Local and global estimates of solutions of Hamilton-Jacobi parabolic equation with absorption
aa r X i v : . [ m a t h . A P ] J u l Local and global estimates of solutions of Hamilton-Jacobiparabolic equation with absorption
Marie Fran¸coise BIDAUT-VERON
Abstract
Here we show new apriori estimates for the nonnegative solutions of the equation u t − ∆ u + |∇ u | q = 0in Q Ω ,T = Ω × (0 , T ) , T ≦ ∞ , where q > , and Ω = R N , or Ω is a smooth bounded domain of R N and u = 0 on ∂ Ω × (0 , T ) . In case Ω = R N , we show that any solution u ∈ C , ( Q R N ,T ) of equation (1.1) in Q R N ,T (inparticular any weak solution if q ≦ , without condition as | x | → ∞ , satisfies the universalestimate |∇ u ( ., t ) | q ≦ q − u ( ., t ) t , in Q R N ,T . Moreover we prove that the growth of u is limited by C ( t + t − / ( q − )(1 + | x | q ′ ) , where C dependson u. We also give existence properties of solutions in Q Ω ,T , for initial data locally integrable orunbounded measures. We give a nonuniqueness result in case q > . Finally we show that besidesthe local regularizing effect of the heat equation, u satisfies a second effect of type L Rloc − L ∞ loc , due to the gradient term. Keywords
Hamilton-Jacobi equation; Radon measures; initial trace; universal bounds.,regularizing effects.
A.M.S. Subject Classification
Contents R N
105 Existence and nonuniqueness results 176 Second local regularizing effect 19 .1 Introduction
Here we consider the nonnegative solutions of the parabolic Hamilton-Jacobi equation u t − ν ∆ u + |∇ u | q = 0 , (1.1)where q > , in Q Ω ,T = Ω × (0 , T ) , where Ω is any domain of R N , ν ∈ (0 , . We study the problemof apriori estimates of the nonnegative solutions, with possibly rough unbounded initial data u ( x,
0) = u ∈ M + (Ω) , (1.2)where we denote by M + (Ω) the set of nonnegative Radon measures in Ω , and M + b (Ω) the subsetof bounded ones. We say that u is a solution of (1.1) if it satisfies (1.1) in Q Ω ,T in the weak senseof distributions, see Section 2. We say that u has a trace u in M + (Ω) if u ( ., t ) converges to u inthe weak ∗ topology of measures:lim t → Z Ω u ( ., t ) ψdx = Z Ω ψdu , ∀ ψ ∈ C c (Ω) . (1.3)Our purpose is to obtain apriori estimates valid for any solution in Q Ω ,T = Ω × (0 , T ), withoutassumption on the boundary of Ω , or for large | x | if Ω = R N . Fisrt recall some known results. The Cauchy problem in Q R N ,T ( P R N ,T ) (cid:26) u t − ν ∆ u + |∇ u | q = 0 , in Q R N ,T ,u ( x,
0) = u in R N , (1.4)is the object of a rich literature, see among them [2],[9], [5], [11], [26],[12], [13], and referencestherein. The first studies concern classical solutions, that means u ∈ C , ( Q R N ,T ) , with smoothbounded initial data u ∈ C b (cid:0) R N (cid:1) : there a unique global solution such that k u ( ., t ) k L ∞ ( R N ) ≦ k u k L ∞ ( R N ) , and k∇ u ( ., t ) k L ∞ ( R N ) ≦ k∇ u k L ∞ ( R N ) , in Q R N ,T , see [2]. Then universal apriori estimates of the gradient are obtained for this solution , by using theBersnstein technique, which consists in computing the equation satisfied by |∇ u | : first from [23], k∇ u ( ., t ) k qL ∞ ( R N ) ≦ k u k L ∞ ( R N ) t , in Q R N ,T , , then from [9], |∇ u ( ., t ) | q ≦ q − u ( ., t ) t , (1.5) k∇ ( u q − q )( ., t ) k L ∞ ( R N ) ≦ Ct − / k u k q − q L ∞ ( R N ) , C = C ( N, q, ν ) . (1.6)Existence and uniqueness was extended to any u ∈ C b (cid:0) R N (cid:1) in [20]; then the estimates (1.6) and(1.5) are still valid, see [5]. In case of nonnegative rough initial data u ∈ L R (cid:0) R N (cid:1) , R ≧ , or u ∈ M + b ( R N ) , the problem was studied in a semi-group formulation [9], [11], [26], then in the2arger class of weak solutions in [12], [13]. Recall that two critical values appear: q = 2 , where theequation can be reduced to the heat equation, and q ∗ = N + 2 N + 1 . Indeed the Cauchy problem with initial value u = κδ , where δ is the Dirac mass at 0 and κ > , has a weak solution u κ if and only if q < q ∗ , see [9], [12]. Moreover as κ → ∞ , ( u κ ) converges to aunique very singular solution Y, see [25], [10], [8], [12]. And Y ( x, t ) = t − a/ F ( | x | / √ t ) , where a = 2 − qq − , (1.7)and F is bounded and has an exponential decay at infinity.In [13, Theorem 2.2] it is shown that for any R ≧ L R - L ∞ properties oftwo types hold for the Cauchy problem in Q R N ,T : one due to the heat operator: k u ( ., t ) k L ∞ ( R N ) ≦ Ct − N R k u k L R ( R N ) , C = C ( N, R, ν ) , (1.8)and the other due to the gradient term, independent of ν ( ν > k u ( ., t ) k L ∞ ( R N ) ≦ Ct − NqR + N ( q − k u k qRqR + N ( q − L R ( R N ) , C = C ( N, q, R ) . (1.9)A great part of the results has been extended to the Dirichlet problem in a bounded domainΩ : ( P Ω ,T ) u t − ∆ u + |∇ u | q = 0 , in Q Ω ,T ,u = 0 , on ∂ Ω × (0 , T ) ,u ( x,
0) = u , (1.10)where u ∈ M + b (Ω), and u ( ., t ) converges to u weakly in M + b (Ω), see [6], [26], [12], [13]. Universalestimates are given in [16], see also [12]. Note that (1.5) cannot hold, since it contradicts the H¨opfLemma.Finally local estimates in any domain Ω were proved in [26]: for any classical solution u in Q Ω ,T and any ball B ( x , η ) ⊂ Ω , there holds in Q B ( x ,η ) ,T |∇ u | ( ., t ) ≦ C ( t − q + η − + η − q − )(1 + u ( ., t )) , C = C ( N, q, ν ) . (1.11) In Section 3 we give local integral estimates of the solutions in terms of the initial data, and a firstregularizing effect, local version of (1.8), see Theorem 3.3.
Theorem 1.1
Let q > . Let u be any nonnegative weak solution of equation (1.1) in Q Ω ,T , andlet B ( x , η ) ⊂⊂ Ω such that u has a trace u ∈ L Rloc (Ω) , R ≧ and u ∈ C ([0 , T ) ; L Rloc (Ω)) . Thenfor any < t ≦ τ < T, sup x ∈ B ( x ,η/ u ( x, t ) ≦ Ct − N R ( t + k u k L R ( B ( x ,η ) ) , C = C ( N, q, ν, R, η, τ ) . If R = 1 , the estimate remains true when u ∈ M + (Ω) (with k u k L ( B ( x ,η ) replaced by R B ( x ,η ) du ).
3n Section 4, we give global estimates of the solutions of (1.1) in Q R N ,T , and this is our mainresult. We show that the universal estimate (1.5) in R N holds without assuming that the solutionsare initially bounded : Theorem 1.2
Let q > . Let u be any classical solution, in particular any weak solution if q ≦ , of equation (1.1) in Q R N ,T . Then |∇ u ( ., t ) | q ≦ q − u ( ., t ) t , in Q R N ,T . (1.12)And we prove that the growth of the solutions is limited, in | x | q ′ as | x | → ∞ and in t − / ( q − as t → Theorem 1.3
Let q > . Let u be any classical solution, in particular any weak solution if q ≦ , of equation (1.1) in Q R N ,T , such that there exists a ball B ( x , η ) such that u has a trace u ∈ M + (( B ( x , η )) . Then u ( x, t ) ≦ C ( q ) t − q − | x − x | q ′ + C ( t − q − + t + Z B ( x ,η ) du ) , C = C ( N, q, η ) . (1.13)In [14], we show that there exist solutions with precisely this type of behaviour of order t − / ( q − | x | q ′ as | x | → ∞ or t →
0. Moreover we prove that the condition on the trace isalways satisfied for q < q ∗ . In Section 5 we complete the study by giving existence results with only local assumptions on u , extending some results of [5] where u is continuous, and [11], [13], where the assumptions areglobal: Theorem 1.4
Let
Ω = R N (resp. Ω bounded).(i) If < q < q ∗ , then for any u ∈ M + (cid:0) R N (cid:1) (resp. M + (Ω) ), there exists a weak solution u of equation (1.1) (resp. of ( D Ω ,T ) ) with trace u .(ii) If q ∗ ≦ q ≦ , then existence still holds for any nonnegative u ∈ L loc (cid:0) R N (cid:1) (resp. L loc (Ω) ).And then u ∈ C ([0 , T ) ; L loc (cid:0) R N (cid:1) (resp. u ∈ C ([0 , T ) ; L loc (Ω)) . (iii) If q > , existence holds for any nonnegative u ∈ L loc (cid:0) R N (cid:1) (resp. L loc (Ω) ) which islimit of a nondecreasing sequence of continuous functions. Moreover we give a result of nonuniqueness of weak solutions in case q >
Theorem 1.5
Assume that q > , N ≥ . Then the Cauchy problem ( P R N , ∞ ) with initial data ˜ U ( x ) = ˜ C | x | | a | ∈ C (cid:0) R N (cid:1) , ˜ C = q − q − N − q − N ) q − q − , admits at least two weak solutions: the stationary solution ˜ U , and a radial self-similar solution ofthe form U ˜ C ( x, t ) = t | a | / f ( | x | / √ t ) , (1.14) where f is increasing on [0 , ∞ ) , f (0) > , and lim η →∞ η −| a | / f ( η ) = ˜ C. Theorem 1.6
Let q > , and let u be any nonnegative classical solution (resp. any weak solutionif q ≦ of equation (1.1) in Q Ω ,T , and let B ( x , η ) ⊂ Ω . Assume that u ∈ L Rloc (Ω) for some R ≧ , R > q − , and u ∈ C ([0 , T ) ; L Rloc (Ω)) . Then for any ε > , and for any τ ∈ (0 , T ) , thenthere exists C = C ( N, q, R, η, ε, τ ) such thatsup B η/ u ( ., t ) ≦ Ct − NqR + N ( q − ( t + k u k L R ( B η ) ) RqqR + N ( q − + Ct − εR +1 − q k u k RR +1 − q L R ( B η ) . (1.15) If q < , the estimates for R = 1 are also valid when u has a trace u ∈ M + (Ω) , with k u k L ( B η ) replaced by R B η du . In conclusion, note that a part of our results could be extended to more general quasilinearoperators, for example to the case of equation involving the p -Laplace operator u t − ν ∆ p u + |∇ u | q = 0with p > , following the results of [13], [4], [21], [19]. We set Q Ω ,s,τ = Ω × ( s, τ ) , for any 0 ≦ s < τ ≦ ∞ , thus Q Ω ,T = Q Ω , ,T . Definition 2.1
Let q > and Ω be any domain of R N . We say that a nonnegative function u is a classical solution of (1.1) in Q Ω ,T if u ∈ C , ( Q Ω ,T ) . We say that u is a weak solution (resp. weak subsolution) of (1.1) in Q Ω ,T , if u ∈ C ((0 , T ); L loc ( Q Ω ,T )) ∩ L loc ((0 , T ); W , loc (Ω)) , |∇ u | q ∈ L loc ( Q Ω ,T ) and u satisfies (1.1) in the distribution sense: Z T Z Ω ( − uϕ t − νu ∆ ϕ + |∇ u | q ϕ ) = 0 , ∀ ϕ ∈ D ( Q Ω ,T ) , (2.1) (resp. Z T Z Ω ( − uϕ t − νu ∆ ϕ + |∇ u | q ϕ ) ≦ , ∀ ϕ ∈ D + ( Q Ω ,T ) . ) (2.2) And then for any < s < t < T, and any ϕ ∈ C ((0 , T ) , C c (Ω)) , Z Ω ( uϕ )( ., t ) − Z Ω ( uϕ )( ., s ) + Z ts Z Ω ( − uϕ t + ν ∇ u. ∇ ϕ + |∇ u | q ϕ ) = 0 (resp. ≦ . (2.3) Remark 2.2
Any weak subsolution u is locally bounded in Q Ω ,T . Indeed, since u is ν -subcaloric,there holds for any ball B ( x , ρ ) ⊂⊂ Ω and any ρ ≦ t < T, sup B ( x , ρ ) × h t − ρ ,t i u ≦ C ( N, ν ) ρ − ( N +2) Z tt − ρ Z B ( x ,ρ ) u. (2.4) Any nonnegative function u ∈ L loc ( Q Ω ,T ) , such that |∇ u | q ∈ L loc ( Q Ω ,T ) , and u satisfies (2.1), is aweak solution and |∇ u | ∈ L loc ( Q Ω ,T )) , u ∈ C ((0 , T ); L sloc ( Q Ω ,T )) , ∀ s ≧ , see [12, Lemma 2.4]. q ≦ , see [12, Theorem 2.9], [13,Corollary 5.14]: Theorem 2.3
Let < q ≦ . Let Ω be any domain in R N . Let u be any weak nonnegative solutionof (1.1) in Q Ω ,T . Then u ∈ C γ, γ/ loc ( Q Ω ,T ) for some γ ∈ (0 , , and for any smooth domains ω ⊂⊂ ω ′ ⊂⊂ Ω , and < s < τ < T, k u k C γ, γ/ ( Q ω,s,τ ) is bounded in terms of k u k L ∞ ( Q ω ′ ,s/ ,τ ) . Thus for any sequence ( u n ) of nonnegative weak solutions of equation (1.1) in Q Ω ,T , uniformlylocally bounded, one can extract a subsequence converging in C , loc ( Q Ω ,T ) to a weak solution u of(1.1) in Q Ω ,T . Remark 2.4
Let q > . From the estimates (1.11), for any sequence of classical nonnegativesolutions ( u n ) of (1.1) in Q Ω ,T , uniformly bounded in L ∞ loc ( Q Ω ,T ) , one can extract a subsequenceconverging in C , loc ( Q R N ,T ) to a classical solution u of (1.1). Remark 2.5
Let us mention some results of concerning the trace, valid for any q > , see [12,Lemma 2.14]. Let u be any nonnegative weak solution u of (1.1) in Q Ω ,T . Then u has a trace u in M + (Ω) if and only if u ∈ L ∞ loc ( [0 , T ) ; L loc (Ω)) , and if and only if |∇ u | q ∈ L loc (Ω × [0 , T )) . Andthen for any t ∈ (0 , T ) , and any ϕ ∈ C c (Ω × [0 , T )) , and any ζ ∈ C c (Ω) , Z Ω u ( ., t ) ϕdx + Z t Z Ω ( − uϕ t + ν ∇ u. ∇ ϕ + |∇ u | q ϕ ) = Z Ω ϕ ( ., du , (2.5) Z Ω u ( ., t ) ζ + Z t Z Ω ( ν ∇ u. ∇ ζ + |∇ u | q ζ ) = Z Ω ζdu . (2.6) If u ∈ L loc (Ω) , then u ∈ C ( [0 , T ) ; L loc (Ω)) . Finally we consider the Dirichlet problem in a smooth bounded domain Ω:( D Ω ,T ) (cid:26) u t − ∆ u + |∇ u | q = 0 , in Q Ω ,T ,u = 0 , on ∂ Ω × (0 , T ) . (2.7) Definition 2.6
We say that a function u is a weak solution of ( D Ω ,T ) if it is a weak so-lution of equation (1.1) such that u ∈ C ((0 , T ); L (Ω)) ∩ L loc ((0 , T ); W , (Ω)) , and |∇ u | q ∈ L loc ((0 , T ); L (Ω)) . We say that u is a classical solution of ( D Ω ,T ) if u ∈ C , ( Q Ω ,T ) ∩ C , (cid:0) Ω × (0 , T ) (cid:1) . Lemma 3.1
Let Ω be any domain in R N , q > , R ≧ . Let u be any nonnegative weak subsolutionof equation (1.1) in Q Ω ,T , such that u ∈ C ((0 , T ); L Rloc (Ω)) . Let ξ ∈ C ((0 , T ); C c (Ω)) , with valuesin [0 , . Let λ > . Then there exists C = C ( q, R, λ ) , such that, for any < s < t ≦ τ < T, R Ω u R ( ., t ) ξ λ + 12 Z τs Z Ω u R − |∇ u | q ξ λ + ν R − Z τs Z Ω u R − |∇ u | ξ λ ≦ R Ω u R ( ., s ) ξ λ + λR Z ts Z Ω u R ξ λ − | ξ t | + C Z ts Z Ω u R − ξ λ − q ′ |∇ ξ | q ′ . (3.1)6 roof. (i) Let R = 1 . Taking ϕ = ξ λ in (2.3), we obtain, since ν ≦ , Z Ω u ( ., t ) ξ λ + Z ts Z Ω |∇ u | q ξ λ ≦ Z Ω u ( s, . ) ξ λ + λ Z ts Z Ω ξ λ − uξ t + λν Z ts Z Ω ξ λ − ∇ u. ∇ ξ ≦ Z Ω u ( ., s ) ξ λ + λ Z ts Z Ω ξ λ − u | ξ t | + 12 Z ts Z Ω |∇ u | q ξ q ′ + C ( q, λ ) Z ts Z Ω ξ λ − q ′ |∇ ξ | q ′ , hence (3.1) follows.(ii) Next assume R > . Consider u δ,n = (( u + δ ) ∗ ϕ n ), where ( ϕ n ) is a sequence of mollifiers,and δ > . Then by convexity, u δ,n is also a subsolution of (1.1):( u δ,n ) t − ν ∆ u δ,n + |∇ u δ,n | q ≦ . Multiplying by u R − δ,n ξ λ and integrating between s and t, and going to the limit as δ → n → ∞ , see [13], we get with different constants C = ( N, q, R, λ ) , independent of ν, R Z Ω u R ( ., t ) ξ λ + ν ( R − Z ts Z Ω u R − |∇ u | ξ λ + Z ts Z Ω u R − |∇ u | q ξ λ ≦ R Z Ω u R ( ., s ) ξ λ + λ Z ts Z B ρ ξ λ − u R | ξ t | + λν Z tθ Z Ω u R − |∇ u | |∇ ξ | ξ λ − ≦ R Z Ω u R ( ., s ) ξ λ + λ Z ts Z B ρ ξ λ − u R | ξ t | + 12 Z τs Z Ω u R − |∇ u | q ξ λ + C ( λ, R ) Z ts Z Ω u R − ξ λ − q ′ |∇ ξ | q ′ , and (3.1) follows again.Then we give local integral estimates of u ( ., t ) in terms of the initial data: Lemma 3.2
Let q > . Let η > . Let u be any nonnegative weak solution of equation (1.1) in Q Ω ,T , with trace u ∈ M + (Ω) , and let B ( x , η ) ⊂⊂ Ω . Then for any t ∈ (0 , T ) , Z B ( x ,η ) u ( x, t ) ≦ C ( N, q ) η N − q ′ t + Z B ( x , η ) du . (3.2) Moreover if u ∈ L Rloc (Ω) (
R > , and u ∈ C ([0 , T ) ; L Rloc (Ω)) , then k u ( ., t ) k L R ( B ( x ,η )) ≦ C ( N, q, R ) η NR − q ′ t + k u k L R ( B ( x , η )) . (3.3) If u ∈ C ( B ( x , η ) × [0 , T )) , then k u ( ., t ) k L ∞ ( B ( x ,η )) ≦ C ( N, q ) η − q ′ t + k u k L ∞ ( B ( x , η )) . (3.4) Proof.
We can assume that 0 ∈ Ω and x = 0 . We take ξ ∈ C c (Ω) , independent of t, withvalues in [0 , , and R = 1 in (3.1), λ = q ′ . Then for any 0 < s < t < T, Z Ω u ( ., t ) ξ q ′ + 12 Z ts Z Ω |∇ u | q ξ q ′ ≦ Z Ω u ( ., s ) ξ q ′ + C ( q ) Z ts Z Ω |∇ ξ | q ′ ≦ Z Ω u ( ., s ) ξ q ′ + C ( q ) t Z Ω |∇ ξ | q ′ . s → , we get Z Ω u ( ., t ) ξ q ′ + 12 Z t Z Ω |∇ u | q ξ q ′ ≦ C ( q ) t Z Ω |∇ ξ | q ′ + Z Ω ξ q ′ du . (3.5)Then taking ξ = 1 in B η with support in B η and |∇ ξ | ≦ C ( N ) /η, Z B η u ( x, t ) ≦ C ( N, q ) η N − q ′ t + Z B η ξ q ′ du , (3.6)hence we get (3.2). Next assume u ∈ L Rloc (Ω) (
R > , and u ∈ C ([0 , T ) ; L Rloc (Ω)) . Then from(3.1), for any 0 < s < t ≦ τ < T, we find, R Ω u R ( ., t ) ξ λ + 12 Z τs Z Ω u R − |∇ u | q ξ λ ≦ R Ω u R ( ., s ) ξ λ + Z ts Z Ω u R − ξ λ − q ′ |∇ ξ | q ′ ≦ R Ω u R ( ., s ) ξ λ + ε Z ts Z B η u R ξ λ + ε − R Z ts Z B η ξ λ − Rq ′ |∇ ξ | Rq ′ . Taking λ = Rq ′ , and ξ as above, we find Z B η u R ( ., t ) ξ Rq ′ ≦ Z B η u R ( ., s ) ξ Rq ′ + ε Z ts Z B η u R ξ Rq ′ + ε − R C ( N ) C Rq ′ ( N ) η N − Rq ′ t. Next we set ̟ ( t ) = sup σ ∈ [ s,t ] R B η u R ( ., σ ) ξ Rq ′ . Then ̟ ( t ) ≦ Z B η u R ( ., s ) ξ Rq ′ + ε ( t − s ) ̟ ( t ) + ε − R C ( N ) C Rq ′ ( N ) η N − Rq ′ t. Taking ε = 1 / t, we get12 Z B η u R ( ., t ) ξ Rq ′ ≦ Z B η u R ( ., s ) ξ rq ′ + C ( N ) C Rq ′ ( N ) η N − Rq ′ t R . Then going to the limit as s → , Z B η u R ( x, t ) ≦ C ( N ) C Rq ′ ( N ) η N − Rq ′ t R + Z B η u R ξ Rq ′ , (3.7)thus (3.3) follows.If u ∈ C ( B ρ × [0 , T )) , then (3.7) holds for any R ≧ , implying k u ( ., t ) k L R ( B η ) ≦ C R ( N ) C q ′ ( N ) η NR − q ′ t + k u k L R ( B η ) , and (3.3) follows as R → ∞ . 8 .2 Regularizing effect of the heat operator We first give a first regularizing effect due to the Laplace operator in Q Ω ,T , for any domain Ω , forclassical or weak solutions in terms of the initial data. Theorem 3.3
Let q > . Let u be any nonnegative weak subsolution of equation (1.1) in Q Ω ,T ,and let B ( x , η ) ⊂ Ω such that u has a trace u ∈ M + ( B ( x , η )) . Then for any τ < T, and any t ∈ (0 , τ ] , sup x ∈ B ( x ,η/ u ( x, t ) ≦ Ct − N ( t + Z B ( x ,η ) du ) , C = C ( N, q, ν, η, τ ) . (3.8) Moreover if u ∈ L Rloc (Ω) (
R > , and u ∈ C ([0 , T ) ; L Rloc (Ω)) , then sup x ∈ B ( x ,η/ u ( x, t ) ≦ Ct − N R ( t + k u k L R ( B ( x ,η )) ) , C = C ( N, q, ν, R, η, τ ) . (3.9) Proof.
We still assume that x = 0 ∈ Ω . Let ξ ∈ C c ( B η ) be nonnegative, radial, with values in[0 , , with ξ = 1 on B η and |∇ ξ | ≦ C ( N ) /η . Since u is ν -subcaloric, from (2.4), for any ρ ∈ (0 , η )such that ρ ≦ t < τ, sup B η/ u ( ., t ) ≦ C ( N, ν ) ρ − ( N +2) Z tt − ρ / Z B η u, (3.10)hence from Lemma 3.2, sup B η/ u ( ., t ) ≦ C ( N, q, ν ) ρ − N ( η N − q ′ t + Z B η du ) . Let k ∈ N such that k η / ≧ τ. For any t ∈ (0 , τ ] , there exists k ∈ N with k ≦ k such that t ∈ (cid:0) kη / , ( k + 1) η / (cid:3) . Taking ρ = t/ ( k + 1) , we findsup B η/ u ( ., t ) ≦ C ( N, q, ν )( k + 1) N t − N ( η N − q ′ t + Z B η du ) ≦ C ( N, q, ν )( η − N τ N + 1) t − N ( η N − q ′ t + Z B η du ) . (3.11)Thus we obtain (3.8). Next assume that u ∈ C ([0 , T ) ; L Rloc ( B η )) , with R >
1. We still approximate u by u δ,n = ( u + δ ) ∗ ϕ n , where ( ϕ n ) is a sequence of mollifiers, and δ > . Since u is ν -subcaloric,then u Rδ,n is also ν -subcaloric. Then for any ρ ∈ (0 , η ) such that ρ ≦ t < τ, we havesup B η/ u Rδ,n ( ., t ) ≦ C ( N, ν ) ρ − ( N +2) Z tt − ρ / Z Bρ/ u Rδ,n , hence as δ → n → ∞ , from Lemma (3.2),sup B η/ u R ( ., t ) ≦ C ( N, ν ) ρ − ( N +2) Z tt − ρ / Z Bρ/ u R ≦ C ( N, q, ν, R )( η − N τ N +1)( η N − Rq ′ t R + Z B η u R ) . (3.12)We deduce (3.9) as above. 9 Global estimates in R N We first show that the universal estimate of the gradient (1.12) implies the estimate (1.13) of thefunction:
Theorem 4.1
Let q > . Let u be a classical solution of equation (1.1) in Q R N ,T . Assume thatthere exists a ball B ( x , η ) such that u has a trace u ∈ M + (( B ( x , η )) . If u satisfies (1.12), thenfor any t ∈ (0 , T ) ,u ( x, t ) ≦ C ( q ) t − q − | x − x | q ′ + C ( t − q − + t + Z B ( x ,η ) du ) , C = C ( N, q, η ) , (4.1) If u ∈ L Rloc (Ω) , R ≧ and u ∈ C ([0 , T ) ; L Rloc (Ω)) , then u ( x, t ) ≦ C ( q ) t − q − | x − x | q ′ + Ct − N R ( t + k u k L R ( B ( x ,η )) ) , C = C ( N, q, R, ν, η ) . (4.2) u ( x, t ) ≦ C ( q ) t − q − | x − x | q ′ + C ( t − q − + t + k u k L R ( B ( x ,η )) ) , C = C ( N, q, R, η ) . (4.3) Proof.
Estimate (1.12) is equivalent to (cid:12)(cid:12)(cid:12) ∇ ( u q ′ ) (cid:12)(cid:12)(cid:12) ( ., t ) ≦ ( q − q ′ q t − q , in Q R N ,T . (4.4)Then with constants C ( q ) only depending of q,u q ′ ( x, t ) ≦ u q ′ ( x , t ) + C ( q ) t − q | x − x | , (4.5)then u ( x, t ) ≦ C ( q )( u ( x , t ) + t − q − | x − x | q ′ ) , (4.6)and, from Theorem 3.3, u ( x , t ) ≦ C ( N, q, R, ν, η ) t − N R ( t + k u k L R ( B ( x ,η )) ) . Therefore (4.2) follows. Also, interverting x and x , for any R ≧ ,u R ( x , t ) ≦ C ( q, R )( u R ( x, t ) + t − Rq − | x − x | Rq ′ ) . Integrating on B ( x , η/ , we get η N u R ( x , t ) ≦ C ( q, R )( Z B ( x ,η/ u R ( ., t ) + t − Rq − η N − Rq ′ );using Lemma 3.2, we deduce u ( x , t ) ≦ C ( N, q, R, η )( t − q − + t + Z B ( x ,η ) du ) , and if u ∈ L Rloc (Ω) , u ( x , t ) ≦ C ( N, q, R, η )( t − q − + t + k u k L R ( B ( x ,η )) ) , and the conclusions follow from (4.6). 10 emark 4.2 In particular, the estimates (4.1)-(4.3) hold for solutions with u ∈ C b ( R N ) , andmore generally for limits a.e. of such solutions, that we can call reachable solutions. Inegality(4.5) was used in [5, Theorem 3.3] for obtaining local estimates of classical of bounded solutions.in Q R N ,T . In order to prove Theorem 1.2, we first give an estimate of the type of (1.13) on a time interval(0 , τ ], with constants depending on τ and ν, which is not obtained from any estimate of the gradient .Our result is based on the construction of suitable supersolutions in annulus of type Q B ρ \ B ρ , ∞ , ρ > . For the construction we consider the function t ∈ (0 , ∞ ) ψ h ( t ) ∈ (1 , ∞ ) , where h > ψ h ) t + h ( ψ qh − ψ h ) = 0 in (0 , ∞ ) , ψ h (0) = ∞ , ψ h ( ∞ ) = 1 , (4.7)given explicitely by ψ h ( t ) = (1 − e − h ( q − t ) − q − ; hence ψ qh − ψ h ≧ , and for any t > , (( q − ht ) − q − ≦ ψ h ( t ) ≦ q − (1 + (( q − ht ) − q − ) . (4.8)since, for x > , x (1 − x/ ≦ − e − x ≦ x, hence x/ ≦ − e − x ≦ x, for x ≦ . Proposition 4.3
Let q > . Then there exists a nonnegative function V defined in Q B × (0 , ∞ ) , such that V is a supersolution of equation (1.1) on Q B \ B , ∞ , , and V converges to ∞ as t → , uniformly on B and converges to ∞ as x → ∂B , uniformly on (0 , τ ) for any τ < ∞ . And V hasthe form V ( x, t ) = e t Φ( | x | ) ψ h ( t ) in Q B , ∞ (4.9) for some h = h ( N, q, ν ) > , where ψ h is given by (4.7), and Φ is a suitable radial functiondepending on N, q, ν, such that − ν ∆Φ + Φ + |∇ Φ | q ≧ in B . (4.10) Proof.
We first construct Φ . Let σ > , such that σ ≧ a = (2 − q ) / ( q − . Let ϕ be the firsteigenfunction of the Laplacian in B such that ϕ (0) = 1 , associated to the first eigenvalue λ , hence ϕ is radial ; let m = min B ϕ > M = min B \ B |∇ ϕ | . Let us take Φ = Φ K = Φ + K, where Φ = γϕ − σ , K > γ > − ν ∆Φ + Φ + |∇ Φ | q = F (Φ ) + K, with F (Φ ) = γϕ − ( σ +2)1 ( γ q − σ q ϕ ( q − a − σ )1 (cid:12)(cid:12) ϕ ′ (cid:12)(cid:12) q + (1 − νσλ ) ϕ − νσ ( σ + 1) ϕ ′ ) . There holds lim r → | ϕ ′ | = c > σ > a we fix γ = 1 , and thenlim r → F (Φ ) = ∞ . If q < σ = a, we get F (Φ ) = γϕ − q ′ ( γ q − a q (cid:12)(cid:12) ϕ ′ (cid:12)(cid:12) q + (1 − νaλ ) ϕ − aq ′ ϕ ′ ) , hence fixing γ > γ ( N, q, ν ) large enough, we still get lim r → F (Φ ) = ∞ . Thus F has a minimum µ in B . Taking K = K ( N, q, ν ) > | µ | we deduce that Φ satisfies (4.10), and lim r → Φ = ∞ . ′ q / Φ = γ q σ q / ( γϕ q + σ ( q − + Kϕ q ( σ +1)1 ) is increasing, then m K = m K ( N, q, ν ) =min [1 , | Φ ′ | q / Φ = | Φ ′ (1) | q / Φ(1) > . We define V by (4.9) and compute V t − ν ∆ V + |∇ V | q = e t (Φ ψ h + Φ( ψ h ) t − ν ∆Φ) + e qt |∇ Φ | q ψ qh ≧ e t (Φ ψ h + Φ ψ t − ν ∆Φ + |∇ Φ | q ψ q ) = e t ( ψ q − ψ h )( |∇ Φ | q − h Φ) . We take h = h ( N, q, ν ) < m K . Then on B \ B we have |∇ Φ | q − h Φ > , and ψ q ≧ ψ h , then V is asupersolution on B \ B . Moreover V is radial and increasing with respect to | x | , thensup B V ( x, t ) = sup ∂B V ( x, t ) = e t Φ(2) ψ h ( t ) ≦ q − e t Φ(2)(1 + (( q − ht ) − q − ) ≦ C ( N, q, ν ) e t Φ(2)(1 + t − q − ) . (4.11) Theorem 4.4
Let u be a classical solution, (in particular any weak solution if q ≦ of equa-tion (1.1) in Q R N ,T . Assume that there exists a ball B ( x , η ) such that u admits a trace u ∈ M + ( B ( x , η )) . (i) Then for any τ ∈ (0 , T ) , and t ≦ τ,u ( x, t ) ≦ C ( t − q − | x − x | q ′ + t − N ( t + Z B ( x ,η ) du )) , C = C ( N, q, ν, η, τ ) , (4.12) (ii) Also if u ∈ C ([0 , T ) ; L Rloc ( B ( x , η ))) ,u ( x, t ) ≦ C ( t − q − | x − x | q ′ + t − N R ( t + k u k L R ( B ( x ,η )) )) , C = C ( N, q, ν, R, η, τ ) , (4.13) if u ∈ C ([0 , T ) × B ( x , η )) , then u ( x, t ) ≦ C ( t − q − | x − x | q ′ + t + sup B ( x ,η ) u ) , C = C ( N, q, ν, η, τ ) . (4.14) Proof.
We use the function V constructed above. We can assume x = 0. For any ρ > , weconsider the function V ρ defined in B ρ × (0 , ∞ ) by V ρ ( x, t ) = ρ − a V ( ρ − x, ρ − t ) . It is a supersolution of the equation (1.1) on B ρ \ B ρ × (0 , ∞ ) , infinite on ∂B ρ × (0 , ∞ ) and on B ρ × { } , and from (4.11)sup B ρ V ρ ( x, t ) = sup ∂B ρ V ρ ( x, t ) ≦ C ( N, q, ν ) ρ − a e tρ Φ(2)(1 + ρ q − t − q − ) ≦ C ( N, q, ν ) ρ q ′ e tρ ( ρ − q − + t − q − ) . (4.15)(i) First suppose that u ∈ C ([0 , T ) × R N )) . Let τ ∈ (0 , T ) , and C ( τ ) = sup Q Bρ,τ u . Then w = C ( τ ) + V ρ is a supersolution in Q = ( B ρ \ B ρ ) × (0 , τ ] , and from the comparison principle weobtain u ≦ C ( τ ) + V ρ in that set. Indeed let ǫ > τ ǫ < ǫ and12 ǫ ∈ (3 ρ − ǫ, ρ ), such that w ( ., s ) ≧ max B ρ u ( ., ǫ ) for any s ∈ (0 , τ ǫ ], and w ( x, t ) ≧ max B ρ × [0 ,τ ] u for any t ∈ (0 , τ ] and r ǫ ≦ | x | < ρ. We compare u ( x, t + ǫ ) with w ( x, t + s ) on [0 , τ − ǫ ] × B r ǫ \ B ρ . And for | x | = ρ, we have u ( x, t + ǫ ) ≦ C ( τ ) ≦ w ( x, t + s ). Then u ( ., t + ǫ ) ≦ w ( ., t + s ) in B r ǫ \ B ρ × (0 , τ − ǫ ]. As s, ǫ → , we deduce that u ≦ w in Q .Hence in B ρ × (0 , τ ) , we find from (4.15) u ≦ C ( τ ) + sup B ρ V ρ ( x, t ) ≦ C ( τ ) + C ρ q ′ e tρ ( ρ − q − + t − q − ) . (4.16)Making t tend to τ, this proves thatsup Q B ρ,τ u ≦ sup Q Bρ,τ u + C ρ q ′ e τρ ( ρ − q − + τ − q − )By induction, we get sup Q B n +1 ρ ,τ u ≦ sup Q B nρ ,τ u + C nq ′ ρ q ′ e τ nρ ((2 n ρ ) − q − + τ − q − ) ≦ sup Q B nρ ,τ u + C nq ′ ρ q ′ e τρ ( ρ − q − + τ − q − );sup Q B n +1 ρ ,τ u ≦ sup Q Bρ u + C (1 + 2 q ′ + .. + 2 nq ′ ) ρ q ′ e τρ ( ρ − q − + τ − q − ) ≦ sup Q Bρ,τ u + C q ′ (2 n ρ ) q ′ e τρ ( ρ − q − + τ − q − ) . For any x ∈ R N such that | x | ≧ ρ, there exists n ∈ N ∗ such that x ∈ B n +1 ρ \ B n ρ , then u ( x, τ ) ≦ sup Q Bρ ,τ u + C q ′ | x | q ′ e τρ ( ρ − q − + τ − q − ) (4.17)thus sup Q R N ,τ u ≦ sup Q Bρ ,τ u + C q ′ | x | q ′ e τρ ( ρ − q − + τ − q − ) . (4.18)(ii) Next we consider any classical solution u in Q R N ,T with trace u in B ( x , η ). We stillassume x = 0 . Then for 0 < ǫ ≦ t ≦ τ, from (3.4) in Lemma 3.2, there holdssup B η / u ( x, t ) ≦ C ( N, q ) η − q ′ t + sup B η u ( x, ǫ ) . Then from (4.18) with ρ = η/
2, we deduce that for any ( x, t ) ∈ Q R N ,ǫ,τ ,u ( x, t ) ≦ C ( N, q ) η − q ′ t + sup B η/ u ( ., ǫ ) + C (1 + ( t − ǫ ) − q − ) | x | q ′ , C = C ( N, q, ν, η, τ ) . Next we take ǫ = t/
2. Then for any t ∈ (0 , τ ] , from (3.8) in Theorem 3.3, u ( x, t ) ≦ C ( N, q, η ) t + Ct − q − | x | q ′ + Ct − N ( t + Z B η du ) . with C = C ( N, q, ν, η, τ ) and we obtain (4.12). And (4.13), (4.14) follow from (3.9) and (3.4).Next we show our main Theorem 1.2. We use a local
Bernstein technique, as in [26]. The idea isto compute the equation satisfied by the function v = u ( q − /q , introduced in [9], and the equationsatisfied by w = |∇ v | , to obtain estimates of w in a cylinder Q B M ,T , M > . The difficulty is thatthis equation involves an elliptic operator w w t − ∆ w + b. ∇ w, where b depends on v, and maybe unbounded. However it can be controlled by the estimates of v obtained at Theorem 4.4. Thenas M → ∞ , we can prove nonuniversal L ∞ estimates of w. Finally we obtain universal estimatesof w by application of the maximum principle in Q R N ,T , valid because w is bounded . First we givea slight improvement of a comparison principle shown in [26, Proposition 2.2].
Lemma 4.5
Let Ω be any domain of R N , and τ, κ ∈ (0 , ∞ ) , A, B ∈ R . Let U ∈ C ([0 , τ ) ; L loc (Ω)) such that U t , ∇ u, D u ∈ L loc (Ω × (0 , τ )) , ess sup Q Ω ,τ U < ∞ , U ≦ B on the parabolic boundary of Q Ω ,τ , and U t − ∆ U ≦ κ (1 + | x | ) |∇ U | + f in Q Ω ,τ where f = f ( x, t ) such that f ( ., t ) ∈ L loc (Ω) for a.e. t ∈ (0 , τ ) and f ≦ on { ( x, t ) ∈ Q Ω ,τ : U ( x, t ) ≧ A } . Then ess sup Q Ω ,τ U ≦ max( A, B ) . Proof.
We set ϕ ( x, t ) = Λ t + ln(1 + | x | ) , Λ > . Then ∇ ϕ = 2 x/ (1 + | x | ) , ≦ ∆ ϕ ≦ N/ (1 + | x | ) ≦ N. Let ε > Y = U − max( A, B ) − εϕ. Taking Λ = 2 √ κ + 2 N, we obtain Y t − ∆ Y − f − κ (1 + | x | ) |∇ Y | ≦ ε ( K (1 + | x | ) |∇ ϕ | − ϕ t + ∆ ϕ ) ≦ ε (2 √ κ + 2 N − Λ) = 0 . Since esssup Q Ω ,τ U < ∞ , for R large enough, and any t ∈ (0 , τ ) , we have Y ( ., t ) ≦ a.e. in Ω ∩{| x | > R } . And Y + ∈ C ([0 , τ ) ; L (Ω)) ∩ W , ((0 , τ ); L (Ω)) , Y + (0) = 0 and Y + ( ., t ) ∈ W , (Ω ∩ B R )for a.e. t ∈ (0 , τ ) , and f Y + ( ., t ) ≦ . Then12 ddt ( Z Ω Y +2 ( ., t ) ≦ − Z Ω (cid:12)(cid:12) ∇ Y + ( ., t ) (cid:12)(cid:12) + κ (1 + R ) Z Ω |∇ Y ( ., t ) | Y + ( ., t ) ≦ κ (1 + R ) Z Ω Y +2 ( ., t ) , hence by integration Y ≦ a.e. in Q Ω ,τ . We conclude as ε → . Proof of Theorem 1.2.
We can assume x = 0 . By setting u ( x, t ) = ν q ′ / U ( x/ √ ν, t ) , forproving (4.4) we can suppose that u is a classical solution of (1.1) with ν = 1 . We set δ + u = v qq − , δ ∈ (0 , . (i) Local problem relative to |∇ v | . Here u is any classical solution u of equation (1.1) ina cylinder Q B M ,T with M > . Then v satisfies the equation v t − ∆ v = 1 q − |∇ v | v − cv |∇ v | q , c = ( q ′ ) q − . (4.19)14etting w = |∇ v | , we define L w = w t − ∆ w + b. ∇ w, b = ( qcvw q − − q − v ) ∇ v. Differentiating (4.19) and using the identity ∆ w = 2 ∇ (∆ w ) . ∇ w + 2 (cid:12)(cid:12) D v (cid:12)(cid:12) , we obtain the equation L w + 2 cw q +22 + 2 (cid:12)(cid:12) D v (cid:12)(cid:12) + 2 q − w v = 0 . (4.20)As in [26], for s ∈ (0 , , we consider a test function ζ ∈ C ( B M/ ) with values in [0 , , ζ = 0for | x | ≥ M/ |∇ ζ | ≦ C ( N, s ) ζ s /M and | ∆ ζ | + |∇ ζ | /ζ ≦ C ( N, s ) ζ s /M in B M/ . We set z = wζ. We have L z = ζ L w + w L ζ − ∇ w. ∇ ζ ≦ ζ L w + w L ζ + (cid:12)(cid:12) D v (cid:12)(cid:12) ζ + 4 w |∇ ζ | ζ . It follows that in Q B M ,T , L z + 2 cw q +22 ζ + 2 q − w v ζ ≦ Cζ s wM + Cζ s w M (cid:12)(cid:12)(cid:12)(cid:12) cqvw q − − q − v (cid:12)(cid:12)(cid:12)(cid:12) ≦ Cζ s ( wM + vw q +12 M + w M v ) , with constants C = C ( N, q, s ) . Since ζ ≦ , from the Young inequality, taking s ≧ max( q +1) , / ( q + 2) , for any ε > ,CM ζ s vw q +12 = CM ζ q +1 q +2 ζ s − q +1 q +2 vw q +12 ≦ εζw q +22 + C ( N, q, ε ) v q +2 M q +2 , and CM ζ s w ≦ εζw q +22 + C ( N, q, ε ) 1 M q +2) q ,CM ζ s w v ≦ δM ζ s w = 1 δM ζ s − q +2 ζ q +2 w ≦ εζw q +22 + C ( N, q, ε ) 1( δM ) q +2 q − . Then with a new C = C ( N, q, δ ) L z + cz q +22 ≦ C ( v q +2 M q +2 + 1 M q +2) q + 1 M q +2 q − ) . (4.21) (ii) Nonuniversal estimates of w. Here we assume that u is a classical solution of (1.1) in whole Q R N ,T , such that u ∈ C ( R N × [0 , T )). From Theorem 4.4, for any τ ∈ (0 , T ) , there holds in Q R N ,τ v ( x, t ) = ( δ + u ( x, t )) q − q ≦ C ( t − q | x | + ( t + sup B η u ) q − q ) , C = C ( N, q, η, τ ) . (4.22)hence for M ≧ M ( q, sup B η u , τ ) ≧ , we deduce v ( x, t ) ≦ Ct − q M, in Q B M ,τ . C = C ( N, q, η, τ, δ ) , there holds in Q B M/ ,τ L z + cz q +22 ≦ Ct − q +2 q . (4.23)Next we consider Ψ( t ) = Kt − /q . It satisfiesΨ t + c Ψ q +22 = ( cK q +22 − q − K ) t − q +2 q ≧ Ct − q +2 q if K ≧ K = K ( N, q, η, τ, δ ) . Fixing ǫ ∈ (0 , T ) such that τ + ǫ < T, there exists τ ǫ ∈ (0 , ǫ ) such thatΨ( θ ) ≧ sup B M z ( ., ǫ ) for any θ ∈ (0 , τ ǫ ) . We have z t ( ., t + ǫ ) − ∆ z ( ., t + ǫ ) + b ( ., t + ǫ ) . ∇ ( z, t + ǫ ) + cz q +22 ( t + ǫ ) ≦ C ( t + ǫ ) − q +2 q ≦ C ( t + θ ) − q +2 q ≦ Ψ t ( t + θ ) + c Ψ q +22 ( t + θ ) . Therefore, setting ˜ z ( ., t ) = z ( ., t + ǫ ) − Ψ( t + θ ) , there holds˜ z ( ., t ) − ∆˜ z ( ., t ) + b ( ., t + ǫ ) . ∇ ˜ z ( ., t ) ≦ V = n ( x, t ) ∈ Q B M/ ,τ + ǫ : ˜ z ( x, t ) ≧ o ; otherwise ˜ z ( ., t ) ≦ t > , and ˜ z ≦ ∂B M/ × [0 , τ ] . Then from Lemma 4.5, we get z ( ., t + ǫ ) ≦ Ψ( t + θ ) in Q B M/ ,τ , since | b | ≦ ( qcvw q − + q − δ w / ) , hence bounded on Q B M/ ,τ + ǫ . Going to the limit as θ, ǫ → , wededuce that z ( ., t ) ≦ Kt − q in Q B M/ ,τ , thus w ( ., t ) ≦ Kt − q in Q B M/ ,τ . Next we go to the limit as M → ∞ and deduce that w ( ., t ) ≦ Kt − q in Q R N ,τ , namely( q ′ ) q |∇ v ( ., t ) | q = |∇ u | q δ + u ( ., t ) ≦ Ct − , C = C ( N, q, η, δ, τ ) . In turn for any ǫ as above, there holds w ∈ L ∞ ( Q R N ,ǫ,T ), that means |∇ v | ∈ L ∞ ( Q R N ,ǫ,τ ) . (iii) Universal estimate (4.4) for u ∈ C ( R N × [0 , T )) : we prove the universal estimate (4.4).Taking again Ψ( t ) = Kt − /q , with now K = K ( N, q ) = q − ( q − /q ′ , we haveΨ t + 2 c Ψ q +22 ≧ (2 cK q +22 − q − K ) t − q +2 q ≧ . And L w + 2 cw q +22 ≦ τ ǫ ∈ (0 , τ ) such that Ψ( θ ) ≧ sup R N w ( ., ǫ )for any θ ∈ (0 , τ ǫ ) . Setting y ( ., t ) = w ( ., t + ǫ ) − Ψ( ., t + θ ) , hence on the set U = (cid:8) ( x, t ) ∈ Q R N ,τ : y ( x, t ) ≧ (cid:9) , there holds in the same way y ( ., t ) − ∆ y ( ., t ) + b ( ., t + ǫ ) . ∇ y ( ., t ) ≦ . Here we only have from (4.22) | b | ≦ ( qcvw q − + 2 q − δ w / ) ≦ κ ǫ (1 + | x | )16n Q R N ,ǫ,τ , for some κ ǫ = κ ǫ ( N, q, η, sup B η u , τ, ǫ ) . It is sufficient to apply Lemma 4.5. We deducethat w ( ., t + ǫ ) ≦ Ψ( t + θ ) on (0 , τ ) . As θ, ǫ → w ( ., t ) ≦ Ψ( t ) = q − ( q − /q ′ t − /q , which shows now that in (0 , T ) |∇ v ( ., t ) | q = ( q ′ ) − q |∇ u | q δ + u ( ., t ) ≦ q − q ( q − ( q − t − . As δ → , we obtain (4.4). (iv) General universal estimate. Here we relax the assumption u ∈ C ( R N × [0 , T )) : Forany ǫ ∈ (0 , T ) such that τ + ǫ < T, we have u ∈ C ( R N × [ ǫ, τ + ǫ )) , then from above, |∇ v ( ., t + ǫ ) | q ≦ q − t , and we obtain (4.4) as ǫ → , on (0 , τ ) for any τ < T, hence on (0 , T ) . Proof of Theorem 1.3.
It is a direct consequence of Theorems 1.2 and 4.1.
First mention some known uniqueness and comparison results, for the Cauchy problem, see [11,Theorems 2.1,4.1,4.2 and Remark 2.1 ],[13, Theorem 2.3, 4.2, 4.25, Proposition 4.26 ], and for theDirichlet problem, see [1, Theorems 3.1, 4.2], [6], [13, Proposition 5.17], [24].
Theorem 5.1
Let
Ω = R N (resp. Ω bounded). (i) Let < q < q ∗ , and u ∈ M b ( R N )( resp. u ∈ M b (Ω) ). Then there exists a unique weak solution u of (1.1) with trace u (resp. a weaksolution of ( D Ω ,T ) , such that lim t → u ( .t ) = u weakly in M b (Ω)) ). If v ∈ M b (Ω) and u ≦ v , and v is the solution associated to v , then u ≦ v. (ii) Let u ∈ L R (Ω) , ≦ R ≦ ∞ . If < q < ( N + 2 R ) / ( N + R ) , or if q = 2 , R < ∞ , there existsa unique weak solution u of (1.1) (resp. ( D Ω ,T ) ) such that u ∈ C ([0 , T ) ; L R (Ω) and u (0) = u . If v ∈ L R (cid:0) R N (cid:1) and u ≦ v , then u ≦ v. If u is nonnegative, then for any < q ≦ , there stillexists at least a weak nonnegative solution u satisfying the same conditions. Next we prove Theorem 1.4. Our proof of (ii) (iii) is based on approximations by nonincreasingsequences. Another proof can be obtained when u ∈ L loc (cid:0) R N (cid:1) and q ≦ , by techniques ofequiintegrability, see [22] for a connected problem. Proof of Theorem 1.4.
Assume Ω = R N (resp. Ω bounded).(i) Case 1 < q < q ∗ , u ∈ M + (cid:0) R N (cid:1) (resp. M + (Ω)): Let u ,n = u x B n (resp. u ,n = u x Ω ′ /n , where Ω n = { x ∈ Ω : d ( x, ∂ Ω) > /n } , for n large enough). From Theorem 5.1, there exists aunique weak solution u n of (1.1) (resp. of ( D Ω ,T )) with trace u ,n , and ( u n ) is nondecreasing; and u n ∈ C , ( Q R N ,T ) since q ≦ . From (3.1), (3.5), for any ξ ∈ C c (Ω) , Z Ω u n ( ., t ) ξ q ′ + 12 Z t Z Ω |∇ u n | q ξ q ′ ≦ Ct Z Ω |∇ ξ | q ′ + Z Ω ξ q ′ du . (5.1)17ence ( u n ) is bounded in L ∞ loc (cid:0) [0 , T ) ; L loc (Ω) (cid:1) , and ( |∇ u n | q ) is bounded in L loc (cid:0) [0 , T ) ; L loc (Ω) (cid:1) . In turn ( u n ) is bounded in L ∞ loc ((0 , T ); L ∞ loc (Ω)) , from Theorem 3.3. From Theorem 2.3, up toa subsequence, ( u n ) converges in C , loc ( Q R N ,T ) (resp. C , loc ( Q Ω ,T ) ∩ C , (cid:0) Ω × (0 , T ) (cid:1) ) to a weaksolution u of (1.1) in Q R N ,T (resp. of ( D Ω ,T )). Also from [3, Lemma 3.3], for any k ∈ [1 , q ∗ ) andany 0 < s < τ < T, k u n k L k (( s,τ ); W ,k ( ω )) ≦ C ( k, ω )( k u n ( s, . ) k L ( ω ) + k|∇ u n | q + |∇ u n | + u n k L ( Q ω,s,τ ) ) , ∀ ω ⊂⊂ Ω(resp. k u n k L k (( s,τ ); W ,k (Ω)) ≦ C ( k, Ω)( k u n ( ., s ) k L (Ω) + k|∇ u n | q k L ( Q Ω ,s,τ ) ) . )hence ( u n ) is bounded in L kloc ([0 , T ) ; W ,kloc ( R N )) (resp. L kloc ([0 , T ) ; W ,k (Ω))). Since q < q ∗ , ( |∇ u n | q ) is equiintegrable in Q B M ,τ for any M > Q Ω ,τ ) and τ ∈ (0 , T ) , then ( |∇ u | q ) ∈ L loc (cid:0) [0 , T ) ; L loc (Ω) (cid:1) . From (2.6), Z Ω u n ( t, . ) ξ + Z t Z Ω |∇ u n | q ξ = − Z t Z Ω ∇ u n . ∇ ξ + Z Ω ξdu . (5.2)As n → ∞ we obtain Z Ω u ( t, . ) ξ + Z t Z Ω |∇ u | q ξ = − Z t Z Ω ∇ u. ∇ ξ + Z Ω ξdu . Thus lim t → R Ω u ( ., t ) ξ = R Ω ξdu , for any ξ ∈ C c (Ω) , hence for any ξ ∈ C + c (Ω); hence u admitsthe trace u . (ii) Case q ∗ ≦ q ≦ . Let us set u ,n = min( u , n ) χ B n (resp. u ,n = min( u , n ) χ Ω ′ /n for n largeenough). Then u ,n ∈ L R (Ω) for any R ≧
1. From Theorem 5.1, the problem admits a solution u n , and it is unique in C ([0 , T ) ; L R (Ω)) for any R > (2 − q ) /N ( q −
1) and then ( u n ) is nondecreasing.As above, ( u n ) is bounded in L ∞ loc (cid:0) [0 , T ) ; L loc (Ω) (cid:1) , ( |∇ u n | q ) is bounded in L loc (cid:0) [0 , T ) ; L loc (Ω) (cid:1) , ( u n ) is bounded in L ∞ loc ((0 , T ); L ∞ loc (Ω)) from Theorem 3.3. From Theorem 2.3, ( u n ) converges in C , loc ( Q Ω ,T ) to a weak solution u of (1.1) in Q Ω ,T , such that u ∈ L ∞ loc (cid:0) [0 , T ) ; L loc (Ω) (cid:1) and |∇ u | q ∈ L loc (cid:0) [0 , T ) ; L loc (Ω) (cid:1) .Then from Remark 2.5, u admits a trace µ ∈ M + (Ω) as t → . Applying (5.2) to u n , since u n ≦ u, we get lim t → Z Ω u ( ., t ) ξ = Z Ω ξdµ ≧ lim t → Z Ω u n ( ., t ) ξ = Z Ω ξdu , for any ξ ∈ C + c (Ω); thus u ≦ µ . Moreover Z Ω u n ( t, . ) ξ + Z t Z Ω |∇ u n | q ξ = Z t Z Ω u n ∆ ξdx + Z Ω ξdu . And ( u n ) is bounded in L k ( Q ω,τ ) for any k ∈ (1 , q ∗ ) ; then for any domain ω ⊂⊂ Ω, ( u n ) convergesstrongly in L ( Q ω,τ ) ; then from the convergence a.e. of the gradients, and the Fatou Lemma, Z R N u ( t, . ) ξ + Z t Z R N |∇ u | q ξ ≦ Z t Z R N u ∆ ξdx + Z R N ξdu . But from Remark 2.5, Z R N u ( t, . ) ξ + Z t Z R N |∇ u | q ξ = Z t Z R N u ∆ ξdx + Z R N ξdµ , µ ≦ u , hence µ = u . Finally we prove the continuity: Let ξ ∈ D + (Ω) and ω ⊂⊂ Ωcontaining the support of ξ. Then z = uξ is solution of the Dirichlet problem z t − ∆ z = g, in Q ω,T ,z = 0 , on ∂ω × (0 , T ) , lim t → z ( ., t ) = ξu , weakly in M b ( ω ) , with g = − |∇ u | q ξ + v ( − ∆ ψ ) − ∇ v. ∇ ψ ∈ L ( Q ω,T ) . The solution is unique, see [6, Proposition2.2]. Since u ∈ L loc (Ω) , there also exists a unique solution such that z ∈ C ([0 , T ) , L ( ω )) from[3, Lemma 3.3], hence u ∈ C ([0 , T ) , L loc (Ω)) . (iii) Case q > . We get the existence as above, by taking for ( u ,n ) a nondecreasing sequencein C b (cid:0) R N (cid:1) (resp. in C (Ω)) , converging to u , and using Remark 2.4 for classical solutions.Next we show the nonuniqueness of the weak solutions when q > a defined at (1.7) is negative, and | a | = ( q − / ( q − < . Proof of Theorem 1.5.
Since q > N ≥ , the function ˜ U is a solution in D ′ (cid:0) R N (cid:1) ofthe stationary equation − ∆ u + |∇ u | q = 0Indeed ˜ U ∈ W ,qloc (cid:0) R N (cid:1) ∩ W , loc (cid:0) R N (cid:1) because N > q ′ , and ˜ U is a classical solution in R N \ { } . Thenit is a weak solution of ( P R N , ∞ ) , and ˜ U C ( Q R N , ∞ ) . Since ˜ U ∈ C (cid:0) R N (cid:1) , from Theorem ?? , orfrom [5], there exists also a classical solution U ˜ C ∈ C , ( Q R N , ∞ ) of the problem, thus U ˜ C = U . More generally, for any
C > , there exists a classical solution U C with trace C | x | | a | . And U C is obtained as the limit of the nondecreasing sequence of the unique solutions U n,C withtrace min( C | x | | a | , n ) , then it is radial. Moreover for any λ > , the function U n,C,λ ( x, t ) = λ − a U n,C ( λx, λ t ) admits the trace min( C | x | | a | , nλ − a ) . Therefore, denoting by k λ,n the integer partof nλ − a , there holds U k λ,n ,C ≤ U n,C,λ ≤ U k λ,n +1 from the comparison principle. And U n,C,λ ( x, t )converges everywhere to λ − a U C ( λx, λ t ) , thus U C ( x, t ) = λ − a U C ( λx, λ t ) , that means U C is self-similar. Then U C has the form (1.14), where f ∈ C ([0 , ∞ )) , f (0) ≧ , f ′ (0) = 0 , lim η →∞ η −| a | / f ( η ) = C, and for any η > , f ′′ ( η ) + ( N − η + η f ′ ( η ) − | a | f ( η ) − (cid:12)(cid:12) f ′ ( η ) (cid:12)(cid:12) q = 0 . (5.3)From the Cauchy-Lipschitz Theorem, we find f (0) > , since f , hence f ′′ (0) > . The function f is increasing: indeed if there exists a first point η > f ′ ( η ) = 0 , then f ′′ ( η ) > , which is contradictory. Here we show the second regularizing effect. We prove an estimate, playing the role of the sub-caloricity estimate (2.4). Our proof follows the general scheme of Stampacchia’s method, developpedby many authors, see [17] and references there in, and [19].First we write estimate (3.1) in another form, and from Gagliardo estimate, we obtain thefollowing: 19 emma 6.1
Let q > . Let η > , r ≧ . Let u be any nonnegative weak subsolution of equation(1.1) in Q Ω ,T . Let B η ⊂⊂ Ω , < θ < τ < T, and ξ ∈ C ((0 , T ) , C c (Ω)) , with values in [0 , , suchthat ξ ( ., t ) = 0 for t ≦ θ . Let λ ≧ max(2 , q ′ ) . Then for any ν ∈ (0 , , sup [ θ,τ ] Z Ω u r ( ., t ) ξ λ + R τθ R Ω u ( q + r − µN ) ξ λ (1+ µN ) (sup t ∈ [ θ,τ ] R Ω u r ξ λrq + r − ) qN ≦ C Z τθ Z Ω ( u r | ξ t | + u r − |∇ ξ | q ′ + u q + r − |∇ ξ | q ) , (6.1) where µ = rq/ ( q + r − , C = C ( N, q, r, λ ) . Proof.
From Remark 2.2, u ∈ L ∞ loc ( Q Ω ,T )) , and hence u q + r − q ξ λq ∈ W ,q ( Q Ω ,θ,t ) and Z tθ Z Ω |∇ ( u q + r − q ξ λq ) | q = Z tθ Z Ω (cid:12)(cid:12)(cid:12)(cid:12) q + r − q u r − q ξ λq ∇ u + λq u q + r − q ξ λ − qq ∇ ξ (cid:12)(cid:12)(cid:12)(cid:12) q ≦ C ( Z tθ Z Ω u r − |∇ u | q ξ λ + Z tθ Z Ω u q + r − |∇ ξ | q ξ λ − q ) , with C = C ( q, r, λ ) . From (3.1), since ν ≦ , we getsup [ θ,τ ] Z Ω u r ( ., t ) ξ λ + Z τθ Z Ω |∇ ( u q + r − q ξ λq ) | q ≦ C Z τθ Z Ω ( u r | ξ t | + u r − |∇ ξ | q ′ + u q + r − |∇ ξ | q ) , (6.2)where C = C ( q, r, λ ) . Next we use a Galliardo type estimate, see [17, Proposition 3.1]: for any µ ≧ , and any w ∈ L ∞ loc ((0 , T ) , L µ (Ω)) ∩ L qloc ((0 , T ) , W ,q (Ω)) , Z τθ Z Ω w q (1+ µN ) ) ≦ C ( Z τθ Z Ω |∇ w | q )( sup t ∈ [ θ,τ ] Z Ω | w | µ ) qN , C = C ( N, q, µ ) . Taking w = u q + r − q ξ λq and µ = qr/ ( q + r − ≧ r ≧ , setting s = 1 + µ/N, it comes Z τθ Z Ω u ( q + r − s ξ λs ≦ C ( Z τθ Z Ω |∇ w | q )( sup t ∈ [ θ,τ ] Z Ω u r ξ λrq + r − ) qN , hence (6.1) follows. Theorem 6.2
Let q > . Let u be any nonnegative weak solution of equation (1.1) in Q Ω ,T . Let B ( x , ρ ) ⊂⊂ Ω . Let
R > q − (in particular any R ≧ if q < . Then there exists C = C ( N, q, R ) such that, for any t, θ such that < t − θ < t < T, sup B ( x , ρ ) × [ t − θ,t ] u ≦ Cθ − N + qqR + N ( q − ( Z tt − θ Z B ( x ,ρ ) u R ) qqR + N ( q − + Cρ − N + q ( q − R + N +1) ( Z tt − θ Z B ( x ,ρ ) u R ) R + N +1 + Cρ − N + qR +1 − q ( Z tt − θ Z B ( x ,ρ ) u R ) R +1 − q . (6.3)20 roof. Since u ∈ C ((0 , T ); L Rloc ( Q Ω ,T )) , by regularization we can assume that u is a classicalsolution in Q Ω ,T . Let t, θ such that 0 < t − θ < t < T. We can assume x = 0 ∈ Ω . By translationof t − θ, we are lead to prove that for any solution in Q Ω , − τ/ ,τ/ ( τ < T ),sup Q Bρ/ , ,θ u ≦ Cθ − N + qqR + N ( q − ( Z θ − θ Z B ρ u R ) qqR + N ( q − + Cρ − N + q ( q − R + N +1) ( Z θ − θ Z B ρ u R ) R + N +1 + Cρ − N + qR +1 − q ( Z θ − θ Z B ρ u R ) R +1 − q . (6.4)For given k > u k = ( u − k ) + . Then u k ∈ C (0 , T ); L Rloc ( Q Ω ,T )) , and u k is a weak subsolutionof equation (1.1), from the Kato inequality. We set ρ n = (1 + 2 − n ) ρ/ , t n = − (1 + 2 − n ) θ/ ,Q n = B ρ n × ( t n , θ ) , Q = B ρ × ( − θ, θ ) , Q ∞ = B ρ/ × ( − θ/ , θ ) ,k n = (1 − − ( n +1) ) k, ˜ k = ( k n + k n +1 ) / . and set M σ = sup Q ∞ u, M = sup Q u. Let ξ ( x, t ) = ξ ( x ) ξ ( t ) where ξ ∈ C c (Ω) , ξ ∈ C ( R ) , withvalues in [0 , ξ = 1 on B ρ n +1 , ξ = 0 on R N \ B ρ n , |∇ ξ | ≦ C ( N )2 n +1 /ρ ; ξ = 1 on [ θ n +1 , ∞ ) , ξ = 0 on ( −∞ , θ n ] , | ξ ,t | ≦ C ( N )2 n +1 /θ. From Lemma 6.1 we get, with µ = qr/ ( q + r − , sup t ∈ [ t n +1 ,θ ] R B ρn +1 u rk n +1 ( ., t ) + R θt n +1 R B ρn +1 u ( q + r − µN ) k n +1 (sup t ∈ [ t n ,θ ] R B ρn u rk n ) qN ≦ CX n , where X n = Z θt n Z B ρn ( u rk n +1 | ζ t | + u r − k n +1 |∇ ξ | q ′ + u q + r − k n +1 |∇ ξ | q )) . Let us define Y n = Z θt n Z B ρn u q + r − k n , Z n = sup t ∈ [ t n ,θ ] Z B ρn u rk n , W n = Z θt n Z B ρn χ { u ≧ k n } . Thus, from the H¨older inequality, Z n +1 + Z − qN n W − µN n +1 Y µN n +1 ≦ CX n . (6.5)Morever, for any γ, β > , Z θt n Z B ρn u γ + βk n ≧ Z θt n Z B ρn ( k n − k n +1 ) γ + β χ { u ≧ k n +1 } ≧ ( k − ( n +2) ) γ + β Z θt n Z B ρn χ { u ≧ k n +1 } ≧ ( k − ( n +2) ) γ + β Z θt n +1 Z B ρn +1 χ { u ≧ k n +1 } , Z θt n Z B ρn u γk n +1 ≦ ( Z θt n Z B ρn u γ + βk n +1 ) γγ + β ( Z θt n Z B ρn χ { u ≧ k n +1 } ) βγ + β ≦ ( Z θt n Z B ρn u γ + βk n )( k − ( n +2) ) β ( Z θt n Z B ρn u γ + βk n ) βγ + β ≤ ( k − ( n +2) ) β Z θt n Z B ρn u γ + βk n . Thus in particular W n +1 ≦ C ( 2 n +1 k ) q + r − Y n , Z θt n Z B ρn u rk n +1 ≦ C ( 2 n +1 k ) q − Y n , Z θt n Z B ρn u r − k n +1 ≦ C ( 2 n +1 k ) q Y n . (6.6)Otherwise X n ≦ Z θt n Z B ρn (2 n +1 θ − u rk n +1 + 2 q ′ ( n +1) ρ − q ′ u r − k n +1 + 2 q ( n +1) ρ − q u q + r − k n +1 ) , then from (6.6), X n ≦ Cb n f ( θ, ρ, k ) Y n , where f ( θ, ρ, k ) = ( θ − k q − + 1 k q ρ − q ′ + ρ − q ) . (6.7)for some b depending on q, r. Then from (6.5), (6.6) and (6.7), Z n +1 ≦ Cb n f ( θ, ρ, k ) Y n , Y µN n +1 ≦ CZ qN n ( 2 n +1 k ) ( q + r − µN b n f ( θ, ρ, k ) Y µN n . Since Y n +1 ≦ Y n , setting α = q/ ( N + µ ) and denoting by b , b some new constants depending on N, q, r, Y n +2 ≦ CZ qN + µ n +1 b n +11 k − ( q + r − µN + µ f NN + µ ( θ, ρ, k ) Y n +1 ≦ C ( b n f ( θ, ρ, k ) Y n ) qN + µ b n +11 k − ( q + r − µN + µ f NN + µ ( θ, ρ, k ) Y n ≦ Cb n f N + qN + µ k − ( q + r − µN + µ Y qN + µ n := Db n Y αn . From [17, Lemma 4.1], Y n → Y α δ /α ≦ D − = C − k ( q + r − µN + µ f − N + qN + µ , that means k qr ≧ cY q (( θ − k q − + 1 k q ρ − q ′ + ρ − q )) N + q . (6.8)For getting (6.8) it is sufficient that k qr +( q − N + q ) ≧ c Y q θ − ( N + q ) , k ( r + N + q ) ≧ ( c /q Y ρ − N + qq − , and k r ≧ c Y ρ − ( N + q ) . Q ∞ u ≦ Cθ − N + qqr +( N + q )( q − ( Z θ − θ Z B ρ u q + r − ) qqr +( N + q )( q − + Cρ − N + q ( q − r + N + q ) ( Z θ − θ Z B ρ u q + r − ) r + N + q + Cρ − N + qr ( Z θ − θ Z B ρ u q + r − ) r . (6.9)If we set q + r − R , we obtain (6.4) for any R ≧ q .Next we consider the case R < q.
From (6.9) we getsup B σρ × ( − θ/ ,θ ) u ≦ Cθ − N + qq +( q − N + q ) ( Z θ Z B ρ u q ) qq +( q − N + q ) + Cρ − N + q ( q − N + q ) ( Z θ − θ Z B ρ u q ) N + q + Cρ − ( N + q ) Z θ − θ Z B ρ u q ≦ Cθ − N + qq +( q − N + q ) ( sup B ρ × ,θ ) u ) q ( q − R ) q +( q − N + q ) ( Z θ − θ Z B ρ u R ) qq +( q − N + q ) + Cρ − N + q ( q − N + q ) ( sup B ρ × ,θ ) u ) q ( q − R )1+ N + q ) ( Z θ − θ Z B ρ u R ) N + q + Cρ − ( N + q ) ( sup B ρ × ,θ ) u ) ( q − R ) Z θ − θ Z B ρ u R . We define˜ ρ n = (1 + 2 − ( n +1) ) ρ, θ n = − (1 + 2 − ( n +1) ) θ, ˜ Q n = B ˜ ρ n × ( θ n , θ ) , M n = sup ˜ Q n u, hence M = sup B ρ/ × ( − θ/ ,θ ) u. We find M n ≦ Cθ − N + qq +( q − N + q ) M q ( q − R ) q +( q − N + q ) n +1 ( Z θ − θ Z B ρ u R ) qq +( q − N + q ) + Cρ − N + q ( q − N + q ) M q ( q − R )1+ N + q n +1 ( Z θ − θ Z B ρ u R ) N + q + Cρ − ( N + q ) M q − Rn +1 Z θ − θ Z B ρ u R . We set I = Cθ − N + qq +( q − N + q ) ( Z θ − θ Z B ρ u R ) qq +( q − N + q ) ,J = Cρ − ( N + q ) Z θ Z B ρ u R , L = Cρ − N + q ( q − N + q ) ( Z θ − θ Z B ρ u R ) N + q . Note that
R > q − , that means q − R < . Then from H¨older inequality, M n ≦ M n +1 + C ( I σ + L δ + J R +1 − q ) , σ = q + ( q − N + q ) N ( q −
1) + qR , δ = 1 + N + qR + N + 1 . M ≦ − n M n + 2 C ( I σ + L δ + J R +1 − q ) , and finally M = sup Q u ≦ C ( I σ + L δ + J R +1 − q ) = Cθ − N + qN ( q − qR ( Z θ − θ Z B ρ u R ) qN ( q − qR + Cρ − N + q ( q − R + N +1) ( Z θ − θ Z B ρ u R ) R + N +1 + Cρ − N + qR +1 − q ( Z θ − θ Z B ρ u R ) R +1 − q , which shows again (6.4). Then (6.4) holds for any R > q − , in particular for any R ≧ q < . Now we prove our second regularing effect due to the effect of the gradient:
Proof of Theorem 1.6.
We assume x = 0 . Let κ > ρ ∈ (0 , η ) such that ρ κ ≦ t < τ, sup B ρ × [ t − ρ κ ,t ] u ≦ Cρ − κ ( N + q ) qR + N ( q − ( Z tt − ρ κ Z B ρ u R ) qqR + N ( q − + Cρ − N + q ( q − R + N +1) ( Z tt − ρ κ Z B ρ u R ) R + N +1 + Cρ − N + qR +1 − q ( Z tt − ρ κ Z B ρ u R ) R +1 − q , where C = C ( N, q, R ) . Now from estimate (3.3) of Lemma 3.2,sup B η/ u ( ., t ) ≦ Cρ − κNqR + N ( q − ( η NR − q ′ t + k u k L R ( B η ) ) RqqR + N ( q − + Cρ − N + q ( q − R + N +1) + κR + N +1 ( η NR − q ′ t + k u k L R ( B η ) ) RR + N +1 + Cρ − ( N + q )+ κR +1 − q ( η NR − q ′ t + k u k L R ( B η ) ) RR +1 − q . Let τ < T, and k ∈ N such that k η κ / ≧ τ. For any t ∈ (0 , τ ] , there exists k ∈ N with k ≦ k suchthat t ∈ ( kη κ / , ( k + 1) η κ / . taking ρ κ = t/ ( k + 1) , we find for any 0 < t < τ, and C = C ( N, q, R ) , sup B η/ u ( ., t ) ≦ C ( 1 + η − κ τt ) NqR + N ( q − ( η NR − q ′ t + k u k L R ( B η ) ) RqqR + N ( q − + C ( 1 + η − κ τt ) N + qκ ( q − − R + N +1 ( η NR − q ′ t + k u k L R ( B η ) ) RR + N +1 + C ( 1 + η − κ τt ) N + qκ − R +1 − q ( η NR − q ′ t + k u k L R ( B η ) ) RR +1 − q . (6.10)If we choose κ such that κε ( N + q ) q ′ ≧ , we obtain, with C = C ( N, q, R, η, ε, τ ) , sup B η/ u ( ., t ) ≦ Ct − NqR + N ( q − ( t + k u k L R ( B η ) ) RqqR + N ( q − + Ct − εR + N +1 ( t + k u k L R ( B η ) ) RR + N +1 + Ct − εR +1 − q ( t + k u k L R ( B η ) ) RR +1 − q (6.11)And in fact the second term can be absorbed by the first one, with a new constant depending on τ, and we finally obtain (1.15). 24 emark 6.3 These estimate in t − N/ ( qR + N ( q − improves the estimate in t − N/ R of the first reg-ularizing effect when q > q ∗ . And it appears to be sharp. Indeed consider for example the partic-ular solutions given in [25] of the form u C ( x, t ) = Ct − a/ f ( | x | / √ t ) , where η f ( η ) is bounded, f ′ (0) = 0 and lim η →∞ η a f ( η ) = C. Then u C is solution of (1.1) in Q R N \{ } , ∞ , with initial data C | x | − a . When a < N, that means q > q ∗ , then | x | − a ∈ L Rloc ( R N ) for any R ∈ [1 , N/a ) , and u C issolution in Q R N , ∞ . We have sup B u ( ., t ) = Cf (0) t − a/ . Taking
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