problem on the domain (0; 1) by an appropriate transformation speci c to the one-dimensional case. For the special case of n=2 and m=0, Daskalopoulos and del Pino [8] have obtained the following de nite result: A weak nonnegative solution to (E1 ) exists if and only R if T ≤ (1=4) R2 f(x) d x ≤ ∞: Their existence results follows from a careful use of Green’s Functions of the balls BR of R2 . Surprisingly, for the very fast di usion case, i.e. when m ¡ 0, the only existence result known is that of Daskalopoulos and del Pino [9] where distributional solutions are shown to exist given intricate conditions on the initial value u0 involving a sequence of Green’s Functions of the balls BR of Rn . As for a priori estimates are concerned, Daskalopoulos and del Pino [7] established the following necessary condition (for nonnegative weak solutions) on the initial value for the range of m ¡ 0 (m ¡ − 1 if n = 1) Z 1 lim u0 (x) d x ≥ CT 1=(1−m) ; r→∞ r n+2=(m−1) | x | ≤ r where C = C(n; m) is a positive constant. This result was later made more precise and its proof much simpli ed in [3]. The purpose of this paper is to demonstrate the existence of classical positive solutions to Cauchy Problem (E) given natural conditions on the initial value u0 . These conditions turn out to be growth and decay restrictions which agree naturally with the above necessary condition of Daskalopoulos and del Pino. Furthermore, all existence results presented in this work hold for the whole range of fast di usion exponents (i.e. m ¡ 1) which covers all the above cases. The demonstrations presented in this work are based on the standard technique of nite domains. That is, the corresponding Initial Boundary Value Problem to (E) is rst solved in a sequence of expanding bounded spatial domains (balls). From these solutions, a converging subsequence is extracted by the usual diagonal process. The limit function is then shown to solve the Cauchy Problem. This method is successfully implemented by the construction of lower and upper barriers bounding all these solutions. The novelty of this work lies in the manner in which each Initial
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Boundary Value Problems is solved, that is by a uni ed method consisting of a Monotone Iteration Method and a Fixed-Point Theorem. In a rst step, a related quasilinear di erential equation to Eq. (E) where the coecient functions on its right-hand side are xed is solved by a Monotone Iteration Method. In a second step, a Fixed-Point Theorem yields the existence of a solution to the fully nonlinear Equation (E) in a set of functions bounded by the upper and lower solutions. This paper is organized as follows: In Section 2, existence theorems for local solutions in time to Di usion Equation (E) are proved while in Section 3, theorems for global solutions are demonstrated. 2. Local solutions Theorem 1. Let n ∈ N and m ¡ 1. Assume that the initial value u0 ∈ C 2; (Rn ) for some number 0 ¡ ¡ 1 and satisÿes the growth and decay conditions: C1 ≤ u0 (x) ≤ C2 (1 + | x |2 )1=(1−m) for all x ∈ Rn (1 + | x |2 )1=(1−m) for some positive constants C1 ≤ C2 . Then; there exist a classical local solution in time to Di usion equation (E). Furthermore; this solution u(x; t) satisÿes the inequality C1 e−tM ≤ u(x; t) ≤ C2 (1 + | x |2 )1=(1−m) e tM (1 + | x |2 )1=(1−m) for all (x; t) ∈ Rn × [0; T ] where e 1 2n : and T = M= 1−m (1 − m) C1 M (1 − m) Proof. The proof is given in ve steps. First, we de ne the following subset of the function space C 2; 1 (BR × [0; T ]) for all R ¿ 0: t) and kukC 1; 0 (BR ×[0;T ]) ≤ L}; KR = {u ∈ C 2; 1 (BR × [0; T ]); v(x; t) ≤ u(x; t) ≤ v(x; where L ¿ 0 is determined after Step 3 and where v(x; t) = C1 (1 + | x |2 )− e−tM ; v(x; t) = C2 (1 + | x |2 ) e tM with constants = = 1=(1 − m): Let u ∈ KR and consider the following Initial Boundary Value Problem in the closed cylinder BR × [0; T ] @v − + u m−1 v + (m − 1)u m−2 |∇v|2 = 0 in BR × [0; T ]; @t (B) v(x; t) = u (x) for all (x; t) ∈ 0 R; T v(x; 0) = u0 (x) where
R; T
n
for all x ∈ BR ;
= {x ∈ R ; | x | = R} × (0; T ).
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Step 1: For each u ∈ KR , there exist a solution v to the Initial Boundary Value Problem (B) with v ≤ v ≤ v. Let Hv = −@v=@t + u m−1 v + (m − 1)u m−2 |∇v|2 . We want to show that v is a lower solution to Problem (B). Clearly, the conditions at t = 0 and at the lateral boundary are satis ed, i.e. v(x; 0) ≤ u0 (x)
for all x ∈ BR ;
v(x; t) ≤ u0 (x)
for all (x; t) ∈
R; T :
Hence, we need only to show that H v ≥ 0: Setting r = | x |, it is then easy to compute that MC1 e−tM − 2 C1 nu m−1 (1 + r 2 )−( +1) e−tM Hv = (1 + r 2 ) + u m−2 {u[4 ( + 1)r 2 C1 (1 + r 2 )−( +2) e−tM ] − (1 − m)[4 2 C12 (1 + r 2 )−2( +1) r 2 ] e−2tM }: Since u ∈ KR and = 1=(1 − m), the last term in the above equation is nonnegative. Hence, we obtain the inequality MC1 e−tM − 2 C1 nu m−1 (1 + r 2 )−( +1) e−tM : Hv ≥ (1 + r 2 ) Again, since u ∈ KR , this inequality further reduces to MC1 e−tM − 2 C1 nC1m−1 (1 + r 2 )− (m−1) et(1−m)M (1 + r 2 )−( +1) e−tM : Hv ≥ (1 + r 2 ) Factoring the right-hand side leads to 2n e−tM C1 m−1 (1−m)TM C M− : e Hv ≥ (1 + r 2 ) (1 − m) 1 In order for v to be a lower solution, the expression in brackets in the above inequality must be nonnegative. This forces 1=(1−m) 2n eTM : C1 ≥ (1 − m)M For this inequality to hold, we choose e 1 2n : and T = M= (1 − m) C11−m M (1 − m) We now show that v is an upper solution of Problem (B) for the same choices of M and T . Clearly, the conditions at t = 0 and at the lateral boundary are satis ed, i.e. v(x; 0) ≥ u0 (x)
for all x ∈ BR ;
v(x; t) ≥ u0 (x)
for all (x; t) ∈
R; T :
Hence, we need only to show that H v ≤ 0.
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Setting r = | x |, straightforward computations show that H v = −MC2 (1 + r 2 ) e tM + 2 C2 nu m−1 (1 + r 2 )( −1) e tM + u m−2 {u[4 ( − 1)r 2 C2 (1 + r 2 )( −2) e tM ] − (1 − m)[4 2 C22 (1 + r 2 )2( −1) r 2 ]e2tM }: Since u ∈ KR and = 1=(1 − m), the last term in the above equation is nonpositive. Hence, we obtain the inequality H v ≤ −MC2 (1 + r 2 ) e tM + 2 C2 nu m−1 (1 + r 2 )( −1) e tM : Again, since u ∈ KR and = 1=(1 − m), the above inequality further reduces to H v ≤ −MC2 (1 + r 2 ) e tM + 2 C2 nC1m−1 (1 + r 2 ) (1−m) et(1−m)M (1 + r 2 )( −1) e tM : Factoring the right-hand side leads to 2n m−1 (1−m)TM tM 2 C e −M : H v ≤ e C2 (1 + r ) (1 − m) 1 In order for v to be an upper solution, the expression in brackets in the above inequality must be nonpositive. This forces 1=(1−m) 2n eTM : C1 ≥ (1 − m)M This is same inequality as the one derived for the lower solution and holds by our previous choices of M = (2n=(1 − m))e=C11−m and T = 1=[M (1 − m)]. Thus, by the Methods of Upper and Lower Solutions devised for quasilinear di erential equations [14, Section 4:2], there exists a solution u ∈ C 2+ ; 1+ =2 (BR × [0; T ]) to Initial Boundary Value Problem (B) satisfying the inequality v ≤ u ≤ v. This concludes Step 1. Step 2: The solution v to Initial Boundary Value Problem (B) is unique. Let v1 and v2 be two solutions to (B). Then, the following Initial Boundary Value Problem holds: −
@(v2 − v1 ) + u m−1 (v2 − v1 ) + (m − 1)u m−2 [|∇v2 |2 − |∇v1 |2 ] = 0 @t
(v2 − v1 )(x; t) = 0
for all (x; t) ∈
(v2 − v1 )(x; 0) = 0
for all x ∈ BR ;
in QR; T ;
R; T ;
where QR; T = BR × (0; T ). This di erential equation can be written as −
@(v2 − v1 ) + u m−1 (v2 − v1 ) + (m − 1)u m−2 [∇(v2 + v1 ) · ∇(v2 − v1 )] = 0 @t
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Let p = (v2 − v1 ). Then, the following system is satis ed: −
@p + u m−1 p + (m − 1)u m−2 [∇(v2 + v1 ) · ∇p] = 0 @t
p=0
for all (x; t) ∈
p=0
for all x ∈ BR :
in QR; T ;
R; T ;
Hence, by the uniqueness theorem for uniformly parabolic di erential equations [17, Section 3:4], we must have p = 0 in QR; T . This concludes Step 2. Step 3: The solution v of Initial Boundary Value Problem (B) satisÿes the estimate kvkC 1; 0 (BR ×[0; T ]) ≤ L4 where L4 does not depend on u ∈ KR or L (constant in KR ). More precisely; L4 = L4 (n; m; R; ; C1 ; C2 ; u0 ). Let u ∈ KR and v be the corresponding solution to u of Initial Boundary Value Problem (B). Consider the following Initial Boundary Value Problem: − (B2 )
@w + u m−1 w = 0 @t
in BR × [0; T ];
w(x; t) = u0 (x)
for all (x; t) ∈
R; T ;
w(x; 0) = u0 (x)
for all x ∈ BR :
By the theory of linear parabolic di erential equations [15, IV.9], there exists a unique solution w ∈ C 2; 1 (BR × [0; T ]) to Problem (B2 ). Set z=v−w. Thus, z satis es the following nonlinear Initial Boundary Value Problem: − (B3 )
@z + u m−1 z + (m − 1)u m−2 |∇z + ∇w|2 = 0 @t
z(x; t) = 0
for all (x; t) ∈
z(x; 0) = 0
for all x ∈ BR :
in BR × [0; T ];
R; T ;
By Lp -estimates for linear parabolic di erential equations [15, IV.9], the solution w of problem (B2 ) satis es the estimate kwkWq2; 1 (Q ) ≤ L1 [ku0 kW 2−2=q (BR ) + ku0 kW 2−2=q;1−1=q ( R
q
q
T)
]
with q = (n + 2)=(1 − ) and where the constant L1 does not depend on the coecient function u directly but its bound (uniform for all u ∈ KR ) in BR × [0; T ]. Thus, L1 = L1 (n; R; m; C1 ; C2 ; ; u0 ). The precise de nitions of the functional spaces Wql and Wql;l=2 where l is not an integer are given in [15, Section II.2 and Section II.3]. Through a Sobolev-type embedding [15, IV.9], we have that kwkC 1+ ; (1+ )=2 (BR ×[0;T ]) ≤ CkwkWq2; 1 (BR ×[0;T ]) ;
G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
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where the constant C = C(n; R; m; :C1 ). From the above estimates, we conclude that kwkC 1+ ; (1+ )=2 (BR ×[0; T ]) ≤ L2 ; where this constant depends only on L2 = L2 (R; n; m; ; C1 ; C2 ; u0 ). A similar estimate is now obtained for the solution z of Problem (B3 ) by using the results of Amann [1]. In the notation of Amann [1] g(x; t; ; ) = (m − 1)[u(x; t)]m−2 | + ∇w(x; t)|2 : Clearly, we have that |g(x; t; ; )| ≤ 2|(m − 1)|[u(x; t)]m−2 max(1; kwk2C 1; 0 (B
R ×[0; T ])m
)[||2 + 1]:
Thus, |g(x; t; ; )| ≤ C[||2 + 1|]; where C = C(R; n; m; ; C1 ; C2 ; u0 ). Hence, by Amann [16, Theorem 2:2], kzkC 1+ ; (1+ )=2 (BR ×[0; T ]) ≤ C(kzkC 0 (Q ) ); T
where the constant C depends on n; m; R; ; C1 ; C2 ; u0 and in an increasing way on the indicated quantities in parantheses. Thus kzkC 1+ ; (1+ )=2 (BR ×[0; T ]) ≤ L3 ; where L3 = L3 (n; m; R; ; C1 ; C2 ; u0 ) since kzkC(BR ×[0; T ]) ≤ C(m; R; C2 ). Since v = w + z, we obtain the following estimate: kvkC 1+ ; (1+ )=2 (BR ×[0; T ]) ≤ L4 ; where L4 = L4 (n; m; R; ; C1 ; C2 ; u0 ). This concludes Step 3. At this point of the proof, we set the constant L in the de nition of KR to have the value L = L4 . Thus by the previous steps, the following mapping is well de ned: : KR → KR ; where v = u is the solution to Initial Boundary Value Problem (B) corresponding to u as its coecient in the di erential equation. Step 4: The mapping has a ÿxed point; that is there is a function u ∈ KR satisfying the following fully nonlinear Boundary Value Problem: −
@u + u m−1 u + (m − 1)u m−2 |∇u|2 = 0 @t
u(x; t) = u0 (x)
for all (x; t) ∈
u(x; 0) = u0 (x)
for all x ∈ BR :
in QR; T ;
R; T ;
This result follows from the extended version of Schauder’s Fixed-Point Theorem [10]. Since KR is clearly a closed convex subset of the functional Banach space C 2; 1 (QR; T ),
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it suces to show that (a) is a continuous mapping, (b) (KR ) is precompact. (a) Let {un } ⊂ KR be such that un → u in C 2; 1 (QR; T ). We need to show that vn → v where vn =un and v=u. This will be the case if we can show that every subsequence of {vn } contains a subsequence which converges to v. Let {vn1 } be a subsequence of {vn }. Clearly, since {un1 } ⊂ KR we have that kun1 kC(Q
R; T )
≤ L5
and kDun1 kC(Q
R; T )
≤L
for all n1 where L5 = L5 (n; m; R; C1 ; C2 ). Hence by Holder-type estimates for linear parabolic equations [15, IV.5], the sequence {vn1 } is bounded in C 2+ ; 1+ =2 (QR; T ), i.e. kvn kC 2+ ; 1+ =2 (Q
R; T )
≤ C[ku0 kC 2+ (BR ) + ku0 kC 2+ ; 1+ =2 (
T)
];
where the constant C is independent of the coecient functions {un1 } because of the above estimates. More precisely, C = C(n; m; R; ; C1 ; C2 ; u0 ). Thus, there exist a subsequence of {vn1 } which converges in C 2; 1 (QR; T ). Let v∞ be this limit function. It follows that v∞ = u. Thus, v∞ = v by uniqueness. (b) The proof of part (b) is similar to that of (a). Step 5: There is a local solution in time u(x; t) to fast di usion equation (E) with t). T = 1=[M (1 − m)] satisfying v(x; t) ≤ u(x; t) ≤ v(x; Let {un } ⊂ KRn be the solutions given in Step 4 to Initial Boundary Value Problem (B) with Rn = n where n ∈ N. Fix R = 1. By local Holder estimates for quasilinear parabolic di erential equations [15, V.3], there is some constant L6 = L6 (n; m; R; C1 ; C2 ; u0 ) such that kDun kC(B1 ×[0; T ]) ≤ L6 for all n ∈ N. Also by contruction of the solutions there is a constant L7 = L7 (m; R; C2 ) such that kun kC(B1 ×[0; T ]) ≤ L7 for all n ∈ N. Thus, local Holder estimates for quasilinear parabolic di erential equations [15, V.5] yield that the sequence {un } is bounded in C 2+ ; 1+ =2 (B1 × [0; T ]), i.e. kun kC 2+ ; 1+ =2 (B1 ×[0; T ]) ≤ L8 ; where L8 = L8 (n; m; R; ; C1 ; C2 ; u0 ). Hence, we can extract from {un } a converging subsequence whose limit satis es Problem (B) except possibly for the lateral boundary condition. Repeating this process for R = 2; 3; 4; : : : ; we obtain a sequence of nested subsequences of {un } each converging to a solution of Problem (B) (except for the lateral boundary condition) in its corresponding cylinder BR × [0; T ]. By employing the standard diagonal process,
G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
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we extract from {un } a subsequence which converges pointwise as well as its partial derivatives ( rst-order time derivative and second-order space derivatives) to a limit function u(x; t) and to its corresponding partial derivatives. This limit function will clearly satisfy di usion equation (E) in the entire domain of Rn × [0; T ]. The previous theorem can be improved to relax the smoothness restriction on the initial condition requiring it only to be locally Holder continuous yielding a classical solution which is continuous in the whole of Rn × [0; T ] but which satis es the di erential equation only in Rn × (0; T ], that is, it satis es the equation (F)
@u = u m−1 u + (m − 1)u m−2 |∇u|2 @t u(x; 0) = u0 (x) for all x ∈ Rn :
in Rn × (0; T ];
Theorem 2. Let n ∈ N and m ¡ 1. Assume that the initial value u0 ∈ C (Rn ) where 0 ¡ ¡ 1 and satisÿes the growth and decay conditions: C1 ≤ u0 (x) ≤ C2 (1 + | x |2 )1=(1−m) (1 + | x |2 )1=(1−m)
for all x ∈ Rn
for some positive constants C1 ≤ C2 . Then; there exist a classical local solution in time to di usion equation (F) where u ∈ C(Rn × [0; T ]) with u(x; 0) = u0 (x). Furthermore; this solution u(x; t) satisÿes the inequality 31=(1−m) (1
C1 e−tM1 ≤ u(x; t) ≤ 31=(1−m) C2 (1 + | x |2 )1=(1−m) e tM1 + | x |2 )1=(1−m)
for all (x; t) ∈ Rn × [0; T ] where M1 = (2n=(1 − m)) (3e=C11−m ) and T = 1=[M1 (1 − m)]. Proof. With k ∈ N, let u0k be the following regularization of u0 : Z ([x − y]k) u0 (y) dy; u0k (x) = k n Rn
where is the standard molli er [10]. It is easy to verify that C1 ≤ u0k (x) ≤ C2 31=(1−m) (1 + | x |2 )1=(1−m) : 31=(1−m) (1 + | x |2 )1=(1−m) for all x ∈ Rn . By Theorem 1, for each k ∈ N there exists a solution u k to Cauchy Problem (E) with initial value u0k satisfying the following growth and decay conditions: 31=(1−m) (1
C1 e−tM1 ≤ u k (x; t) ≤ 31=(1−m) C2 (1 + | x |2 )1=(1−m) e tM1 + | x |2 )1=(1−m)
for all (x; t) ∈ Rn × [0; T ] where M1 = (2n=(1 − m)) (3e=C11−m ) and T = 1=[M1 (1 − m)].
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Since the solutions {uk } are unifomly bounded above and below, the Holder-type estimates for quasilinear parabolic di erential equations [15, V.1, V.3, V.5] yield that for each R ¿ 0, we have that kuk kC 2+; 1+=2 (BR ×[1=R;T ]) ≤ C; where = (n; m; R; C1 ; C2 ) with 0 ¡ ¡ 1 and C = C(n; m; R; C1 ; C2 ). By the standard diagonal process, we can extract a converging subsequence of {u k } which converges to a solution of Eq. (F). The continuity of this solution in Rn × [0; T ] follows immediatly from the uniform convergence of the subsequence (diagonal subsequence) {u k } in BR × [0; T ] and the regularizations of the initial value. The uniform convergence follows from the fact that the initial values u0k are locally Holder continuous in Rn uniformly in k. Thus, we have by Holder-type estimates for quasilinear parabolic di erential equations [15, V.5] that kuk kC ; =2 (BR ×[0; T ]) ≤ C; where = (n; m; R; C1 ; C2 ) with 0 ¡ ¡ 1 and C = C(n; m; R; ; C1 ; C2 ). The smoothness restriction on the initial condition can be further relaxed by requiring it to be only locally Holder integrable yielding a classical solution satisfying the di erential equation in Rn × (0; T ] but converging to the initial value u0 in L1loc (Rn ). Theorem 3. Let n ∈ N and m ¡ 1. Assume that the initial value u0 ∈ W1; loc (Rn ) where 0 ¡ ¡ 1 and satisÿes the growth and decay conditions: C1 ≤ u0 (x) ≤ C2 (1 + | x |2 )1=(1−m) (1 + | x |2 )1=(1−m)
for all x ∈ Rn
for some positive constants C1 ≤ C2 . Then; there exist a classical local solution in time u to di usion equation (F) such that u → u0 in L1loc (Rn ). Furthermore; this solution u(x; t) satisÿes the inequality C1 e−tM1 ≤ u(x; t) ≤ 31=(1−m) C2 (1 + | x |2 )1=(1−m) e tM1 31=(1−m) (1 + | x |2 )1=(1−m) for all (x; t) ∈ Rn × [0; T ] where M1 = (2n=(1 − m)) (3e=C11−m ) and T = 1=[M1 (1 − m)]. Remark. The space W1; loc (Rn ) is de ned in the usual way, i.e. W1; loc (Rn ) = {f; f ∈ W1 ( ) for all bounded domains }. Refer to Ladyzenskaja et al. [15, II.2] for the de nition of the space W1 ( ) when 0 ¡ ¡ 1. Proof. The demonstration is identical to that of Theorem 2 up the point of showing the convergence in L1loc (Rn ) of the solution to the initial value. This follows from the uniform convergence of a subsequence (diagonal subsequence) of {u k } in BR × [0; T ] and the regularizations of the initial value u0 . The uniform convergence follows from the fact that u0 ∈ W1; loc implies that the initial values u0k are locally Holder continuous
G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
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uniformly in k and hence imply the same Holder-type estimate as in the proof of Theorem 2, i.e. kuk kC ; =2 (BR ×[0; T ]) ≤ C where = (n; m; R; C1 ; C2 ) with 0 ¡ ¡ 1 and C = C(n; m; R; ; C1 ; C2 ). The last two theorems illustrate the instantaneous smoothing e ect of fast di usion equations as in the case of the linear parabolic di erential equations. The reason for this is that the initial value u0 and the solutions contructed in these theorems are locally bounded above and below by positive functions. 3. Global solutions In this section, global solutions in time to the Fast Di usion Equation are shown to exist, i.e. classical solutions to the following Cauchy Problem: @u = u m−1 u + (m − 1)u m−2 |∇u|2 in Rn × [0; ∞); (G) @t u(x; 0) = u0 (x) for all x ∈ Rn : Theorem 4. Let n ∈ N and m ¡ 1. Assume that the initial value u0 ∈ C 2; (Rn ) for some number 0 ¡ ≤ 1 and satisÿes the growth and decay conditions: C1 ≤ u0 (x) ≤ C2 (1 + | x |2 )(1−)=(1−m) for all x ∈ Rn 2 (1 + | x | )(1−)=(1−m) for some positive constants C1 ≤ C2 and 0 ¡ ¡ 1. Then; there exist a classical solution to Di usion equation (G). Proof. The proof is given into ve steps. First, we de ne the following subset of the function space C 1; 2 (BR × [0; 1]) for all R ¿ 0: KR∗ = {u ∈ C 2; 1 (BR × [0; 1]); vN (x; t) ≤ u(x; t) ≤ vN (x; t) and kukC 1; 0 (BR×[0; T ]) ≤ L∗ }; where L∗ ¿ 0 is determined after Step 2 and where vN (x; t) = C1 (N + | x |2 )− e−tM ; vN (x; t) = C2 (N + | x |2 ) e tM with constants = =(1−)=(1−m); M =(2n(1−)=(1−m)) (1=C11−m ) and N = eM (1−m)= . Let u ∈ KR∗ and consider the following Initial Boundary Value Problem in the closed cylinder BR × [0; 1]: @v − + u m−1 v + (m − 1)u m−2 |∇v|2 = 0 in BR × [0; 1]; @t (B4 ) v(x; t) = u (x) for all (x; t) ∈ 0 R;1 ; v(x; 0) = u0 (x) where
R;1
n
for all x ∈ BR ;
= {x ∈ R ; | x | = R} × (0; 1).
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Step 1: For each u ∈ KR∗ , there is a unique solution v ∈ KR∗ to Initial Boundary Value Problem (B4 ). As in Theorem 1, set Hv = −(@v=@t) + u m−1 v + (m − 1)u m−2 |∇v|2 . We want to show that vN is a lower solution of Initial Boundary Value Problem (B4 ). Clearly, the conditions at t = 0 and at the lateral boundary are satis ed, i.e. vN (x; 0) ≤ u0 (x)
for all x ∈ BR ;
vN (x; t) ≤ u0 (x)
for all (x; t) ∈
R;1 :
Hence, we need only to show that H vN ≥ 0: Setting r = | x |, it is then easy to compute that H vN =
MC1 e−tM − 2 C1 nu m−1 (N + r 2 )−( +1) e−tM (N + r 2 ) +u m−2 {u[4 ( + 1)r 2 C1 (N + r 2 )−( +2) e−tM ] −(1 − m)[4 2 C12 (N + r 2 )−2( +1) r 2 ] e−2tM }:
Since u ∈ KR∗ and =(1−)=(1−m), the last term in the above equation is nonnegative. Hence, we obtain the inequality H vN ≥
MC1 e−tM − 2 C1 nu m−1 (N + r 2 )−( +1) e−tM : (N + r 2 )
Again, since u ∈ KR∗ , this inequality further reduces to H vN ≥
MC1 e−tM − 2 C1 nC1m−1 (N + r 2 )− (m−1) (N + r 2 ) ×et(1−m)M (N + r 2 )−( +1) e−tM :
Factoring the right-hand side leads to 1 e−tM C1 2n(1 − ) m−1 (1−m)M e C M − : H vN ≥ (N + r 2 ) (1 − m) 1 (N + r 2 ) In order for vN to be a lower solution, we require that the expression in brackets in the above inequality be nonnegative. This forces
2n(1 − ) C1 ≥ (1 − m)M
1=(1−m)
1 eM : (N + r 2 )=(1−m)
For this inequality to hold, we choose M=
2n(1 − ) 1 (1 − m) C11−m
and
N = eM (1−m)= :
G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
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We now show that vN is an upper solution of Initial Boundary Value Problem (B2 ). Clearly, the conditions at t = 0 and at the lateral boundary are satis ed, i.e. vN (x; 0) ≥ u0 (x)
for all x ∈ BR ;
vN (x; t) ≥ u0 (x)
for all (x; t) ∈
R;1 :
Hence, we need only to show that H vN ≤ 0. Setting r = | x |, we compute H vN = −MC2 (N + r 2 ) e tM + 2 C2 nu m−1 (N + r 2 )( −1) e tM +u m−2 {u[4 ( − 1)r 2 C2 (N + r 2 )( −2) e tM ] −(1 − m)[4 2 C22 (N + r 2 )2( −1) r 2 ] e2tM }: Since u ∈ KR∗ and =(1−)=(1−m), the last term in the above equation is nonpositive. Hence, we obtain the inequality H vN ≤ −MC2 (N + r 2 ) e tM + 2 C2 nu m−1 (N + r 2 )( −1) e tM : Again, since u ∈ KR∗ and = (1 − )=(1 − m), the above inequality further reduces to H vN ≤ −MC2 (N + r 2 ) e tM + 2 C2 nC1m−1 ×(N + r 2 ) (1−m) et(1−m)M (N + r 2 )( −1) e tM : Factoring the right-hand side leads to 1 tM 2 2n(1 − ) m−1 (1−m)M C e −M : H vN ≤ e C2 (N + r ) (1 − m) 1 (N + r 2 ) In order for vN to be an upper solution, we require that the expression in brackets in the above inequality be nonpositive. This forces 1=(1−m) 1 2n(1 − ) eM : C1 ≥ 2 (1 − m)M (N + r )=(1−m) This is same inequality as the one derived for the lower solution and holds by our previous choice of M = (2n(1 − )=(1 − m)) (1=C11−m ) and N = eM (1−m)= . Thus, by the Methods of Upper and Lower Solutions devised for quasilinear di erential equations [14, Section 4:2], there exists a solution u ∈ C 2+ ; 1+ =2 (BR × [0; 1]) to Initial Boundary Value Problem (B4 ) satisfying vN ≤ u ≤ vN . The proof of the uniqueness of the solution is identical to that of Step 2 of Theorem 1. This concludes Step 1. Step 2: The solution v of the Initial Boundary Value Problem (B4 ) satisÿes the estimate kvkC 1; 0 (BR ×[0; T ]) ≤ L9 ; where L9 does not depend on u ∈ KR∗ or L∗ (constant in KR∗ ). More precisely; L9 = L9 (n; m; R; ; C1 ; C2 ; u0 ).
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G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
The proof is identical to that of Step 3 of Theorem 1. We now set the value of the constant L∗ appearing in KR∗ to be L∗ = L9 . By the previous steps, the following mapping is well de ned: : KR∗ → KR∗
by v = u;
where v is the solution to Initial Boundary Value Problen (B4 ) with u as its coecient. Step 3: The mapping has a ÿxed point; that is there is a function u ∈ KR∗ satisfying the following fully nonlinear Boundary Value Problem: @u + u m−1 u + (m − 1)u m−2 |∇u|2 = 0 @t u(x; t) = u0 (x) for all (x; t) ∈ R;1 ;
−
u(x; 0) = u0 (x)
in BR × [0; 1];
for all x ∈ BR :
The proof is identical to that of Step 4 of Theorem 1 with T = 1. Step 4: There is a local solution in time u(x; t) to fast di usion equation (E) with t). T = 1 satisfying v(x; t) ≤ u(x; t) ≤ v(x; The proof is identical to that of Step 5 of Theorem 1. Step 5: There is a solution (global in time) to Eq. (G). We begin by extending the solution obtained in Step 4 and de ned on Rn × [0; 1] to Rn × [0; 2]. First, we notice that (1 +
C1∗ 2 | x | )(1−)=(1−m)
≤ u(x; 1) ≤ C2∗ (1 + | x |2 )(1−)=(1−m)
for all x ∈ Rn ;
where C1∗ = C1 =(N eM ) and C2∗ = C2 eM N . Let u1 = u1 (x; t) be solution to di usion equation (E) corresponding to the initial value u1 (x; 0) = u(x; 1). Set u(x; t) = u1 (x; t − 1) for all (x; t) ∈ Rn × [1; 2]. It is now clear that u(x; t) thus extended satis es di usion equation (E) in Rn × [0; 2]. Repeating this process a countably in nite number of times, we obtain a positive solution to the Di usion equation (G). Similar theorems to Theorems 2 and 3 where the smoothness of the initial value is relaxed can be demonstrated for global solutions in time. Their demonstrations are identical. 4. Conclusion It would seem natural to demonstrate the existence to the fast di usion equation by the use of semigroup methods for evolution equations in abstract Banach spaces. Indeed, such methods were succesful in the study of the porous medium equation or slow di usion equation (i.e. Equation (E) with m ¿ 1) [2]. In particular, the Generation Theorem of Crandall and Liggett [6] would seem appropriate for such a nonlinear discontinuous operator as that of the fast di usion operator u m . But this operator in
G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
589
the range of m ¡ 0 could only be shown to be accretive in L1 (BR ) which is not re exive in which case, the Crandall–Liggett Theorem does not guarantee the existence of a strong solution. It is uncertain whether this operator is accretive in any useful re exive Banach Space such as L2 (BR ). Thus, a uni ed method consisting of a Monotone Iteration Method and a Fixed-Point Theorem was constructed and shown to be more
exible and appropriate to the problem at hand. The method of proof presented in this paper may be applied to many other strongly nonlinear di erential equations of second order both of parabolic and elliptic-type on bounded and unbounded domains. Systems of nonlinear di erential equations appearing in mathematical biology and chemistry may very likely also be treated by this method as perhaps the special cases of nonlinear boundary conditions when dealing with Boundary and Initial Boundary Value Problems. In a forthcoming paper by this author, this technique is applied to the slow di usion equation yielding local and global existence results analogous to those of this work. Acknowledgements I would like to thank Professor Manuel del Pino of the Universidad de Chile and Professor Panagiota Daskalopoulos of the University of California at Irvine for generously making their most recent work available to me. References [1] H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z. 150 (1976) 281–295. [2] P. Benilan, M.G. Crandall, M. Pierre, Solutions of the porous medium equation in Rn under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1) (1984) 51–87. [3] G. Bernard, A Priori Estimates for very fast di usion equations in Rn , Methods Appl. Sci., submitted. [4] J.G. Berryman, Evolution of a stable pro le for a class of nonlinear di usion equations with xed boundaries, J. Math. Phys. 18 (1977) 2108–2115. [5] J.G. Berryman, C.J. Holland, Asymptotic behaviour of the nonlinear di erential equation nt = (n−1 nx )x , J. Math. Phys. 23 (1982) 983–987. [6] M.G. Crandall, T.M. Liggett, Generation of semi-groups of nonlinear transformations on general banach spaces, Amer. J. Math. 93 (1971) 265–298. [7] P. Daskalopoulos, M.A. del Pino, On fast di usion nonlinear heat equations and a related singular elliptic problem, Indiana Univ. Math. J. 43 (2) (1994) 703–728. [8] P. Daskalopoulos, M.A. del Pino, On a singular di usion equation, Comm. Anal. Geom. 3 (3) (1995) 523–542. [9] P. Daskalopoulos, M.A. del Pino, On nonlinear parabolic equations of very fast di usion, Arch. Rational Mech. Anal. 137 (1997) 363–380. [10] D. Gilbarg, N.S. Trudinger, Elliptic Partial Di erential Equations of Second Order, 2nd Edition (Grundlehren der mathematischen Wissenschaften;224), Springer, Berlin, 1983. [11] M.A. Herrero, M. Pierre, The Cauchy Problem for ut = u m when 0 ¡ m ¡ 1, Trans. Amer. Math. Soc. 291 (1985) 145–158. [12] H.G. Kaper, G.K. Leaf, S. Reich, Convergence of semigroups with an application to the Carleman equation, Math. Methods Appl. Sci. 2 (1980) 303–308. [13] T.G. Kurtz, Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc. 186 (1973) 259–272.
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[14] G.S. Ladde, V. Ladshmikantham, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Di erential Equations, Pitman Advanced Publishing Program, Boston, 1985. [15] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, Amer. Math. Soc., Providence, RI, 1968. [16] H.P. McKean, The central limit theorem for Carleman’s equation, Israel J. Math. 21 (1975) 54–92. [17] M. Protter, H. Weinberger, Maximum Principles in Di erential Equations, Prentice-Hall, Englewood Cli s, NJ, 1967. [18] P. Takac, A fast di usion equation which generates a monotone semi ow I: local existence and uniqueness, Di erential Integral Equations 4 (1) (1991) 151–174. [19] P. Takac, A fast di usion equation which generates a monotone semi ow II: global existence and asymptotic behavior, Di erential Integral Equations 4 (1) (1991) 175–187.
www.elsevier.nl/locate/na
Existence theorems for fast di usion equations Guy Bernard ∗ Department of Mathematics, Eberly College of Science, The Pennsylvania State University, University Park, PA 16802, USA Received 9 November 1998; accepted 6 April 1999
Keywords: Nonlinear di erential equations; Fast di usion equations; Parabolic di erential equations
1. Introduction Nonlinear di usion equations of the form @u = ∇ · (u m−1 ∇u); @t where ∇ is the gradient operator and m ¡ 1 are often referred to as fast di usion equations. They arise in many applications in physics, chemistry and engineering sciences. Perhaps the most important of these occur in gas kinetics theory and in plasma physics. In gas kinetics, the above equation (with m = 0) results from the uid dynamical limit of Carleman’s equations [12,13,16]. In plasma physics [4,5], this equation (with 0 ¡ m ¡ 1) models the geometry-free case of plasma di using across a magnetic eld. Central to the understanding of this equation is its corresponding Cauchy Problem. Hence, this paper focusses on positive solutions to the Cauchy Problem for the fast di usion equation, that is (E1 )
@u = ∇ · (u m−1 ∇u) in Rn × [0; T ]; @t u(x; 0) = u0 (x) for all x ∈ Rn
∗ Tel.: +902-585-1382. E-mail address: [email protected] (G. Bernard).
0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 9 ) 0 0 2 2 0 - 5
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G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
or equivalently (E)
@u = u m−1 u + (m − 1)u m−2 |∇u|2 @t u(x; 0) = u0 (x) for all x ∈ Rn
in Rn × [0; T ];
where u0 ≥ 0 is the initial value, 0 ¡ T ¡ ∞ and m ¡ 1. The question of existence to the fast di usion equation in Rn has been treated by only a few writers. Global solutions in time have been demonstrated by Herrero and Pierre [11] for the restricted case of 0 ¡ m ¡ 1. Distributional solutions are shown to exist by a semigroup method for any intial value u0 ∈ L1loc (Rn ). For such nonnegative initial values, the positive solutions obtained are classical solutions if (n − 2)+ =n ¡ m ¡ 1: For the case of m = 0, few existence theorems are known for the Cauchy Problem. In the one-dimensional case i.e. n = 1, Takac [18,19] demonstrated the existence of solutions but with a Neumann condition at −∞ and a Dirichlet condition at ∞. Here the Cauchy Problem is transformed into a Boundary Value Problem on the domain (0; 1) by an appropriate transformation speci c to the one-dimensional case. For the special case of n=2 and m=0, Daskalopoulos and del Pino [8] have obtained the following de nite result: A weak nonnegative solution to (E1 ) exists if and only R if T ≤ (1=4) R2 f(x) d x ≤ ∞: Their existence results follows from a careful use of Green’s Functions of the balls BR of R2 . Surprisingly, for the very fast di usion case, i.e. when m ¡ 0, the only existence result known is that of Daskalopoulos and del Pino [9] where distributional solutions are shown to exist given intricate conditions on the initial value u0 involving a sequence of Green’s Functions of the balls BR of Rn . As for a priori estimates are concerned, Daskalopoulos and del Pino [7] established the following necessary condition (for nonnegative weak solutions) on the initial value for the range of m ¡ 0 (m ¡ − 1 if n = 1) Z 1 lim u0 (x) d x ≥ CT 1=(1−m) ; r→∞ r n+2=(m−1) | x | ≤ r where C = C(n; m) is a positive constant. This result was later made more precise and its proof much simpli ed in [3]. The purpose of this paper is to demonstrate the existence of classical positive solutions to Cauchy Problem (E) given natural conditions on the initial value u0 . These conditions turn out to be growth and decay restrictions which agree naturally with the above necessary condition of Daskalopoulos and del Pino. Furthermore, all existence results presented in this work hold for the whole range of fast di usion exponents (i.e. m ¡ 1) which covers all the above cases. The demonstrations presented in this work are based on the standard technique of nite domains. That is, the corresponding Initial Boundary Value Problem to (E) is rst solved in a sequence of expanding bounded spatial domains (balls). From these solutions, a converging subsequence is extracted by the usual diagonal process. The limit function is then shown to solve the Cauchy Problem. This method is successfully implemented by the construction of lower and upper barriers bounding all these solutions. The novelty of this work lies in the manner in which each Initial
G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
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Boundary Value Problems is solved, that is by a uni ed method consisting of a Monotone Iteration Method and a Fixed-Point Theorem. In a rst step, a related quasilinear di erential equation to Eq. (E) where the coecient functions on its right-hand side are xed is solved by a Monotone Iteration Method. In a second step, a Fixed-Point Theorem yields the existence of a solution to the fully nonlinear Equation (E) in a set of functions bounded by the upper and lower solutions. This paper is organized as follows: In Section 2, existence theorems for local solutions in time to Di usion Equation (E) are proved while in Section 3, theorems for global solutions are demonstrated. 2. Local solutions Theorem 1. Let n ∈ N and m ¡ 1. Assume that the initial value u0 ∈ C 2; (Rn ) for some number 0 ¡ ¡ 1 and satisÿes the growth and decay conditions: C1 ≤ u0 (x) ≤ C2 (1 + | x |2 )1=(1−m) for all x ∈ Rn (1 + | x |2 )1=(1−m) for some positive constants C1 ≤ C2 . Then; there exist a classical local solution in time to Di usion equation (E). Furthermore; this solution u(x; t) satisÿes the inequality C1 e−tM ≤ u(x; t) ≤ C2 (1 + | x |2 )1=(1−m) e tM (1 + | x |2 )1=(1−m) for all (x; t) ∈ Rn × [0; T ] where e 1 2n : and T = M= 1−m (1 − m) C1 M (1 − m) Proof. The proof is given in ve steps. First, we de ne the following subset of the function space C 2; 1 (BR × [0; T ]) for all R ¿ 0: t) and kukC 1; 0 (BR ×[0;T ]) ≤ L}; KR = {u ∈ C 2; 1 (BR × [0; T ]); v(x; t) ≤ u(x; t) ≤ v(x; where L ¿ 0 is determined after Step 3 and where v(x; t) = C1 (1 + | x |2 )− e−tM ; v(x; t) = C2 (1 + | x |2 ) e tM with constants = = 1=(1 − m): Let u ∈ KR and consider the following Initial Boundary Value Problem in the closed cylinder BR × [0; T ] @v − + u m−1 v + (m − 1)u m−2 |∇v|2 = 0 in BR × [0; T ]; @t (B) v(x; t) = u (x) for all (x; t) ∈ 0 R; T v(x; 0) = u0 (x) where
R; T
n
for all x ∈ BR ;
= {x ∈ R ; | x | = R} × (0; T ).
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G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
Step 1: For each u ∈ KR , there exist a solution v to the Initial Boundary Value Problem (B) with v ≤ v ≤ v. Let Hv = −@v=@t + u m−1 v + (m − 1)u m−2 |∇v|2 . We want to show that v is a lower solution to Problem (B). Clearly, the conditions at t = 0 and at the lateral boundary are satis ed, i.e. v(x; 0) ≤ u0 (x)
for all x ∈ BR ;
v(x; t) ≤ u0 (x)
for all (x; t) ∈
R; T :
Hence, we need only to show that H v ≥ 0: Setting r = | x |, it is then easy to compute that MC1 e−tM − 2 C1 nu m−1 (1 + r 2 )−( +1) e−tM Hv = (1 + r 2 ) + u m−2 {u[4 ( + 1)r 2 C1 (1 + r 2 )−( +2) e−tM ] − (1 − m)[4 2 C12 (1 + r 2 )−2( +1) r 2 ] e−2tM }: Since u ∈ KR and = 1=(1 − m), the last term in the above equation is nonnegative. Hence, we obtain the inequality MC1 e−tM − 2 C1 nu m−1 (1 + r 2 )−( +1) e−tM : Hv ≥ (1 + r 2 ) Again, since u ∈ KR , this inequality further reduces to MC1 e−tM − 2 C1 nC1m−1 (1 + r 2 )− (m−1) et(1−m)M (1 + r 2 )−( +1) e−tM : Hv ≥ (1 + r 2 ) Factoring the right-hand side leads to 2n e−tM C1 m−1 (1−m)TM C M− : e Hv ≥ (1 + r 2 ) (1 − m) 1 In order for v to be a lower solution, the expression in brackets in the above inequality must be nonnegative. This forces 1=(1−m) 2n eTM : C1 ≥ (1 − m)M For this inequality to hold, we choose e 1 2n : and T = M= (1 − m) C11−m M (1 − m) We now show that v is an upper solution of Problem (B) for the same choices of M and T . Clearly, the conditions at t = 0 and at the lateral boundary are satis ed, i.e. v(x; 0) ≥ u0 (x)
for all x ∈ BR ;
v(x; t) ≥ u0 (x)
for all (x; t) ∈
R; T :
Hence, we need only to show that H v ≤ 0.
G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
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Setting r = | x |, straightforward computations show that H v = −MC2 (1 + r 2 ) e tM + 2 C2 nu m−1 (1 + r 2 )( −1) e tM + u m−2 {u[4 ( − 1)r 2 C2 (1 + r 2 )( −2) e tM ] − (1 − m)[4 2 C22 (1 + r 2 )2( −1) r 2 ]e2tM }: Since u ∈ KR and = 1=(1 − m), the last term in the above equation is nonpositive. Hence, we obtain the inequality H v ≤ −MC2 (1 + r 2 ) e tM + 2 C2 nu m−1 (1 + r 2 )( −1) e tM : Again, since u ∈ KR and = 1=(1 − m), the above inequality further reduces to H v ≤ −MC2 (1 + r 2 ) e tM + 2 C2 nC1m−1 (1 + r 2 ) (1−m) et(1−m)M (1 + r 2 )( −1) e tM : Factoring the right-hand side leads to 2n m−1 (1−m)TM tM 2 C e −M : H v ≤ e C2 (1 + r ) (1 − m) 1 In order for v to be an upper solution, the expression in brackets in the above inequality must be nonpositive. This forces 1=(1−m) 2n eTM : C1 ≥ (1 − m)M This is same inequality as the one derived for the lower solution and holds by our previous choices of M = (2n=(1 − m))e=C11−m and T = 1=[M (1 − m)]. Thus, by the Methods of Upper and Lower Solutions devised for quasilinear di erential equations [14, Section 4:2], there exists a solution u ∈ C 2+ ; 1+ =2 (BR × [0; T ]) to Initial Boundary Value Problem (B) satisfying the inequality v ≤ u ≤ v. This concludes Step 1. Step 2: The solution v to Initial Boundary Value Problem (B) is unique. Let v1 and v2 be two solutions to (B). Then, the following Initial Boundary Value Problem holds: −
@(v2 − v1 ) + u m−1 (v2 − v1 ) + (m − 1)u m−2 [|∇v2 |2 − |∇v1 |2 ] = 0 @t
(v2 − v1 )(x; t) = 0
for all (x; t) ∈
(v2 − v1 )(x; 0) = 0
for all x ∈ BR ;
in QR; T ;
R; T ;
where QR; T = BR × (0; T ). This di erential equation can be written as −
@(v2 − v1 ) + u m−1 (v2 − v1 ) + (m − 1)u m−2 [∇(v2 + v1 ) · ∇(v2 − v1 )] = 0 @t
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G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
Let p = (v2 − v1 ). Then, the following system is satis ed: −
@p + u m−1 p + (m − 1)u m−2 [∇(v2 + v1 ) · ∇p] = 0 @t
p=0
for all (x; t) ∈
p=0
for all x ∈ BR :
in QR; T ;
R; T ;
Hence, by the uniqueness theorem for uniformly parabolic di erential equations [17, Section 3:4], we must have p = 0 in QR; T . This concludes Step 2. Step 3: The solution v of Initial Boundary Value Problem (B) satisÿes the estimate kvkC 1; 0 (BR ×[0; T ]) ≤ L4 where L4 does not depend on u ∈ KR or L (constant in KR ). More precisely; L4 = L4 (n; m; R; ; C1 ; C2 ; u0 ). Let u ∈ KR and v be the corresponding solution to u of Initial Boundary Value Problem (B). Consider the following Initial Boundary Value Problem: − (B2 )
@w + u m−1 w = 0 @t
in BR × [0; T ];
w(x; t) = u0 (x)
for all (x; t) ∈
R; T ;
w(x; 0) = u0 (x)
for all x ∈ BR :
By the theory of linear parabolic di erential equations [15, IV.9], there exists a unique solution w ∈ C 2; 1 (BR × [0; T ]) to Problem (B2 ). Set z=v−w. Thus, z satis es the following nonlinear Initial Boundary Value Problem: − (B3 )
@z + u m−1 z + (m − 1)u m−2 |∇z + ∇w|2 = 0 @t
z(x; t) = 0
for all (x; t) ∈
z(x; 0) = 0
for all x ∈ BR :
in BR × [0; T ];
R; T ;
By Lp -estimates for linear parabolic di erential equations [15, IV.9], the solution w of problem (B2 ) satis es the estimate kwkWq2; 1 (Q ) ≤ L1 [ku0 kW 2−2=q (BR ) + ku0 kW 2−2=q;1−1=q ( R
q
q
T)
]
with q = (n + 2)=(1 − ) and where the constant L1 does not depend on the coecient function u directly but its bound (uniform for all u ∈ KR ) in BR × [0; T ]. Thus, L1 = L1 (n; R; m; C1 ; C2 ; ; u0 ). The precise de nitions of the functional spaces Wql and Wql;l=2 where l is not an integer are given in [15, Section II.2 and Section II.3]. Through a Sobolev-type embedding [15, IV.9], we have that kwkC 1+ ; (1+ )=2 (BR ×[0;T ]) ≤ CkwkWq2; 1 (BR ×[0;T ]) ;
G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
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where the constant C = C(n; R; m; :C1 ). From the above estimates, we conclude that kwkC 1+ ; (1+ )=2 (BR ×[0; T ]) ≤ L2 ; where this constant depends only on L2 = L2 (R; n; m; ; C1 ; C2 ; u0 ). A similar estimate is now obtained for the solution z of Problem (B3 ) by using the results of Amann [1]. In the notation of Amann [1] g(x; t; ; ) = (m − 1)[u(x; t)]m−2 | + ∇w(x; t)|2 : Clearly, we have that |g(x; t; ; )| ≤ 2|(m − 1)|[u(x; t)]m−2 max(1; kwk2C 1; 0 (B
R ×[0; T ])m
)[||2 + 1]:
Thus, |g(x; t; ; )| ≤ C[||2 + 1|]; where C = C(R; n; m; ; C1 ; C2 ; u0 ). Hence, by Amann [16, Theorem 2:2], kzkC 1+ ; (1+ )=2 (BR ×[0; T ]) ≤ C(kzkC 0 (Q ) ); T
where the constant C depends on n; m; R; ; C1 ; C2 ; u0 and in an increasing way on the indicated quantities in parantheses. Thus kzkC 1+ ; (1+ )=2 (BR ×[0; T ]) ≤ L3 ; where L3 = L3 (n; m; R; ; C1 ; C2 ; u0 ) since kzkC(BR ×[0; T ]) ≤ C(m; R; C2 ). Since v = w + z, we obtain the following estimate: kvkC 1+ ; (1+ )=2 (BR ×[0; T ]) ≤ L4 ; where L4 = L4 (n; m; R; ; C1 ; C2 ; u0 ). This concludes Step 3. At this point of the proof, we set the constant L in the de nition of KR to have the value L = L4 . Thus by the previous steps, the following mapping is well de ned: : KR → KR ; where v = u is the solution to Initial Boundary Value Problem (B) corresponding to u as its coecient in the di erential equation. Step 4: The mapping has a ÿxed point; that is there is a function u ∈ KR satisfying the following fully nonlinear Boundary Value Problem: −
@u + u m−1 u + (m − 1)u m−2 |∇u|2 = 0 @t
u(x; t) = u0 (x)
for all (x; t) ∈
u(x; 0) = u0 (x)
for all x ∈ BR :
in QR; T ;
R; T ;
This result follows from the extended version of Schauder’s Fixed-Point Theorem [10]. Since KR is clearly a closed convex subset of the functional Banach space C 2; 1 (QR; T ),
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it suces to show that (a) is a continuous mapping, (b) (KR ) is precompact. (a) Let {un } ⊂ KR be such that un → u in C 2; 1 (QR; T ). We need to show that vn → v where vn =un and v=u. This will be the case if we can show that every subsequence of {vn } contains a subsequence which converges to v. Let {vn1 } be a subsequence of {vn }. Clearly, since {un1 } ⊂ KR we have that kun1 kC(Q
R; T )
≤ L5
and kDun1 kC(Q
R; T )
≤L
for all n1 where L5 = L5 (n; m; R; C1 ; C2 ). Hence by Holder-type estimates for linear parabolic equations [15, IV.5], the sequence {vn1 } is bounded in C 2+ ; 1+ =2 (QR; T ), i.e. kvn kC 2+ ; 1+ =2 (Q
R; T )
≤ C[ku0 kC 2+ (BR ) + ku0 kC 2+ ; 1+ =2 (
T)
];
where the constant C is independent of the coecient functions {un1 } because of the above estimates. More precisely, C = C(n; m; R; ; C1 ; C2 ; u0 ). Thus, there exist a subsequence of {vn1 } which converges in C 2; 1 (QR; T ). Let v∞ be this limit function. It follows that v∞ = u. Thus, v∞ = v by uniqueness. (b) The proof of part (b) is similar to that of (a). Step 5: There is a local solution in time u(x; t) to fast di usion equation (E) with t). T = 1=[M (1 − m)] satisfying v(x; t) ≤ u(x; t) ≤ v(x; Let {un } ⊂ KRn be the solutions given in Step 4 to Initial Boundary Value Problem (B) with Rn = n where n ∈ N. Fix R = 1. By local Holder estimates for quasilinear parabolic di erential equations [15, V.3], there is some constant L6 = L6 (n; m; R; C1 ; C2 ; u0 ) such that kDun kC(B1 ×[0; T ]) ≤ L6 for all n ∈ N. Also by contruction of the solutions there is a constant L7 = L7 (m; R; C2 ) such that kun kC(B1 ×[0; T ]) ≤ L7 for all n ∈ N. Thus, local Holder estimates for quasilinear parabolic di erential equations [15, V.5] yield that the sequence {un } is bounded in C 2+ ; 1+ =2 (B1 × [0; T ]), i.e. kun kC 2+ ; 1+ =2 (B1 ×[0; T ]) ≤ L8 ; where L8 = L8 (n; m; R; ; C1 ; C2 ; u0 ). Hence, we can extract from {un } a converging subsequence whose limit satis es Problem (B) except possibly for the lateral boundary condition. Repeating this process for R = 2; 3; 4; : : : ; we obtain a sequence of nested subsequences of {un } each converging to a solution of Problem (B) (except for the lateral boundary condition) in its corresponding cylinder BR × [0; T ]. By employing the standard diagonal process,
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we extract from {un } a subsequence which converges pointwise as well as its partial derivatives ( rst-order time derivative and second-order space derivatives) to a limit function u(x; t) and to its corresponding partial derivatives. This limit function will clearly satisfy di usion equation (E) in the entire domain of Rn × [0; T ]. The previous theorem can be improved to relax the smoothness restriction on the initial condition requiring it only to be locally Holder continuous yielding a classical solution which is continuous in the whole of Rn × [0; T ] but which satis es the di erential equation only in Rn × (0; T ], that is, it satis es the equation (F)
@u = u m−1 u + (m − 1)u m−2 |∇u|2 @t u(x; 0) = u0 (x) for all x ∈ Rn :
in Rn × (0; T ];
Theorem 2. Let n ∈ N and m ¡ 1. Assume that the initial value u0 ∈ C (Rn ) where 0 ¡ ¡ 1 and satisÿes the growth and decay conditions: C1 ≤ u0 (x) ≤ C2 (1 + | x |2 )1=(1−m) (1 + | x |2 )1=(1−m)
for all x ∈ Rn
for some positive constants C1 ≤ C2 . Then; there exist a classical local solution in time to di usion equation (F) where u ∈ C(Rn × [0; T ]) with u(x; 0) = u0 (x). Furthermore; this solution u(x; t) satisÿes the inequality 31=(1−m) (1
C1 e−tM1 ≤ u(x; t) ≤ 31=(1−m) C2 (1 + | x |2 )1=(1−m) e tM1 + | x |2 )1=(1−m)
for all (x; t) ∈ Rn × [0; T ] where M1 = (2n=(1 − m)) (3e=C11−m ) and T = 1=[M1 (1 − m)]. Proof. With k ∈ N, let u0k be the following regularization of u0 : Z ([x − y]k) u0 (y) dy; u0k (x) = k n Rn
where is the standard molli er [10]. It is easy to verify that C1 ≤ u0k (x) ≤ C2 31=(1−m) (1 + | x |2 )1=(1−m) : 31=(1−m) (1 + | x |2 )1=(1−m) for all x ∈ Rn . By Theorem 1, for each k ∈ N there exists a solution u k to Cauchy Problem (E) with initial value u0k satisfying the following growth and decay conditions: 31=(1−m) (1
C1 e−tM1 ≤ u k (x; t) ≤ 31=(1−m) C2 (1 + | x |2 )1=(1−m) e tM1 + | x |2 )1=(1−m)
for all (x; t) ∈ Rn × [0; T ] where M1 = (2n=(1 − m)) (3e=C11−m ) and T = 1=[M1 (1 − m)].
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Since the solutions {uk } are unifomly bounded above and below, the Holder-type estimates for quasilinear parabolic di erential equations [15, V.1, V.3, V.5] yield that for each R ¿ 0, we have that kuk kC 2+; 1+=2 (BR ×[1=R;T ]) ≤ C; where = (n; m; R; C1 ; C2 ) with 0 ¡ ¡ 1 and C = C(n; m; R; C1 ; C2 ). By the standard diagonal process, we can extract a converging subsequence of {u k } which converges to a solution of Eq. (F). The continuity of this solution in Rn × [0; T ] follows immediatly from the uniform convergence of the subsequence (diagonal subsequence) {u k } in BR × [0; T ] and the regularizations of the initial value. The uniform convergence follows from the fact that the initial values u0k are locally Holder continuous in Rn uniformly in k. Thus, we have by Holder-type estimates for quasilinear parabolic di erential equations [15, V.5] that kuk kC ; =2 (BR ×[0; T ]) ≤ C; where = (n; m; R; C1 ; C2 ) with 0 ¡ ¡ 1 and C = C(n; m; R; ; C1 ; C2 ). The smoothness restriction on the initial condition can be further relaxed by requiring it to be only locally Holder integrable yielding a classical solution satisfying the di erential equation in Rn × (0; T ] but converging to the initial value u0 in L1loc (Rn ). Theorem 3. Let n ∈ N and m ¡ 1. Assume that the initial value u0 ∈ W1; loc (Rn ) where 0 ¡ ¡ 1 and satisÿes the growth and decay conditions: C1 ≤ u0 (x) ≤ C2 (1 + | x |2 )1=(1−m) (1 + | x |2 )1=(1−m)
for all x ∈ Rn
for some positive constants C1 ≤ C2 . Then; there exist a classical local solution in time u to di usion equation (F) such that u → u0 in L1loc (Rn ). Furthermore; this solution u(x; t) satisÿes the inequality C1 e−tM1 ≤ u(x; t) ≤ 31=(1−m) C2 (1 + | x |2 )1=(1−m) e tM1 31=(1−m) (1 + | x |2 )1=(1−m) for all (x; t) ∈ Rn × [0; T ] where M1 = (2n=(1 − m)) (3e=C11−m ) and T = 1=[M1 (1 − m)]. Remark. The space W1; loc (Rn ) is de ned in the usual way, i.e. W1; loc (Rn ) = {f; f ∈ W1 ( ) for all bounded domains }. Refer to Ladyzenskaja et al. [15, II.2] for the de nition of the space W1 ( ) when 0 ¡ ¡ 1. Proof. The demonstration is identical to that of Theorem 2 up the point of showing the convergence in L1loc (Rn ) of the solution to the initial value. This follows from the uniform convergence of a subsequence (diagonal subsequence) of {u k } in BR × [0; T ] and the regularizations of the initial value u0 . The uniform convergence follows from the fact that u0 ∈ W1; loc implies that the initial values u0k are locally Holder continuous
G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
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uniformly in k and hence imply the same Holder-type estimate as in the proof of Theorem 2, i.e. kuk kC ; =2 (BR ×[0; T ]) ≤ C where = (n; m; R; C1 ; C2 ) with 0 ¡ ¡ 1 and C = C(n; m; R; ; C1 ; C2 ). The last two theorems illustrate the instantaneous smoothing e ect of fast di usion equations as in the case of the linear parabolic di erential equations. The reason for this is that the initial value u0 and the solutions contructed in these theorems are locally bounded above and below by positive functions. 3. Global solutions In this section, global solutions in time to the Fast Di usion Equation are shown to exist, i.e. classical solutions to the following Cauchy Problem: @u = u m−1 u + (m − 1)u m−2 |∇u|2 in Rn × [0; ∞); (G) @t u(x; 0) = u0 (x) for all x ∈ Rn : Theorem 4. Let n ∈ N and m ¡ 1. Assume that the initial value u0 ∈ C 2; (Rn ) for some number 0 ¡ ≤ 1 and satisÿes the growth and decay conditions: C1 ≤ u0 (x) ≤ C2 (1 + | x |2 )(1−)=(1−m) for all x ∈ Rn 2 (1 + | x | )(1−)=(1−m) for some positive constants C1 ≤ C2 and 0 ¡ ¡ 1. Then; there exist a classical solution to Di usion equation (G). Proof. The proof is given into ve steps. First, we de ne the following subset of the function space C 1; 2 (BR × [0; 1]) for all R ¿ 0: KR∗ = {u ∈ C 2; 1 (BR × [0; 1]); vN (x; t) ≤ u(x; t) ≤ vN (x; t) and kukC 1; 0 (BR×[0; T ]) ≤ L∗ }; where L∗ ¿ 0 is determined after Step 2 and where vN (x; t) = C1 (N + | x |2 )− e−tM ; vN (x; t) = C2 (N + | x |2 ) e tM with constants = =(1−)=(1−m); M =(2n(1−)=(1−m)) (1=C11−m ) and N = eM (1−m)= . Let u ∈ KR∗ and consider the following Initial Boundary Value Problem in the closed cylinder BR × [0; 1]: @v − + u m−1 v + (m − 1)u m−2 |∇v|2 = 0 in BR × [0; 1]; @t (B4 ) v(x; t) = u (x) for all (x; t) ∈ 0 R;1 ; v(x; 0) = u0 (x) where
R;1
n
for all x ∈ BR ;
= {x ∈ R ; | x | = R} × (0; 1).
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G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
Step 1: For each u ∈ KR∗ , there is a unique solution v ∈ KR∗ to Initial Boundary Value Problem (B4 ). As in Theorem 1, set Hv = −(@v=@t) + u m−1 v + (m − 1)u m−2 |∇v|2 . We want to show that vN is a lower solution of Initial Boundary Value Problem (B4 ). Clearly, the conditions at t = 0 and at the lateral boundary are satis ed, i.e. vN (x; 0) ≤ u0 (x)
for all x ∈ BR ;
vN (x; t) ≤ u0 (x)
for all (x; t) ∈
R;1 :
Hence, we need only to show that H vN ≥ 0: Setting r = | x |, it is then easy to compute that H vN =
MC1 e−tM − 2 C1 nu m−1 (N + r 2 )−( +1) e−tM (N + r 2 ) +u m−2 {u[4 ( + 1)r 2 C1 (N + r 2 )−( +2) e−tM ] −(1 − m)[4 2 C12 (N + r 2 )−2( +1) r 2 ] e−2tM }:
Since u ∈ KR∗ and =(1−)=(1−m), the last term in the above equation is nonnegative. Hence, we obtain the inequality H vN ≥
MC1 e−tM − 2 C1 nu m−1 (N + r 2 )−( +1) e−tM : (N + r 2 )
Again, since u ∈ KR∗ , this inequality further reduces to H vN ≥
MC1 e−tM − 2 C1 nC1m−1 (N + r 2 )− (m−1) (N + r 2 ) ×et(1−m)M (N + r 2 )−( +1) e−tM :
Factoring the right-hand side leads to 1 e−tM C1 2n(1 − ) m−1 (1−m)M e C M − : H vN ≥ (N + r 2 ) (1 − m) 1 (N + r 2 ) In order for vN to be a lower solution, we require that the expression in brackets in the above inequality be nonnegative. This forces
2n(1 − ) C1 ≥ (1 − m)M
1=(1−m)
1 eM : (N + r 2 )=(1−m)
For this inequality to hold, we choose M=
2n(1 − ) 1 (1 − m) C11−m
and
N = eM (1−m)= :
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We now show that vN is an upper solution of Initial Boundary Value Problem (B2 ). Clearly, the conditions at t = 0 and at the lateral boundary are satis ed, i.e. vN (x; 0) ≥ u0 (x)
for all x ∈ BR ;
vN (x; t) ≥ u0 (x)
for all (x; t) ∈
R;1 :
Hence, we need only to show that H vN ≤ 0. Setting r = | x |, we compute H vN = −MC2 (N + r 2 ) e tM + 2 C2 nu m−1 (N + r 2 )( −1) e tM +u m−2 {u[4 ( − 1)r 2 C2 (N + r 2 )( −2) e tM ] −(1 − m)[4 2 C22 (N + r 2 )2( −1) r 2 ] e2tM }: Since u ∈ KR∗ and =(1−)=(1−m), the last term in the above equation is nonpositive. Hence, we obtain the inequality H vN ≤ −MC2 (N + r 2 ) e tM + 2 C2 nu m−1 (N + r 2 )( −1) e tM : Again, since u ∈ KR∗ and = (1 − )=(1 − m), the above inequality further reduces to H vN ≤ −MC2 (N + r 2 ) e tM + 2 C2 nC1m−1 ×(N + r 2 ) (1−m) et(1−m)M (N + r 2 )( −1) e tM : Factoring the right-hand side leads to 1 tM 2 2n(1 − ) m−1 (1−m)M C e −M : H vN ≤ e C2 (N + r ) (1 − m) 1 (N + r 2 ) In order for vN to be an upper solution, we require that the expression in brackets in the above inequality be nonpositive. This forces 1=(1−m) 1 2n(1 − ) eM : C1 ≥ 2 (1 − m)M (N + r )=(1−m) This is same inequality as the one derived for the lower solution and holds by our previous choice of M = (2n(1 − )=(1 − m)) (1=C11−m ) and N = eM (1−m)= . Thus, by the Methods of Upper and Lower Solutions devised for quasilinear di erential equations [14, Section 4:2], there exists a solution u ∈ C 2+ ; 1+ =2 (BR × [0; 1]) to Initial Boundary Value Problem (B4 ) satisfying vN ≤ u ≤ vN . The proof of the uniqueness of the solution is identical to that of Step 2 of Theorem 1. This concludes Step 1. Step 2: The solution v of the Initial Boundary Value Problem (B4 ) satisÿes the estimate kvkC 1; 0 (BR ×[0; T ]) ≤ L9 ; where L9 does not depend on u ∈ KR∗ or L∗ (constant in KR∗ ). More precisely; L9 = L9 (n; m; R; ; C1 ; C2 ; u0 ).
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G. Bernard / Nonlinear Analysis 43 (2001) 575 – 590
The proof is identical to that of Step 3 of Theorem 1. We now set the value of the constant L∗ appearing in KR∗ to be L∗ = L9 . By the previous steps, the following mapping is well de ned: : KR∗ → KR∗
by v = u;
where v is the solution to Initial Boundary Value Problen (B4 ) with u as its coecient. Step 3: The mapping has a ÿxed point; that is there is a function u ∈ KR∗ satisfying the following fully nonlinear Boundary Value Problem: @u + u m−1 u + (m − 1)u m−2 |∇u|2 = 0 @t u(x; t) = u0 (x) for all (x; t) ∈ R;1 ;
−
u(x; 0) = u0 (x)
in BR × [0; 1];
for all x ∈ BR :
The proof is identical to that of Step 4 of Theorem 1 with T = 1. Step 4: There is a local solution in time u(x; t) to fast di usion equation (E) with t). T = 1 satisfying v(x; t) ≤ u(x; t) ≤ v(x; The proof is identical to that of Step 5 of Theorem 1. Step 5: There is a solution (global in time) to Eq. (G). We begin by extending the solution obtained in Step 4 and de ned on Rn × [0; 1] to Rn × [0; 2]. First, we notice that (1 +
C1∗ 2 | x | )(1−)=(1−m)
≤ u(x; 1) ≤ C2∗ (1 + | x |2 )(1−)=(1−m)
for all x ∈ Rn ;
where C1∗ = C1 =(N eM ) and C2∗ = C2 eM N . Let u1 = u1 (x; t) be solution to di usion equation (E) corresponding to the initial value u1 (x; 0) = u(x; 1). Set u(x; t) = u1 (x; t − 1) for all (x; t) ∈ Rn × [1; 2]. It is now clear that u(x; t) thus extended satis es di usion equation (E) in Rn × [0; 2]. Repeating this process a countably in nite number of times, we obtain a positive solution to the Di usion equation (G). Similar theorems to Theorems 2 and 3 where the smoothness of the initial value is relaxed can be demonstrated for global solutions in time. Their demonstrations are identical. 4. Conclusion It would seem natural to demonstrate the existence to the fast di usion equation by the use of semigroup methods for evolution equations in abstract Banach spaces. Indeed, such methods were succesful in the study of the porous medium equation or slow di usion equation (i.e. Equation (E) with m ¿ 1) [2]. In particular, the Generation Theorem of Crandall and Liggett [6] would seem appropriate for such a nonlinear discontinuous operator as that of the fast di usion operator u m . But this operator in
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the range of m ¡ 0 could only be shown to be accretive in L1 (BR ) which is not re exive in which case, the Crandall–Liggett Theorem does not guarantee the existence of a strong solution. It is uncertain whether this operator is accretive in any useful re exive Banach Space such as L2 (BR ). Thus, a uni ed method consisting of a Monotone Iteration Method and a Fixed-Point Theorem was constructed and shown to be more
exible and appropriate to the problem at hand. The method of proof presented in this paper may be applied to many other strongly nonlinear di erential equations of second order both of parabolic and elliptic-type on bounded and unbounded domains. Systems of nonlinear di erential equations appearing in mathematical biology and chemistry may very likely also be treated by this method as perhaps the special cases of nonlinear boundary conditions when dealing with Boundary and Initial Boundary Value Problems. In a forthcoming paper by this author, this technique is applied to the slow di usion equation yielding local and global existence results analogous to those of this work. Acknowledgements I would like to thank Professor Manuel del Pino of the Universidad de Chile and Professor Panagiota Daskalopoulos of the University of California at Irvine for generously making their most recent work available to me. References [1] H. Amann, Existence and multiplicity theorems for semi-linear elliptic boundary value problems, Math. Z. 150 (1976) 281–295. [2] P. Benilan, M.G. Crandall, M. Pierre, Solutions of the porous medium equation in Rn under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1) (1984) 51–87. [3] G. Bernard, A Priori Estimates for very fast di usion equations in Rn , Methods Appl. Sci., submitted. [4] J.G. Berryman, Evolution of a stable pro le for a class of nonlinear di usion equations with xed boundaries, J. Math. Phys. 18 (1977) 2108–2115. [5] J.G. Berryman, C.J. Holland, Asymptotic behaviour of the nonlinear di erential equation nt = (n−1 nx )x , J. Math. Phys. 23 (1982) 983–987. [6] M.G. Crandall, T.M. Liggett, Generation of semi-groups of nonlinear transformations on general banach spaces, Amer. J. Math. 93 (1971) 265–298. [7] P. Daskalopoulos, M.A. del Pino, On fast di usion nonlinear heat equations and a related singular elliptic problem, Indiana Univ. Math. J. 43 (2) (1994) 703–728. [8] P. Daskalopoulos, M.A. del Pino, On a singular di usion equation, Comm. Anal. Geom. 3 (3) (1995) 523–542. [9] P. Daskalopoulos, M.A. del Pino, On nonlinear parabolic equations of very fast di usion, Arch. Rational Mech. Anal. 137 (1997) 363–380. [10] D. Gilbarg, N.S. Trudinger, Elliptic Partial Di erential Equations of Second Order, 2nd Edition (Grundlehren der mathematischen Wissenschaften;224), Springer, Berlin, 1983. [11] M.A. Herrero, M. Pierre, The Cauchy Problem for ut = u m when 0 ¡ m ¡ 1, Trans. Amer. Math. Soc. 291 (1985) 145–158. [12] H.G. Kaper, G.K. Leaf, S. Reich, Convergence of semigroups with an application to the Carleman equation, Math. Methods Appl. Sci. 2 (1980) 303–308. [13] T.G. Kurtz, Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc. 186 (1973) 259–272.
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[14] G.S. Ladde, V. Ladshmikantham, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Di erential Equations, Pitman Advanced Publishing Program, Boston, 1985. [15] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, Amer. Math. Soc., Providence, RI, 1968. [16] H.P. McKean, The central limit theorem for Carleman’s equation, Israel J. Math. 21 (1975) 54–92. [17] M. Protter, H. Weinberger, Maximum Principles in Di erential Equations, Prentice-Hall, Englewood Cli s, NJ, 1967. [18] P. Takac, A fast di usion equation which generates a monotone semi ow I: local existence and uniqueness, Di erential Integral Equations 4 (1) (1991) 151–174. [19] P. Takac, A fast di usion equation which generates a monotone semi ow II: global existence and asymptotic behavior, Di erential Integral Equations 4 (1) (1991) 175–187.