Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption
aa r X i v : . [ m a t h . A P ] A ug Singular solutions of some nonlinear parabolicequations with spatially inhomogeneousabsorption
Andrey Shishkov
Institute of Applied Mathematics and Mechanics of NAS of Ukraine , R. Luxemburg str. 74, 83114 Donetsk, Ukraine
Laurent V´eron
Laboratoire de Math´ematiques et Physique Th´eorique , Universit´e Fran¸cois-Rabelais, 37200 Tours, France
Abstract
We study the limit behaviour of solutions of ∂ t u − ∆ u + h ( | x | ) | u | p − u = 0 in R N × (0 , T )with initial data kδ when k → ∞ , where h is a positive nondecreasing function and p > h ( r ) = r β , ( β > −
2) we prove that the limit function u ∞ is an explicit very singularsolution. If lim inf r → r ln(1 /h ( r )) > u ∞ has a persistent singularity at (0 , t ) ( t ≥ R r r ln(1 /h ( r )) dr < ∞ , u ∞ has a pointwise singularity localized at (0 , . . Key words . Parabolic equations, Saint-Venant principle, very singular solutions, razor blade, Keller-Osserman estimates,asymptotic expansions.
Consider ∂ t u − ∆ u + h ( x ) | u | p − u = 0 in Q T := R N × (0 , T ) , (1.1)with p > h is a nonnegative measurable function defined in R N . It is well known that if Z Z Q T h ( x ) E p ( x, t ) dx dt < ∞ , (1.2)where E ( x, t ) = (4 πt ) − N/ e −| x | / t is the heat kernel, then, for any k > u = u k to (1.1 ) satisfying initial condition u ( .,
0) = kδ (1.3)in the sense of measures in R N . Furthermore the mapping k u k is increasing. If it assumed that h is positive essentially locally bounded from above and from below in R N \ { } , then the set { u k } is also bounded in the C loc ( Q T \ { × (0 , ∞ ) } )-topology. Thus there exist u ∞ := lim k →∞ u k and1 u ∞ is a solution of (1.1 ) in Q T \ { × (0 , ∞ ) } . Furthermore u ∞ is continuous in Q T \ { × [0 , ∞ ) } and vanishes on R N \ { } × { } . We shall prove that only two situations can occur:(i) Either u ∞ (0 , t ) is finite for every t > u ∞ is a solution of (1.1 ) in Q T . Such a solutionwhich has a pointwise singularity at (0 ,
0) is called a very singular solution (abr. V.S.S.)(ii) Or u ∞ (0 , t ) = ∞ for every t > u ∞ is a solution of (1.1 ) in Q T \ { × (0 , ∞ ) } only. Sucha solution with a persistent singularity is called a razor blade (abr. R. B.).In the well-known article [4], Brezis, Peletier and Terman proved in 1985 that u ∞ is a V.S.S., if h ( x ) ≡
1. Furthermore they showed that u ∞ ( x, t ) = t − / ( p − f ( x/ √ t ) for ( x, t ) ∈ Q T where f isthe unique positive (and radial) solution of the problem − ∆ f − η. ∇ f − p − f + | f | p − f = 0 in R N lim | η |→∞ | η | / ( p − f ( η ) = 0 . (1.4)Their proof of existence and uniqueness relied on shooting method in ordinary differential equations(abr. O.D.E.). The already mentioned self-similar very singular solutions of the problem (1.4 )was discovered independently in [6] too. Later on, a new proof of existence, has been given byEscobedo and Kavian [8] by a variational method in a weighted Sobolev space. More preciselythey proved that the following functional v J ( v ) = 12 Z R N (cid:18) |∇ v | − p − v + 2 p + 1 | v | p +1 (cid:19) K ( η ) dη (1.5)achieves a nontrivial minimum in H K ( R N ), where K ( η ) = e | η | / .In this article we first study equation (1.1 ) when h ( x ) = | x | β ( β ∈ R ). Looking for self-similarsolutions under the form u ( x, t ) = t − (2+ β ) / p − f ( x/ √ t ), we are led to − ∆ f − η. ∇ f − β p − f + | η | β | f | p − f = 0 in R N f ∈ H loc ( R N ) ∩ L p +1 loc ( R N ; | η | β dη ) ∩ C ( R N \ { } )lim | η |→∞ | η | (2+ β ) / ( p − f ( η ) = 0 , (1.6)and the associated functional v J ( v ) = 12 Z R N (cid:18) |∇ v | − β p − v + 2 p + 1 | η | β | v | p +1 (cid:19) K ( η ) dη. (1.7)We prove the following Theorem A
I- Assume β ≤ N ( p − − ; then there exists no nonzero solution to (1.6 ).II- Assume β > N ( p − − ; then there exists a unique positive solution f ∗ to (1.6 ). One of the key arguments in the study of isolated singularities of (1.1 ) is the following a prioriestimate | u ( x, t ) | ≤ ˜ c ( t + | x | ) (2+ β ) / p − ∀ ( x, t ) ∈ Q T (1.8)valid for any p > β > −
2. The remarkable aspect of this proof is that it is based upon theauxiliary construction of the maximal solution of (1.1 ) under a selfsimilar form. Next we give twoproofs of II, one based upon scaling transformations and asymptotic analysis of O.D.E., combiningideas from [4], [5] and [10], and the second based on variational methods, extending some ideasfrom [8] and valid in a more general context. As a consequence we prove
Theorem B
Assume β > N ( p − − , then u ∞ ( x, t ) = t − (2+ β ) / p − f ∗ ( x/ √ t ) . It must be noticed that, if β ≤ N ( p − − u k does not exist, and more precisely, the isolatedsingularities of solutions of (1.1 ) are removable.Next we consider the case of more degenerate potentials h ( x ): h ( x ) | x | α → | x | → ∀ α > . (1.9)In the set of such potentials we find the borderline which separates the above mentioned twopossibilities (i) — (V.S.S.) and (ii) — (R.B). Remark that in the case of flat potentials like (1.9),the corresponding solution u ∞ ( x, t ) does not have self-similar structure and we haveto find somealternative techniques for the study of the structure of u ∞ . Main results of the paper are thefollowing two statements. Theorem C (sufficient condition of V.S.S. solution)
Assume that the function h is continuousand positive in R N \ { } and verifies the following flatness condition | x | ln (cid:18) h ( x ) (cid:19) ≤ ω ( | x | ) ⇔ h ( x ) ≥ e − ω ( | x | ) / | x | ∀ x ∈ R N , (1.10) where the function ω ≥ is nondecreasing, satisfies the following Dini-like condition Z ω ( s ) dss < ∞ , (1.11) and the additional technical condition sω ′ ( s ) ≤ (2 − α ) ω ( s ) near , (1.12) for some α ∈ (0 , . Then u ∞ ( x, t ) < ∞ for any ( x, t ) ∈ Q T . Furthermore there exists positiveconstants C i ( i = 1 , , ), depending only on N , α and p , such that Z R N u ∞ ( x, t ) dx ≤ C t exp h C (cid:0) Φ − ( C t ) (cid:1) − i ∀ t > , (1.13) where Φ − is the inverse function of Φ( τ ) := Z τ ω ( s ) s ds. Notice that (1.11 )-(1.12 ) is satisfied if h ( x ) ≥ Ce −| x | θ − for some θ > Theorem D (sufficient condition of R.B. solution)
Assume h is continuous and positive in R N \ { } and satisfies lim inf x → | x | ln (cid:18) h ( x ) (cid:19) > ⇔ ∃ ω = const > h ( x ) ≤ exp (cid:18) − ω | x | (cid:19) . (1.14) Then u ∞ (0 , t ) = ∞ for any t > , and t u ∞ ( x, t ) is increasing. If we denote U ( x ) =lim t →∞ u ∞ ( x, t ) , then U is the minimal large solution of − ∆ u + h ( x ) u p = 0 in R N \ { } , (1.15) i.e. the smallest solution of (1.15 ) which satisfies Z B ǫ u ( x ) dx = ∞ ∀ ǫ > . (1.16)Theorem C is proved by some new version of local energy method. A similar variant of thismethod was used in [1] for the study of extinction properties of solutions of nonstationary diffusion-absorption equations.Theorem D is obtained by constructing local appropriate sub-solutions. The monotonicity andthe limit property of u ∞ are characteristic of razor blades solutions [16].A natural question which remains unsolved is to characterize u ∞ if the potential h ( x ) satisfies h ( x ) ≈ exp (cid:18) − ω ( | x | ) | x | (cid:19) , where ω ( s ) → s → Z ω ( s ) dss = ∞ . This article is the natural continuation of [12], [14] where (1.1 ) is replaced by ∂ t u − ∆ u + h ( t ) | u | p − u = 0 in Q T . (1.17)In equation (1.17 ), the function h ∈ C ([0 , T ]) is positive in (0 , T ] and vanishes only at t = 0. Inthe particular case h ( t ) = t β ( β > u k exists if and only if 1 < p < β ) /N , and u ∞ is anexplicit very singular solution. If h ( t ) ≥ e − ω ( t ) /t where ω is positive, nondecreasing and satisfies Z p ω ( s ) dss = ∞ , then u ∞ has a pointwise singularity at (0 , h is stronger, namelylim inf t → t ln h ( t ) > −∞ , it is proved that the singularity of u k propagates along the axis t = 0; at end, u ∞ is nothing elsethan the (explicit) maximal solution Ψ( t ) of the O.D.E.Ψ ′ + h ( t )Ψ p = 0 in (0 , ∞ ) . (1.18)A very general and probably difficult open problem generalizing (1.1 ) and (1.17 ) is to studythe propagation phenomenon of singularities starting from (0 ,
0) when (1.1 ) is replaced by ∂ t u − ∆ u + h ( x, t ) | u | p − u = 0 in Q T , (1.19)where h ∈ C ( Q T ) is nonnegative and vanishes only on a curve Γ ⊂ Q T starting from (0 , u ∞ has a pointwise singularity at (0 , u ∞ is singular along Γ or a connected part of Γ containing (0 , R N × R . This could serve as a startingmodel for nonlinear heat propagation in inhomogeneous fissured media.Our paper is organized as follows: 1 Introduction - 2 The power case - 3 Pointwise singularities -4 Existence of razor blades. Acknowledgements
The authors have been supported by INTAS grant Ref. No : 05-1000008-7921.
In this section we assume that h ( x ) = | x | β with β ∈ R and the equation under consideration is thefollowing ∂ t u − ∆ u + | x | β | u | p − u = 0 in Q T := R N × (0 , T ) (2.1)with p >
1. By a solution we mean a function u ∈ C , ( Q T ). Let E ( x, t ) = (4 πt ) − N/ e −| x | / t bethe heat kernel in Q T and E [ φ ] the heat potential of a function (or measure) φ defined by E [ φ ]( x, t ) = 1(4 πt ) N/ Z R N e −| x − y | / t φ ( y ) dy. (2.2)If there holds Z Z Q T E p ( x, t ) | x | β dx dt < ∞ , (2.3)it is easy to prove (see [12, Prop 1.2], and [18, Th 6.12]), that for any k ∈ R , there exists a uniquefunction u = u k ∈ L ( B R × (0 , T )) ∩ L p ( B R × (0 , T ); | x | β dx ) such that Z Z Q T (cid:16) − u∂ t ζ − u ∆ ζ + | x | β | u | p − uζ (cid:17) dx dt = kζ (0 , , (2.4)for any ζ ∈ C , ( R N × [0 , T )). By the maximum principle k u k is increasing. Next, it isstraightforward that (2.3 ) is fulfilled as soon as β > max { N ( p − − − N } (2.5) a priori estimate and the maximal solution In order to prove an a priori estimate, we introduce the auxiliary N dimensional equation in thevariable η = x/ √ t − ∆ f − η. ∇ f − γf + | η | β | f | p − f = 0 , (2.6)where γ = (2 + β ) / p − Proposition 2.1
Let a > and β ∈ R ; then there exists a unique nonnegative function F a ∈ H loc ( B a ) ∩ L p +1 loc ( B a ; | η | β dη ) solution of (2.6 ) and satisfying lim | η |→ a F a ( η ) = ∞ . (2.7) Furthermore a F a is decreasing.Proof. Set K ( η ) = e | η | / . Then (2.6 ) becomes − K − div ( K ∇ f ) − γf + | η | β | f | p − f = 0 . (2.8) Step 1- Boundary behaviour.
First we claim thatlim | η |→ a ( a − | η | ) / ( p − F a ( η ) = (cid:18) p + 1) a pβ ( p − (cid:19) / ( p − . (2.9)Actually, if 0 < b < | η | < a , u satisfies − K − div ( K ∇ F a ) − γF a + CF pa ≤ C = min { a β , b β } . We perform a standard variant of the two-sides estimate method usedin [17] : we set Γ := B ρ \ B b with b < ρ < a , α = ( ρ − b ) / z the solution of (cid:26) z ′′ − Cz p = 0 in ( − α, α ) z ( − α ) = z ( α ) = ∞ . (2.10)Then z is an even function and is computed by the formula Z ∞ z ( t ) ds p s p +1 − z (0) p +1 = s Cp + 1 ( α − t ) ∀ t ∈ [0 , α ) . (2.11)Notice also that lim α → z ( t ) = ∞ , uniformly on ( − α, α ) andlim t → α ( t − α ) / ( p − z ( t ) = (cid:18) p + 1) C p ( p − (cid:19) / ( p − . (2.12)We set Z ( η ) = z ( | η | − ( ρ + b ) /
2) and we look for a super-solution in Γ under the form w = M Z ( η )( M > − K − div ( K ∇ w ) − γw + Cw p = M (cid:18) ( M p − − Cz p − (cid:18) N − | η | + | η | (cid:19) z ′ − γz (cid:19) . Since z ′ ( t ) = s Cp + 1 p z p +1 ( t ) − z (0) p +1 < C ∗ z ( p +1) / ( t ) , with C ∗ = s Cp + 1 , we derive − K − div ( K ∇ w ) − γw + Cw p ≥ M (cid:18) ( M p − − Cz p − (cid:18) N − b + a (cid:19) C ∗ z ( p +1) / − γz (cid:19) (2.13)on { η : ( ρ − b ) / | η | < ρ } ; and the same inequality holds true on { η : ρ < | η | < ( ρ − b ) / } , upto interverting a and b . For any M >
1, we can choose b > b < ρ < a , theright-hand side of (2.13 ) is positive and maximum principle applies in B ρ \ B b . Thus M Z ≥ F a inΓ. Furthermore, the previous comparison still holds if we take ρ = a , which implies α = ( a − b ) / C lim sup | η |→ a ( a − | η | ) / ( p − F a ( η ) ≤ M (cid:18) p + 1)min { a pβ , b pβ } ( p − (cid:19) / ( p − . (2.14)Because M > < b < a are arbitrary, we derivelim sup | η |→ a ( a − | η | ) / ( p − F a ( η ) ≤ (cid:18) p + 1) a pβ ( p − (cid:19) / ( p − . (2.15)For the estimate from below we notice that u satisfies − K − div ( K ∇ F a ) − γF a + ˜ CF pa ≥ { η : b < | η | < a } , with ˜ C = max { a β , b β } . Taking now α = a − b , we denote by ˜ z the positivesolution of ˜ z ′′ + γ ˜ z − ˜ C ˜ z p = 0 in (0 , α )˜ z (0) = 0˜ z ( α ) = ∞ . (2.16)Then ˜ z is computed by the formula Z ∞ ˜ z ( t ) ds q ˜ z ′ (0) − γs + 2 ˜ Cs p +1 / ( p + 1) = α − t ∀ t ∈ [0 , α ) , (2.17)and formula (2.12 ) is valid provided C be replaced by ˜ C . We fix A ∈ ∂B a with coordinates( a, , ..., w ( η ) = M ˜ z ( η − b ) with 0 < M <
1. Then − K − div ( K ∇ ˜ w ) − γ ˜ w + ˜ C ˜ w p = ˜ M (cid:16) ( ˜ M p − − z p − η w ′ (cid:17) ≤ , since ˜ w ′ ≥
0. Applying again the maximum principle, we derive ˜ w ( η ) ≤ F a in B a ∩{ η : b < η < a } .But clearly the direction η is arbitrary and can be replaced by any radial direction. Thuslim inf | η |→ a ( a − | η | ) / ( p − F a ( η ) ≥ ˜ M (cid:18) p + 1)max { a pβ , b pβ } ( p − (cid:19) / ( p − . (2.18)In turn, (2.18 ) implieslim inf | η |→ a ( a − | η | ) / ( p − F a ( η ) ≥ (cid:18) p + 1) a pβ ( p − (cid:19) / ( p − , (2.19)and (2.9 ) follows from (2.15 ) and (2.19 ). Step 2- Uniqueness. If F ′ is another nonnegative solution of (2.6 ) satisfying the same boundaryblow-up conditions, then for any ǫ > F ′ ǫ = (1 + ǫ ) F ′ is a super solution. Thus, for δ > Z Z B a (cid:18) − div ( K ∇ F a ) F a + δ + div ( K ∇ F ′ ǫ ) F ′ ǫ + δ + | η | β (cid:18) F pa F a + δ − F ′ ǫp F ′ ǫ + δ (cid:19) K (cid:19) (( F a + δ ) − ( F ′ ǫ + δ ) ) + dη ≤ γ Z Z B a (cid:18) F a F a + δ − F ′ ǫ F ′ ǫ + δ (cid:19) (( F a + δ ) − ( F ′ ǫ + δ ) ) + Kdη.
By monotonicity (cid:18) F pa F a + δ − F ′ ǫp F ′ ǫ + δ (cid:19) (( F a + δ ) − ( F ′ ǫ + δ ) ) + ≥ , and 0 ≤ (cid:18) F a F a + δ − F ′ ǫ F ′ ǫ + δ (cid:19) (( F a + δ ) − ( F ′ ǫ + δ ) ) + ≤ (( F a + δ ) − ( F ′ ǫ + δ ) ) + . By Lebesgue’s theorem, since (2.9 ) implies that (( F a + δ ) − ( F ′ ǫ + δ ) ) + has compact support in B a , lim δ → Z Z B a (cid:18) F a F a + δ − F ′ ǫ F ′ ǫ + δ (cid:19) (( F a + δ ) − ( F ′ ǫ + δ ) ) + Kdη = 0 . Using Green formula, we obtain
Z Z B a (cid:18) − div ( K ∇ F a ) F a + δ + div ( K ∇ F ′ ǫ ) F ′ ǫ + δ (cid:19) (( F a + δ ) − ( F ′ ǫ + δ ) ) + Kdη = Z Z F a ≥ F ′ ǫ (cid:12)(cid:12)(cid:12)(cid:12) ∇ F a − F a + δF ′ ǫ + δ ∇ F ′ ǫ (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∇ F ′ ǫ − F ′ ǫ + δF a + δ ∇ F a (cid:12)(cid:12)(cid:12)(cid:12) ! Kdη ≥ . Letting δ →
0, we derive, by Fatou’s theorem,
Z Z F a ≥ F ′ ǫ (cid:0) F p − a − F ′ ǫp − (cid:1) ( F a − F ′ ǫ ) Kdη ≤ . Thus F a ≤ F ′ ǫ . Since ǫ is arbitrary, F a ≤ F ′ . The reverse inequality is the same. The monotonicityof a F a is proved in a similar way, by the previous form of maximum principle. Step 3- Existence with finite boundary value.
We shall first prove the existence of a positivesolution w k of (2.6 ) with boundary value equal to k > a , and we shall let k → ∞ in order to obtain one solution satisfying (2.7 ). We denote by J a the functional definedover H ( B a ) ∩ L p +1 ( B a ; | η | β dη ) by J a ( w ) = 12 Z B a (cid:18) |∇ w | − γw + 1 p + 1 | η | β | w | p +1 (cid:19) K ( η ) dη. Let k > κ ∈ C ( B a ) with 0 ≤ κ ( η ) ≤ k , supp ( κ ) ⊂ B a \ B a/ , κ ( η ) ≡ k on B a \ B a/ . If v ∈ H ( B a ) ∩ L p +1 ( B a ; | η | β dη ) and w := v + κ , then J a ( w ) = J a ( v + κ ) ≥ J a ( v ) + J a ( κ ) + Z B a (cid:0) ∇ v. ∇ κ − γvκ − | η | β | v | p κ (cid:1) K ( η ) dη. Since γ ≤ λ a , it follows from Cauchy-Schwarz and H¨older-Young inequalities that J a ( w ) ≥ (1 − ǫ ) J a ( v ) − p p ǫ p J a ( κ )for 0 < ǫ <
1. Because lim a → λ a = ∞ , there exists a ∈ (0 , ∞ ] such that, for any 0 < a < a , J a ( v ) is bounded from below on H ( B a ) ∩ L p +1 ( B a ; | η | β dη ). Thus there exists a minimizer w k suchthat w k = v + κ with v in the above space; w k is a solution of (2.6 ) and w k | ∂B a = k . Furthermore w k is positive. Notice that if γ ≤ a = ∞ , in which case there exists a solution w k for any k > a >
0. The uniqueness of w k >
0, is a consequence of the monotonicity of the mapping k w k that we prove by a similar argument as in Step 2: if k < k ′ , there holds Z Z w k >w k ′ (cid:16) w p − k − w p − k ′ )( w k − w k ′ ) (cid:17) | η | β Kdη ≤ , which implies w k < ˜ w k . Uniqueness and radiality follows immediately, thus w k solves the differen-tial equation − w ′′ − (cid:18) N − r + r (cid:19) w ′ − γw + r β w p = 0 on (0 , a ) w ( a ) = k and w ∈ H rad ( B a ) ∩ L p +1 rad ( B a ; | η | β dη ) . (2.20)Next we shall assume γ >
0, equivalently β > −
2. If w k is a positive solution of (2.20 ) and λ > λ < λw k is a super-solution (resp. a sub-solution) larger (resp. smaller) than w k . Notethat β > − w k (0) > β > − w ′ k (0) = 0. Thus, by [13], there existsa solution w λk with boundary data λk , and this solution is positive because w k ≤ w λk ≤ λw k (resp. λw k ≤ w λk ≤ w k ). Consequently, the set A of the positive ˜ a such that there exists apositive solution of (2.20 ) on (0 , a ) for any a < ˜ a is not empty and independent of k . Furthermore,if for some ˜ a > k >
0, there exists some positive w k solution of (2.20 ) on 0 , ˜ a ),then for any 0 < a < ˜ a and any k >
0, there exists a positive solution w k of (2.20 ). Since r max { k, ( γ + a − β ) / ( p − } is a super-solution, there holds w k ( r ) ≤ max { k, ( γ + a − β ) / ( p − } ∀ r ∈ [0 , a ] . (2.21)Let us assume that a ∗ = sup A < ∞ . Because of (2.21 ) and local regularity of solutions of ellipticequations, for any ǫ, ǫ ′ > w ′ k ( a ) is bounded uniformly with respect if ǫ ≤ a < a ∗ − ǫ ′ . But since(2.20 ) implies a N − e a / w ′ k ( a ) = ǫ N − e ǫ / w ′ k ( ǫ ) + Z aǫ ( r β w pk − γw k ) r N − e r / dr,w ′ k ( a ) is actually uniformly bounded on [ ǫ, a ∗ ). It follows from the local existence and uniquenesstheorem that there exists δ >
0, independent of a < a ∗ such that there exists a unique solution z defined on [ a, a + δ ] to − z ′′ − (cid:18) N − r + r (cid:19) z ′ − γz + r β z p = 0 on (0 , a ) z ( a ) = k, z ′ ( a ) = w ′ k ( a ) , (2.22)and δ and k > z > a, a + δ ]. This leads to the existence of a positivesolution to (2.20 ) on [0 , a + δ ]. If a ∗ − a < δ , which contradicts the maximality of a ∗ . Therefore a ∗ = ∞ . Step 4- End of the proof.
We have already seen that k w k is increasing. By Step 1, we knowthat, for any a >
0, and some b < a , there holds w k ( | η | ) ≤ C ( a − | η | ) − / ( p − on B a \ B b . (2.23)In particular w k ( b ) ≤ C ∗ = C ∗ ( a, b, p, N )Next w k ( r ) ≤ max { C ∗ , ( γ + b − β ) / ( p − } ∀ r ∈ [0 , b ] . (2.24)Combining (2.23 ) and (2.24 ) implies that w k is locally uniformly bounded on [0 , a ). Since k w k is increasing, the existence of F a := w ∞ = lim k →∞ w k follows. The fact that a F a decreases isa consequence of the fact that F a ′ is finite on ∂B a for any a < a ′ . (cid:3) Remark.
In the sequel we set F ∞ = lim a →∞ F a . Then F ∞ is a nondecreasing, nonnegative solutionof (2.6 ). Using asymptotic analysis, is is easy to prove that there holds:(i) if β = 0 F ∞ ( η ) = (cid:18) p − (cid:19) / ( p − | η | − β/ ( p − (1 + ◦ (1)) as | η | → ∞ ; (2.25)(ii) if β = 0, F ∞ ( η ) ≡ (cid:18) p − (cid:19) / ( p − . (2.26)Furthermore, if β > −
2, it follows by the strict maximum principle that F a (0) = min { F a ( η ) : | η | < a } >
0. This observation plays a fundamental role for obtaining estimate from above.
Proposition 2.2
Assume p > and β > − . Then any solution u of (2.1 ) in Q T which verifies lim t → u ( x, t ) = 0 ∀ x = 0 , (2.27) satisfies | u ( x, t ) | ≤ min n c ∗ | x | − (2+ β ) / ( p − ; t − (2+ β ) / p − F ∞ ( x/ √ t ) o ∀ ( x, t ) ∈ Q T \ { } , (2.28) where c ∗ = c ∗ ( N, p, β ) . Proof.
Let ǫ > a > P a,ǫ = { ( x, t ) : t > ǫ, | x | / √ t − ǫ < a } . By the previous remarkmin F a >
0, thus the function W ( x, t ) = ( t − ǫ ) − (2+ β ) / p − F a ( | x | / √ t − ǫ ), which is a solution of(2.1 ) in P a,ǫ tends to infinity on the boundary on P a,ǫ ; since u is finite in Q T ∩ P a,ǫ , W dominates u in this domain. Letting successively ǫ → a → ∞ yields to u ≤ F ∞ . The estimate frombelow is similar. Next we consider x ∈ R N \ { } , then v = | u | satisfies (by Kato’s inequality) ∂ t v − ∆ v + C ( x ) v p ≤ B | x | / ( x ) × (0 , T ) , where C ( x ) = max { ( | x | / β ; (3 | x | / β } . It is easy to construct a function under the form w ( y ) =Λ (cid:0) | x | − | x − y | (cid:1) − / ( p − which satisfies (cid:26) − ∆ w + C ( x ) w p = 0 in B | x | / ( x )lim | x − y |→| x | / w = ∞ , with Λ = Λ( x ) = c ∗ | x | (2 − β ) / ( p − , c ∗ = c ∗ ( N, p, β ) >
0. Using (2.27 ), it follows from Lebesgue’stheorem that u ( y, t ) ≤ w ( y ) in B | x | / ( x ) × [0 , T ), thus u ( x, t ) ≤ w ( x ) = c ∗ | x | − (2+ β ) / ( p − . Estimatefrom below is similar. (cid:3) The construction of the first part of the proof of Proposition 2.2 (estimate in P a,ǫ ) shows that,without condition (2.27 ), equation (2.1 ) admits a maximal solution u M . Proposition 2.3
Assume p > and β > − . Then any solution u to (2.1 ) satisfies | u ( x, t ) | ≤ u M ( x, t ) := t − (2+ β ) / p − F ∞ ( x/ √ t ) ∀ ( x, t ) ∈ Q T \ { } . (2.29)As a variant of (2.28 ), we have the following Keller-Osserman type parabolic estimate whichextends the classical one due to Brezis and Friedman in the case β = 0 (see [3]). Proposition 2.4
Under the assumptions of Proposition 2.2 there holds | u ( x, t ) | ≤ ˜ c ( | x | + t ) (2+ β ) / p − ∀ ( x, t ) ∈ Q T \ { } , (2.30) with ˜ c = ˜ c ( N, p, β ) .Proof. Assume | x | ≤ t , then1( | x | + t ) (2+ β ) / p − ≥ − (2+ β ) / p − t − (2+ β ) / p − ≥ − (2+ β ) / p − min { F ∞ ( η ) : | η | ≤ } t − (2+ β ) / p − F ∞ ( x/ √ t ) . (2.31)Assume | x | ≥ t , then 1( | x | + t ) (2+ β ) / p − ≥ − (2+ β ) / ( p − | x | − (2+ β ) / ( p − . (2.32)Combining (2.31 ) and (2.32 ) gives (2.30 ). (cid:3) Theorem 2.5
Assume p > and − < β ≤ N ( p − − . Then any solution u to (2.1 ) whichsatisfies (2.27 ) is identically .Proof. If − (2 + β ) / ( p −
1) + N − > −
1, equivalently β < N ( p − −
2, the function x x | − (2+ β ) / ( p − is locally integrable in R N , thus u ( ., t ) → L loc ( R N ) as t →
0. For ǫ > R = R ( ǫ ) such that u ( x, t ) ≤ ǫ for any | x | ≥ R and t >
0. Thus u ( x, t + τ ) ≤ ǫ + E [ uχ BR u ( ., τ )]( x, t ) ∀ t > , τ > x ∈ R N , (2.33)where E [ φ ] denotes the heat potential of the measure φ (see (2.2 )). Letting successively τ → ǫ →
0, yields to u ≤
0. In the same way u ≥
0. In the case β = N ( p − − | u ( x, t ) | ≤ ˜ c ( | x | + t ) N/ . From this estimate, the proof of [3, Th 2, Steps 5, 6] applies and we recall briefly the steps(i) By choosing positive test functions φ n which vanish in V n = { ( x, t ) : | x | + t ≤ n − } and areconstant on V ′ n = { ( x, t ) : | x | + t ≥ n − } , we first prove that, for any ρ > Z Z B ρ × (0 ,T ) (cid:0) | u ( x, t ) | + | x | β | u | p (cid:1) dxdt < ∞ . (2.34)Thus, using the same test function, we derive that the identity Z Z Q T (cid:16) − u∂ t ζ − u ∆ ζ + | x | β | u | p − uζ (cid:17) dx dt = 0 , (2.35)holds for any ζ ∈ C , ( R N × [0 , T )). The uniqueness yields to u = 0. (cid:3) Proof of Theorem A- case I.
In the case − < β ≤ N ( p − −
2, the result is a consequence ofTheorem 2.5. Next we assume β ≤ −
2. If f is a solution of (1.6 ), it satisfies f ( η ) = ◦ ( | η | − (2+ β ) / ( p − ) as | η | → ∞ . If β = −
2, the equation becomes − ∆ f − η. ∇ f + | η | − | f | p − f = 0 , and f ( η ) → f ≤
0. Similarly f ≥ β < −
2, for ǫ > η ǫ | η | − (2+ β ) / ( p − = ψ ( η ) belongs to W , loc ( R N ) since β < − − ∆ ψ − η. ∇ ψ − β p − ψ + | η | β | ψ | p − ψ = ǫr − (2+ β ) / ( p − − (cid:18)(cid:18) βp − (cid:19) (cid:18) βp − − N (cid:19) + ǫ p − (cid:19) . Therefore, either if N ≥ N = 1 and β ≤ − ( p + 1), ψ is a super-solution of (1.6 ) for any ǫ > N = 1 and − ( p + 1) < β < − f ′′ + r f ′ + 2 + β p − f − r β | f | p − f = 0 on R + , f ( r ) = A β,p r − (2+ β ) / ( p − . Furthermore, if f ≥ φ ′′ + r φ ′ + 2 + β p − φ = 0Noticing that this equation has a solution φ which has the same behaviour at infinity than theexplicit solution of (1.4 ), namely φ ( r ) = cr − (2+ β ) / ( p − (1 + ◦ (1)) , by standard methods (see e.g. [10, Prop A1]), the second solution φ behaves in the following way φ ( r ) = cr (2+ β ) / ( p − − e − r / (1 + ◦ (1)) as r → ∞ . Consequently, by the maximum principle, any solution f of (1.4 ) on R such that f ( r ) = ◦ ( φ ( r ))at infinity, verifies | f ( r ) | ≤ C | r | (2+ β ) / ( p − − e − r / for | r | ≥ . (2.36)Using the equation, we obtain that f ′ ( r ) = e r / Z ∞ r (cid:18) s β | f ( s ) | p − f ( s ) − βp − f ( s ) (cid:19) ds, thus | f ′ ( r ) | ≤ Cr (2+ β ) / ( p − − e − r / for | r | ≥ . (2.37)Since f ∈ H loc ( R ), we derive that for any n ∈ N ∗ , Z n − n (cid:18) f ′ − βp − f (cid:19) e r / dr ≤ e n / ( f ( n ) f ′ ( n ) − f ( − n ) f ′ ( − n )) . Because of (2.36 ) and (2.36 ), this last term tends to 0 as n → ∞ . Therefore Z ∞−∞ (cid:18) f ′ − βp − f (cid:19) e r / dr = 0 = ⇒ f = 0 , which end the proof. (cid:3) Remark.
The method of proof used in the case N = 1 and − p − < β < − β ≤ −
2. But it relies strongly on the fact that f ∈ H loc , while the othermethods use only f ∈ W , loc ( R N ) . Proposition 2.6
Assume β > max { N ( p − − − N } . Then for any k > there exists a uniquesolution u k of (2.1 ) with initial data kδ . Furthermore k u k is increasing and u ∞ := lim k →∞ u k satisfies u ∞ ( x, t ) = t − (2+ β ) / p − f ∞ ( x/ √ t ) , where f ∞ is positive, radially symmetric and satisfies ( − ∆ f ∞ − η. ∇ f ∞ − γf ∞ + | η | β f p ∞ = 0 in R N lim | η |→∞ | η | (2+ β ) / ( p − f ∞ ( η ) = 0 . (2.38) Proof.
The existence of u k and the monotonicity of k u k has already been seen. By the uniformcontinuity of the u k in any compact subset of ¯ Q T \ { (0 , } , the function u ∞ satisfieslim t → u ∞ ( x, t ) = 0 ∀ x = 0 . (2.39)3For ℓ > u is defined in Q ∞ , we set T ℓ [ u ]( x, t ) := ℓ (2+ β ) / p − u ( √ ℓx, ℓt ) . (2.40)If u satisfies equation (2.1 ) in Q ∞ , T ℓ [ u ] satisfies it too. Because of uniqueness T ℓ [ u k ] = u ℓ (2+ β ) / p − − N/ k . (2.41)Using the continuity of u T ℓ [ u ] and the definition of u ∞ , we can let k → ∞ in (2.41 ) and derive(by taking ℓt = 1 and replacing t by ℓ ), T ℓ [ u ∞ ] = u ∞ = ⇒ u ∞ ( x, t ) = t − (2+ β ) / p − u ∞ ( x/ √ t, . (2.42)Setting f ∞ ( η ) = u ∞ ( x/ √ t,
1) with η = x/ √ t , it is straightforward that f ∞ satisfies (2.38 ) (usingin particular 2.39 ). Furthermore f ∞ is radial and positive as the u k are. (cid:3) Lemma 2.7
The function f ∞ satisfies f ∞ ( η ) = c | η | γ − N e −| η | / (cid:0) ◦ ( | η | − ) (cid:1) as | η | → ∞ , (2.43) for some c = c N,p,β > . Furthermore f ′∞ ( η ) = − c c | η | γ +1 − N e −| η | / (cid:0) ◦ ( | η | − ) (cid:1) as | η | → ∞ . (2.44) Proof.
Set r = | η | and denote f ∞ ( η ) = f ∞ ( r ). Then f ∞ satisfies, f ′′∞ + (cid:18) N − r + r (cid:19) f ′∞ + γf ∞ − r β | f ∞ | p − f ∞ = 0 on (0 , ∞ ) , (2.45)and lim r →∞ r γ f ∞ ( r ) = 0. We consider the auxiliary equation f ′′ + (cid:18) N − r + r (cid:19) f ′ + γf = 0 on (0 , ∞ ) . (2.46)By [10, Prop A1], (2.46 ) admits two linearly independent solutions defined on (0 , ∞ ), y and y such that y ( r ) = r − γ (1 + ◦ (1)) and y ( r ) = r γ − N e − r / (1 + ◦ (1)) , (2.47)as r → ∞ . Next we choose R > R, ∞ ) and the y j are positive on the same interval. For δ > Y δ = δy + f ∞ ( R ) y /y ( R )is a supersolution for (2.45 ). Furthermore f ∞ ( r ) = ◦ ( Y δ ) at infinity. Letting δ → f ∞ ( r ) ≤ f ∞ ( R ) y ( R ) y ( r ) ∀ r ≥ R. (2.48)Using (2.47 ) we derive 0 ≤ f ∞ ( η ) ≤ C | η | γ − N e −| η | / ∀ | η | ≥ . Plugging this estimate into (2.45 ), we derive (2.43 ) from standard perturbation theory for secondorder linear differential equation [2, p. 132-133]. Finally, (2.44 ) follows directly from (2.43 ) and(2.45 ). (cid:3)
An alternative proof of the existence of f ∞ is linked to calculus of variations. In the case β = 0,this was performed by Escobedo and Kavian [8]. This construction is based upon the study of thefollowing functional J ( v ) = 12 Z R N (cid:18) |∇ v | − γv + 2 p + 1 | η | β | v | p +1 (cid:19) K ( η ) dη, (2.49)defined over the functions in H K ( R N ) ∩ L p +1 | η | β K ( R N ).4 Proposition 2.8
Assume p > and β > N ( p − − . Then there exists a positive function ˜ f ∞ ∈ H K ( R N ) ∩ L p +1 | η | β K ( R N ) satisfying − ∆ ˜ f ∞ − η. ∇ ˜ f ∞ − γ ˜ f ∞ + | η | β ˜ f p ∞ = 0 in R N . (2.50)We recall that the eigenvalues of − K − div ( K ∇ . ) are the λ k = ( N + k ) /
2, with k ∈ N andthe eigenspaces H k are generated by D α φ where φ ( η ) = K − ( η ) = e −| η | / and | α | = k . It isstraightforward to check that J is C . In order to apply Ekeland Lemma, we have just to provethat J is bounded from below in H K ( R N ). As we shall see it later on, the proof is easy when β < N ( p − /
2, and more difficult when β ≥ N ( p − / Lemma 2.9
For any v ∈ H K ( R N ) , there holds Z R N (cid:0) N + | η | (cid:1) v K ( η ) dη ≤ Z R N |∇ v | K ( η ) dη. Proof.
We borrow the proof to Escobedo and Kavian. Put w = v √ K . Then √ K ∇ v = ∇ w − w η. Hence Z R N |∇ v | K ( η ) dη = Z R N (cid:18) |∇ w | − w ∇ w.η + 14 w | η | (cid:19) dη. Because − Z R N w ∇ w.ηdη = N Z R N w dη, there holds Z R N |∇ v | K ( η ) dη = Z R N (cid:18) |∇ w | + N w + 14 w | η | (cid:19) dη. This implies the formula. (cid:3)
Lemma 2.10
Let p > and β < N ( p − / . For any ǫ > there exists C = C ( ǫ, p ) > and R = R ( ǫ, p ) > such that Z R N v K ( η ) dη ≤ ǫ Z R N |∇ v | K ( η ) dη + C (cid:18)Z R N | v | p +1 | η | β K ( η ) (cid:19) /p +1 . Proof.
For
R > Z | η |≤ R v K ( η ) dη ≤ Z | η |≤ R | v | p +1 | η | β K ( η ) dη ! / ( p +1) Z | η |≤ R | η | − β/ ( p − K ( η ) dη ! ( p − / ( p +1) . Since β < N ( p − / ⇐⇒ N > β/ ( p − Z | η |≤ R | η | − β/ ( p − K ( η ) dη ! ( p − / ( p +1) = C ( R, N, p ) . By Lemma 2.9 Z | η |≥ R v K ( η ) dη ≤ R Z R N |∇ v | K ( η ) dη. ǫ = 4 R − . (cid:3) It follows from the previous Lemmas that J is bounded from below in the space H K ( R N ) ∩ L p +1 | η | β K ( R N ) whenever N ( p − / − < β < N ( p − /
2. Next we consider the case β >
Lemma 2.11
Assume β > . The functional J is bounded from below on the set X = n v ∈ H K ( R N ) ∩ L p +1 | η | β K ( R N ) : v ≥ , v radial and decreasing o . Proof.
For 0 < δ < R , we write J ( v ) = J δ,R ( v ) + J ′ δ,R ( v ) + J ′′ δ,R ( v ) where J δ,R ( v ) = 12 Z | η |≤ δ (cid:18) |∇ v | − γv + 2 p + 1 | η | β | v | p +1 (cid:19) K ( η ) dη,J ′ δ,R ( v ) = 12 Z δ< | η |
Let v be a radially symmetric function in H K ( R N ) ∩ L p +1 | η | β K ( R N ) . Then there existsa radially symmetric decreasing function ˜ v ∈ H K ( R N ) ∩ L p +1 | η | β K ( R N ) such that J (˜ v ) ≤ J ( v ) .Proof. We define the two curves C = (cid:8) ( s, x ) ∈ R + × R + : − − γx + ( p + 1) − s β x p +1 = 0 (cid:9) = n x = (cid:0) − ( p + 1) γs − β (cid:1) / ( p − o , C = (cid:8) ( s, x ) ∈ R + × R + : − γx + s β x p = 0 (cid:9) = n x = (cid:0) γs − β (cid:1) / ( p − o . For fixed s > x
7→ − − γx +( p +1) − s β x p +1 vanishes at x = 0. It has the followingproperties:(i) it is decreasing for 0 < x < (cid:0) γs − β (cid:1) / ( p − ,(ii) it achieves a minimum at x s = (cid:0) γs − β (cid:1) / ( p − ,(iii) and it is increasing for x > (cid:0) γs − β (cid:1) / ( p − with infinite limit. Furthermore it vanishes at˜ x s = (cid:0) − ( p + 1) γs − β (cid:1) / ( p − .Let v be a radially symmetric positive function. By approximation of radial elements in H ,K ( R N ) ∩ L p +1 | η | β K ( R N ), we can assume that v is C with nondegenerate isolated extrema.We can also assume that the graph of v has at most a countable of intersections with C , a < a < a ... < a k < ... , that the set of points { a k } is discrete, that all the intersectionsare transverse and that, for every j ≥ v ( s ) < (cid:0) γs − β (cid:1) / ( p − on ( a j , a j +1 ) , where a = 0, and v ( s ) > (cid:0) γs − β (cid:1) / ( p − on ( a j +1 , a j +2+1 ) . The modifications of the function v is performed by local modification on each interval ( a k , a k +1 ): Step 1 - The construction of ˜ v on ( a j , a j +1 ) is as follows. Let α < α < ... be the sequence oflocal extrema of v , with v ( α i +1 ) local minimum and v ( α i +2 ) local maximum. By extension, since v ′ ( a j +1 ) > − β/ ( p − γ / ( p − a − ( β + p − / ( p − j +1 , v ( a j +1 ) is a local maximum of v on ( a j , a j +1 ).If max { ( α i +1 ) : i ≥ } ≤ v ( a j +1 ), then ˜ v = max { v, v ( a j +1 ) } .If max { v ( α i +1 ) : i ≥ } > v ( a j +1 ), we define the increasing sequence { α i d +1 } by v ( α i +1 ) = max { v ( α i +1 ) : i ≥ } ,v ( α i +1 ) max { v ( α i +1 ) : i > i } , and by induction, v ( α i d +1 ) max { v ( α i +1 ) : i > i d − } . Thus we can assume that the local maxima of v are less than v ( a j +1 ) on the last interval( α i d +1 , a j +1 ). Next we define the function ˜ v by ˜ v = max { v, v ( α i +1 } on ( a j , α i +1 ), ˜ v =max { v, v ( α i +1 } on ( α i +1 , α i +1 ). By induction, ˜ v = max { v, v ( α i d − +1 } on ( α i d − +1 , α i d +1 ).Finally ˜ v = max { v, v ( a j +1 ) } on the last interval ( α i d +1 , a j +1 ). The function ˜ v is Lipschitzcontinuous, nonincreasing and, because v ( s ) ≤ ˜ v ( s ) ≤ (cid:0) γs − β (cid:1) / ( p − , there holds Z a j ≤| η |≤ a j +1 (cid:18) |∇ ˜ v | − γ ˜ v + 2 p + 1 | η | β | ˜ v | p +1 (cid:19) K ( η ) dη ≤ Z a j ≤| η |≤ a j +1 (cid:18) |∇ v | − γv + 2 p + 1 | η | β | v | p +1 (cid:19) K ( η ) dη. (2.51) Step 2 - The construction of ˜ v on ( a j +1 , a j +2 ) follows the same principle. Let β < β < ... < β d be the sequence of local minima of v on this interval. Furthermore v ( a j +1 ) is the minimum of v on ( a j +1 , a j +2 ) and v ′ ( a j +2 ) < − β/ ( p − γ / ( p − a − ( β + p − / ( p − j +2 .7On ( a j +1 , β ) we set ˜ v = min { v, v ( a j +1 ) } . On ( β , β ), ˜ v = min { v, ˜ v ( β ) } . By induction ˜ v =min { v, ˜ v ( β i ) } on ( β i , β i +1 ). On the last interval ( β d , b j +2 ), ˜ v = min { v, ˜ v ( β d ) } . Because ˜ v ≤ v on this interval and x
7→ − − γx + ( p + 1) − s β x p +1 is increasing above the curve C , we obtainsimilarly Z a j +1 ≤| η |≤ a j +2 (cid:18) |∇ ˜ v | − γ ˜ v + 2 p + 1 | η | β | ˜ v | p +1 (cid:19) K ( η ) dη ≤ Z a j +1 ≤| η |≤ a j +2 (cid:18) |∇ v | − γv + 2 p + 1 | η | β | v | p +1 (cid:19) K ( η ) dη. (2.52)By construction ˜ v is nonincreasing. Combining (2.51 ) and (2.52 ), we obtain J (˜ v ) ≤ J (˜ v ). (cid:3) Proof of Proposition 2.8.
It follows from the previous lemmas that J is bounded from below on X and the function φ = K − belongs to X . Furthermore J ( tφ ) = ( N − γ ) t Z K − ( η ) dη + | t | p +1 p + 1 Z φ p ( η ) dη. Since β > N ( p − − ⇐⇒ N − γ <
0, the infimum m of J over radially symmetric functionsis negative but finite and achieved by a decreasing function. Let { v n } ⊂ X a sequence such that J ( v n ) ↓ m . Then { v n } remains bounded in H K ( R N ) ∩ L p +1 | η | β K ( R N ). Up to a subsequence we canassume that v n converges weakly in H K ( R N ) and in L p +1 | η | β K ( R N ) and strongly in L K ( R N ) to somefunction v . Moreover this convergence holds a.e., and, since v n ∈ X the same holds with v . Goingto the limit in the functional yields to J ( v ) ≤ lim inf n →∞ J ( v n ) = m ;thus v is a critical point. (cid:3) The following uniqueness result holds.
Proposition 2.13
Assume p > and β > N ( p − − . Then f ∞ = ˜ f ∞ . Furthermore f ∞ is theunique positive solution of (2.38 ).Proof. We first prove that ˜ f ∞ is the unique positive radial solution of (2.50 ) belonging to H K ( R N ) ∩ L p +1 | η | β K ( R N ). We denote r = | η | and ˜ f ∞ ( η ) = ˜ f ∞ ( r ). Let ˆ f be another solution inthe same class. Thus there exists { r n } converging to ∞ such that ˆ f ( r n ) →
0. For ǫ >
0, set˜ f ǫ = ˜ f ∞ + ǫ . For n ≥ n , large enough, w + ( r n ) = 0, thus, as in the proof of Proposition 2.1, Z Z B rn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ˆ f − ˆ f ˜ f ǫ ∇ ˜ f ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ˜ f ǫ − ˜ f ǫ ˆ f ∇ ˆ f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Kdη + γ Z Z B rn ǫ ˜ f ǫ ( ˆ f − ˜ f ǫ ) + Kdη + Z Z B rn | η | β ( ˆ f p − − ˜ f p − ǫ )( ˆ f − ˜ f ǫ ) + Kdη ≤ . We let successively r n → ∞ with Fatou’s lemma, and ǫ → ǫ/ ˜ f ǫ ≤ f − ˜ f ǫ ) + ≤ ˆ f + ˜ f ∞ ∈ L K ( R N ). We get Z Z R N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ˆ f − ˆ f ˜ f ∞ ∇ ˜ f ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ ˜ f ∞ − ˜ f ∞ ˆ f ∇ ˆ f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | η | β ( ˆ f p − − ˜ f p − ∞ )( ˆ f − ˜ f ∞ ) + Kdη ≤ , f ≤ ˜ f ∞ . In the same way ˜ f ∞ ≤ ˆ f . By Lemma 2.7, f ∞ ∈ H K ( R N ) ∩ L p +1 | η | β K ( R N ).Thus f ∞ = ˜ f ∞ . (cid:3) We end this section with a classification result
Theorem 2.14
Assume p > and β > N ( p − − and let u be a positive solution of (2.1 )which satisfies (2.27 ). Then,(i) either there exists k ≥ such that u = u k ,(i) or u = u ∞ .Proof. Because of (2.27 ), the initial trace tr ( u ) of u is is a outer regular Borel measure concentratedat 0 (see [12]). Then either the initial trace is a Radon measure, say kδ , and we get (i), orlim t → Z B ǫ u ( x, t ) dx = ∞ , (2.53)for every ǫ >
0. This implies u ≥ u ∞ as in [11]. Notice that, in this article, this estimate isperformed in the case β = 0, but the proof in the general case is the same. In order to prove that u ≤ u ∞ , we consider, for ǫ >
0, the minimal solution v := v ǫ of (cid:26) ∂ t v − ∆ v + | x | β | v | p − v = 0 in Q T tr ( v ) = ν ¯ B ǫ , (2.54)where ν ¯ B ǫ is the outer regular Borel measure such that ν ¯ B ǫ ( E ) = 0 for any Borel set E ⊂ R N suchthat E ∩ ¯ B ǫ = ∅ , and ν ¯ B ǫ ( E ) = ∞ otherwhile. This solution is constructed as the limit, when m → ∞ of the solution v ǫ,m of (2.1 ) verifying v ǫ,m ( .,
0) = mχ ¯ Bǫ . Clearly u ≤ v ǫ . Furthermore,for any ℓ > T ℓ [ v ǫ,m ] = v ǫ/ √ ℓ,mℓ (2+ β ) / p − = ⇒ T ℓ [ v ǫ ] = v ǫ/ √ ℓ = ⇒ T ℓ [ v ] = v , (2.55)where v = lim ǫ → v ǫ . This, and the fact that lim t → v ( x, t ) = 0 for every x ∈ R N \ { } , implythat v ( x, t ) = t − (2+ β ) / p − f ∞ ( x/ √ t ) = u ∞ ( x, t ). At the end, since u ≤ v ǫ = ⇒ u ≤ v , it follows u ≤ u ∞ . (cid:3) Our study of the singularity set of the solution u ∞ in the case of strongly degenerate potential(1.9) is based on some variant of the local energy estimate (abr. L.E.E.) method. First the L.E.E.method for the study of singular solutions of quasilinear parabolic equations was used in [15].Adaption of this method to the study of conditions of removability of the point singularities ofsolutions of the quasilinear parabolic equations of diffusion-strong absorption type was given in [9].In [14] there was elaborated a variant of the L.E.E. method, which allowed to find sharp conditionson the time dependent absorption potential, guaranteing existence of very singular solutions of theCauchy problem to diffusion-strong absorption type equation with point singularity set. Here weprovide a new application of the L.E.E. method in describing the transformation of V.S.S solutioninto the R.B. solution in terms of the flatness of the absorption potential in the space variables.We consider the sequence of the Cauchy problems u t − ∆ u + h ( | x | ) | u | p − u = 0 in R N × (0 , T ) , p > , (3.1) u ( x,
0) = u ,k ( x ) = M k exp( − − µ N k ) δ k ( x ) , (3.2)9where δ k is a regularized Dirac measure: δ k ∈ C ( R N ) , δ k ⇀ δ weakly in the sense of measures as k → ∞ , supp δ k ⊂ { x : | x | ≤ exp( − µ k ) } ∀ k ∈ N , (3.3)where the constant µ > M k = exp exp k ∀ k ∈ N . (3.4)Without loss of generality we suppose that k δ k k L ( R N ) ≤ exp( µ N k ) . (3.5)We write the potential h in the equation (3.1) under the form, h ( s ) = exp( − ω ( s ) s − ) ∀ s ≥ , (3.6)where ω ( s ) ≥ , ∞ ). Theorem 3.1
Let the function ω ( s ) defined in (3.6) satisfy additionally the following Dini-likecondition Z d ω ( s ) s − ds ≤ d < ∞ , d = const > , (3.7) and the following technical condition sω ′ ( s ) ω ( s ) ≤ − α ∀ s ∈ (0 , s ) , s > , < α = const < Then the following a priori estimate of solutions u k of the problem (3.1) , (3.2) , (3.5) , holds uni-formly with respect to k ∈ N , Z R N | u k ( x, t ) | dx ≤ C t exp " C (cid:18) Φ − (cid:18) tC (cid:19)(cid:19) − , (3.9) where the constants C > , C > , C > do not depend on k . Here Φ − ( s ) is the inversefunction to s Φ( s ) := Z s ω ( r ) r dτ. Let us define the following families of domains B ( s ) := { x : | x | < s } , Ω( s ) := R N \ B ( s ) ,Q t t ( s ) := Ω( s ) × ( t , t ) , ∀ s > , ∀ ≤ t < t ≤ T. Let u ( x, t ) ≡ u k ( x, t ) be a solution of the problem (3.1), (3.2) under consideration. We introducethe energy functions I ( s, τ ) := Z τ Z Ω( s ) (cid:0) |∇ x u | + h ( | x | ) | u | p +1 (cid:1) dx dt, (3.10) J ( s, t ) = Z Ω( s ) | u ( x, t ) | dx, E ( s, t ) = Z B ( s ) | u ( x, t ) | dx. (3.11)0 Lemma 3.2
The energy functions J ( s, t ) , I ( s, t ) defined by (3.10) , (3.11) corresponding to anarbitrary solution u = u k of problem (3.1) , (3.2) satisfy the following a priori estimate J ( s, t ) + I ( s, t ) ≤ ctg ( s ) := ct (cid:18)Z s r − ( N − p − p +3 h ( r ) p +3 dr (cid:19) − p +3 p − , ∀ s ≥ exp( − µ k ) . (3.12) uniformly with respect to k ∈ N . By c, c i we denote different positive constants, which depend on known parameters N, p, α , d only, but their value may change from lines to lines. Proof.
Multiplying equation (3.1) by u and integrating in Q t t ( s ), we obtain the following startingrelation after standard computations,2 − Z Ω( s ) | u ( x, t ) | dx + Z Z Q t t ( s ) (cid:0) |∇ x u | + h ( | x | ) | u | p +1 (cid:1) dx dt == 2 − Z Ω( s ) | u ( x, t ) | dx + Z t t Z | x | = s u ∂u∂n dσ dt := R + R . (3.13)Let us estimate R from above. Using Holder’s and Young’s inequalities we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z | x | = s u ( x, t ) ∂u∂n dσ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cs ( N − p − p +1) Z | x | = s |∇ x u | dσ ! / Z | x | = τ | u | p +1 dσ ! p +1 ≤≤ cs ( N − p − p +1) h ( s ) − p − Z | x | = s (cid:0) |∇ x u | + h ( s ) | u | p +1 (cid:1) dσ ! p +32( p +1) . Integrating in t , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z τ Z | x | = s u ∂u∂n dσ dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cs ( N − p − p +1) h ( s ) − p − τ p − p +1) × Z τ Z | x | = s (cid:0) |∇ x u | + h ( s ) | u | p +1 (cid:1) dσ dt ! p +32( p +1) . (3.14)It is easy to see that − dds I ( s, τ ) = Z τ Z | x | = s (cid:0) |∇ x u | + h ( s ) | u | p +1 (cid:1) ds, − dds J ( s, t ) ≥ . Therefore because of the property (3.3) satisfied by u ,k , and estimate (3.14), we derive the followinginequality from relation (3.13) with t = t, t = 0 , s ≥ exp( − µ k ), J ( s, t ) + I ( s, t ) ≤ c t p − p +1) h ( s ) − p +1 s ( N − p − p +1) (cid:18) − dds ( I ( s, t ) + J ( s, t )) (cid:19) p +32( p +1) . (3.15)Solving this ordinary differential inequality (abr. O.D.I.) with respect to the function I ( s, t ) + J ( s, t ), we deduce that estimate (3.12) holds for arbitrary s ≥ exp( − µ k ). (cid:3) Next, we define s k > g ( s k ) = M ε k = exp( ε exp k ) , (3.16)1where 0 < ε < s k ≥ exp( − µ k ) := s k ∀ k > k ( ε , α , ν , p ) . (3.17)Using [1, Lemma A1], it follows from the definitions (3.6)of function h ( . ) and (3.12) of function g ( . ), that the next estimate holds, (cid:18) α p + 3 (cid:19) p +3 p − g ( s ) ≤ g ( s ) ≤ (cid:18) p + 3 (cid:19) p +3 p − g ( s ) , (3.18)where g ( s ) = s N − − p +3) p − ω ( s ) p +3 p − exp (cid:16) p − ω ( s ) s (cid:17) , α is constant from condition (3.8). Thefollowing simpler estimate follows from (3.18):exp (cid:18) ω ( s ) s p −
1) (1 − ν ) (cid:19) ≤ g ( s ) ≤ exp (cid:18) ω ( s ) s p −
1) (1 + ν ) (cid:19) , (3.19)for any s ∈ (0 , s ), where s = s ( ν ) → ν →
0. As a consequence of definition (3.16) of s k ,and using (3.19), we get, ω ( s k ) s k − ν )( p − ≤ ε exp k. (3.20)Integrating (3.8), we deduce that ω satisfies ω ( s ) ≥ s − α ∀ s > . (3.21)Combining (3.21) and (3.20) we derive: s k ≥ (cid:18) − ν ) ε ( p − (cid:19) α exp (cid:18) − kα (cid:19) . (3.22)Next we define µ from (3.2) and set µ = 2 α − . It follows from (3.22) that (3.17) is satisfied forall k > k = k ( ε , α , ν , p ). As result we derive that estimate (3.12) obtained in Lemma 3.2 isvalid for s = s k , i.e. J ( s k , t ) + I ( s k , t ) ≤ ctg ( s k ) ∀ k ≥ k = k ( ε , α , ν , p ) . (3.23)In order to find estimates characterizing the behaviour of the energy function E ( s k , t ) withrespect to the variable t >
0, we introduce the nonnegative cut-off function ϕ k ∈ C ( R ) defined by ϕ k ( s ) = 1 if s < s k , ϕ k ( s ) = 0 if s ≥ s k , ϕ ′ k ( s ) ≤ cs − k . (3.24)Multiplying (3.1) by u k ϕ k ( | x | ) and integrating with respect to x , we get2 − ddt Z R N u ( x, t ) ϕ k ( | x | ) dx + Z R N |∇ x ( uϕ k ) | dx + Z R N h ( | x | ) ϕ k | u | p +1 dx ≤ Z R N u ( x, t ) |∇ x ϕ k ( | x | ) | dx := R . (3.25)By (3.24) and (3.23), we obtain R ≤ c s − k Z s k < | x | < s k | u ( x, t ) | dx ≤ c s − k J ( s k , t ) ≤ c s − k tg ( s k ) . (3.26)2Using (3.25), (3.26) and Poincar´e’s inequality we derive the following differential inequality, ddt (cid:18)Z R N u ( x, t ) ϕ k dx (cid:19) + d s − k Z B (2 s k ) u ( x, t ) ϕ k dx ≤ cs − k tg ( s k ) , d > . (3.27)We set ψ k ( t ) := Z R N | u k ( x, t ) | ϕ k ( | x | ) dx, and obtain the following O.D.I. from (3.27), ψ ′ k ( t ) + d s − k ψ k ( t ) ≤ cs − k tg ( s k ) . (3.28)We rewrite (3.28) under the form ψ ′ k ( t ) + d s − k ψ k ( t ) + 2 − (cid:0) d s − k ψ k ( t ) − cs − k tg ( s k ) (cid:1) ≤ . (3.29)Using the relations (3.2), (3.5) satisfied by u k, , we see that ψ k verifies, ψ k (0) ≤ Z R N | u k, ( x ) | dx ≤ M k . (3.30)At last, we define the t k by t k = γω ( s k ) (3.31)where ω is the function in (3.6) and γ > Lemma 3.3
There exists a constant γ > , which does not depend on k , such that any solution ψ k of problem (3.29) , (3.30) satisfies the following a priori estimate ψ k ( t k ) ≤ d − c t k g ( s k ) ∀ k > k ( ε , ν ) , (3.32) for some t k ≤ t k , where t k is defined by (3.31) .Proof. Let us assume that (3.32) is not true, and for any γ > k ≥ k such that ψ k ( t ) > d − ctg ( s k ) ∀ t : 0 < t < γω ( s k ) ≡ t k . (3.33)This relation combined with (3.29) implies the following inequality, ψ ′ k ( t ) + d s − k ψ k ( t ) ≤ ∀ t : 0 < t ≤ γω ( s k ) . Solving this O.D.I. and using (3.30), we get ψ k ( t ) ≤ ψ k (0) exp (cid:18) − d t s k (cid:19) ≤ M k exp (cid:18) − d t s k (cid:19) ∀ t ≤ γω ( s k ) . (3.34)We derive easily the next estimate from (3.34) and (3.33) M k exp (cid:18) − d γω ( s k )2 s k (cid:19) ≥ d − cg ( s k ) γω ( s k ) . (3.35)3Using (3.16) and (3.4), we deduce from this last inequality,(1 − ε ) exp k ≥ d γω ( s k )2 s k + ln(2 d − cγ ) − ln( ω ( s k ) − ) . (3.36)Similarly to (3.20), it follows, from (3.19) and the definition (3.16) of s k , that there holds ω ( s k ) s k ν )( p − ≥ ε exp k. (3.37)Using this estimate and (3.36), we derive(1 − ε ) exp k ≥ d γ ( p − ε ν ) exp k + ln( d − cγ ) − ln( ω ( s k )) − . (3.38)Noticing that (3.21) implies ln( ω ( s k )) − ≤ (2 − α ) ln( s − k ) , (3.39)and (3.22) can be writen under the formln( s − k ) ≤ α ln (cid:18) ε ( p − − ν ) (cid:19) + kα , (3.40)we deduce the following inequality from (3.39), (3.40) and (3.38),(1 − ε ) exp k ≥ d γ ( p − ε ν ) exp k + ln(2 d − cγ ) − (2 − α ) kα − (2 − α ) α ln (cid:18) ε ( p − − ν ) (cid:19) . (3.41)If we define γ by the equality(1 − ε ) = d γ ( p − ε ν ) ⇔ γ = (1 − ε )(1 + ν )8 d ( p − ε := γ , (3.42)then inequality (3.41) yields to(2 − α ) α k ≥ (1 − ε ) exp k + ln(2 d − cγ ) − (2 − α ) α ln (cid:18) ε ( p − − ν ) (cid:19) . It is clear that we can find k = k ( ε , ν ) < ∞ such that the last inequality becomes impossible for k ≥ k , contradiction. Consequently, (3.33) does not hold for γ = γ and estimate (3.32) is truewith γ = γ . (cid:3) Proof of Theorem 3.1 . Comparing definition (3.11) of E ( s, t ) and definition of ψ k , we easily seethat E ( s k , t ) ≤ ψ k ( t ) ⇒ E ( s k , t k ) ≤ ψ k ( t k ) . (3.43)Therefore, using estimates (3.12), (3.32) and (3.43), we obtain Z R N | u k ( x, t k ) | dx = E ( s k , t k ) + J ( s k , t k ) ≤ ( d − c + c ) t k g ( s k ) . (3.44)Next we estimate the right-hand side of (3.44). Using (3.16), (3.31) and inequality (3.32), we get t k g ( s k ) ≤ γ ω ( s k ) M ε k ≤ γ ω ( s ) exp( ε exp k ) , (3.45)4where γ is defined by (3.42)and s > cd − + c ) t k g ( s k ) ≤ exp (cid:20)(cid:18) ε + ln( γ ω ( s )( c + cd − ))exp k (cid:19) exp k (cid:21) . (3.46)Let k be the smallest integer such thatln (cid:0) γ ω ( s )( c + cd − ) (cid:1) ≤ ε exp k , (3.47)equivalently k = (cid:2) ln (cid:0) ε − ln (cid:0) γ ω ( s )( c + cd − ) (cid:1)(cid:1)(cid:3) + 1 , where [ a ] denote integer part of a . Then it follows from (3.46)( cd − + c ) t k g ( s k ) ≤ exp(2 ε exp k ) ∀ k > k . (3.48)If we fix ε such that 2 ε ≤ e − , (3.49)then the next estimate follows from (3.44) and (3.45)–(3.49) Z R N | u k ( x, t k ) | dx ≤ M k − , (3.50)for all k ≥ max { k , k, k } , where k is from (3.17 ), k – from (3.32), and k from (3.47). Estimate(3.50) is the final step of the first round of computations. For the second round, we begin bydefiniting s k − analogously to s k : g ( s k − ) = M ε k − = exp( ε exp( k − . (3.51)From estimate (3.12) we obtain J ( s k − , t ) + I ( s k − , t ) ≤ ctg ( s k − ) , (3.52)since s k − > s k . Analogously to ϕ k , we define the function ϕ k − and set ψ k − ( t ) := Z R N | u k ( x, t ) | | ϕ k − ( x ) | dx. In the same way as (3.28), the following O.D.I. follows ψ ′ k − ( t ) + d s − k − ψ k − ( t ) ≤ cs − k − tg ( s k − ) ∀ t > t k . (3.53)Using (3.50), we derive ψ k − ( t k ) ≤ M k − , t k ≤ t k . (3.54)If we analyze the Cauchy problem (3.53), (3.54) similarly as problem (3.28), (3.30)) was analyzedin Lemma 3.3, we obtain the following a priori estimate for ψ k − ( t ), ψ k − ( t k + t k − ) ≤ d − c ( t k + t k − ) g ( s k − ) , (3.55)where t k − ≤ t k − := γ ω ( s k − ) , γ is from (3.42). It is clear that E ( s k − , t ) ≤ ψ k − ( t ) ∀ t ≥ t k , E ( s k − , t k + t k − ) ≤ ψ k − ( t k + t k − ) ≤ d − c ( t k + t k − ) g ( s k − ) . (3.56)From (3.52 ), we deduce J ( s k − , t k + t k − ) + I ( s k − , t k + t k − ) ≤ c ( t k + t k − ) g ( s k − ) . (3.57)Summing estimates (3.56) and (3.57) we obtain Z R N | u k ( x, t k + t k − ) | dx ≤ ( cd − + c )( t k + t k − ) g ( s k − ) , (3.58)and we use this last estimate for performing a similar third round of computations. Iterating thisprocess j times, we deduce Z R N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u k x, k − j X i = k t i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ ( cd − + c ) k − j X i = k t i ! g ( s k − j ) . (3.59)In particular, we can take j = k − l , where l ∈ N satisfies l ≥ l := max { k , k, k } . (3.60)Then we obtain: Z R N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u k x, l X i = k t i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dx ≤ ( cd − + c ) l X i = k t i ! g ( s l ) . (3.61)Next, we have to estimate from above the sum of the t i for which there holds l X i = k t i ≤ l X i = k γ ω ( s i ) , (3.62)where s i is defined by g ( s i ) = M ε i . By the same way as in (3.37), we obtain s i ≤ ν ) ω ( s i )( p − ε exp( − i ) ≤ ν ) ω ( s )( p − ε exp( − i ) ∀ i ≥ l , where l is the integer appearing in (3.60 ), and from this inequality follows s i ≤ (cid:18) ν ) ω ( s )( p − ε (cid:19) / exp (cid:18) − i (cid:19) := C exp (cid:18) − i (cid:19) . (3.63)Therefore, using the monotonicity of the function ω , we derive l X i = k ω ( s i ) ≤ l X i = k ω (cid:18) C exp (cid:18) − i (cid:19)(cid:19) ≤ − Z l − k ω (cid:16) C exp (cid:16) − s (cid:17)(cid:17) ds ≤ Z C exp( − l − ) C exp( − k ) y − ω ( y ) dy ≤ Z C exp( − l − )0 y − ω ( y ) dy := 2Φ (cid:18) C exp( − l −
12 ) (cid:19) . (3.64)6As a consequence of (3.62) and (3.64), we get l X i = k t i ≤ l X i = ∞ t i ≤ γ Φ (cid:18) C exp( − l −
12 ) (cid:19) := T l . (3.65)The Dini condition (3.7) implies that T l → l → ∞ . Next, we deduce from (3.61) that Z R N | u k ( x, T l ) | dx ≤ C T l g ( s l ) , C = cd − + c ∀ k ≥ l ≥ l . (3.66)Using the fact that s l : g ( s l ) = M ε l and (3.66), we derive Z R N | u k ( x, T l ) | dx ≤ C T l exp( ε exp l ) . (3.67)Because (3.65) implies exp l = eC (cid:18) Φ − (cid:18) T l γ (cid:19)(cid:19) − , (3.68)we get the following inequality by plugging last relationship into (3.67): Z R N | u k ( x, T l ) | dx ≤ C T l exp " e · ε C (cid:18) Φ − (cid:18) T l γ (cid:19)(cid:19) − ∀ l ≥ l . At last, combining last estimate with (3.68 ), we obtain Z R N | u k ( x, t ) | dx ≤ C t exp " e · ε C (cid:18) Φ − (cid:18) t γ (cid:19)(cid:19) − ∀ t > , which ends the proof. (cid:3) Example 3.4
Assume ω ( s ) = s − α , < α < . Then Φ( s ) = Z s s − α ds = s − α − α ⇒ Φ − ( s ) = (2 − α ) − α s − α . Consequently, estimate (3.9 ) reads as follows, Z R N | u k ( x, t ) | dx ≤ C t exp " C (cid:18) C − α (cid:19) − α t − − α ∀ t > . In this section we consider equation (1.1 ) with potential h ( | x | ) of the form (3.6 ) with the limitingfunction ω ( | x | ) := | x | ℓ ( | x | ) , namely, we study the equation ∂ t u − ∆ u + e − ℓ ( | x | ) | u | p − u = 0 , in R N × (0 , ∞ ) (4.1)where ℓ ∈ C ( R N ) is positive, nonincreasing and lim r → ℓ ( r ) = ∞ . Our main result is the following7 Theorem 4.1
Assume p > and ℓ satisfies lim inf x → | x | ℓ ( x ) > . (4.2) Then the solution u k of the problem (1.1 ), (1.3 ), exists for any k > and u ∞ := lim k →∞ is asolution of (4.1 ) in Q ∞ \ { } × R + with the following properties, lim t → u ∞ ( x, t ) = 0 ∀ x = 0 and lim x → u ∞ u ( x, t ) = ∞ ∀ t > . (4.3) Furthermore t u ∞ ( x, t ) is increasing and lim t →∞ u ∞ ( x, t ) = U ( x ) for every x = 0 where U = lim k →∞ U k and U k solves − ∆ U k + e − ℓ ( x ) U pk = kδ in D ′ ( R N ) . (4.4) Proof.
By assumption (4.2 ), property (1.2 ) is fulfilled. Thus for k > u := u k solution of (4.1 ), (1.3 ). Moreover, for any k > U k of (4.4 ) (see [18]); themapping k U k is increasing and U = lim k →∞ U k exists, because of Keller-Osserman estimate. U is the minimal solution of − ∆ V + e − ℓ ( x ) V p = 0 in R N \ { } , (4.5)verifying Z B ǫ V ( x ) dx = ∞ ∀ ǫ > . (4.6)If we denote by ¯ U the maximal solution of (4.5 ), it is classical that ¯ U = lim ǫ → ¯ U ǫ where (cid:26) − ∆ ¯ U ǫ + e − ℓ ( x ) ¯ U pǫ = 0 in R N \ ¯ B ǫ lim | x |→ ǫ ¯ U ǫ ( x ) = ∞ . (4.7)Since any u k is bounded from above by ¯ U , the local equicontinuity of the u k in ¯ Q T \ { (0 , } impliesthat u ∞ satisfies lim t → u ∞ ( x, t ) = 0 for all x = 0. Step 1: Formation of the razor blade . The Case 1: 1 < p < /N . For ǫ > e − ℓ ( | x | ) ≤ e − ℓ ( ǫ ) for | x | ≤ ǫ . Therefore ∂ t u − ∆ u + e − ℓ ( ǫ ) | u | p − u ≥ , in B ǫ × (0 , ∞ ) . (4.8)and u ≥ v ǫ in B ǫ × (0 , T ) where v ǫ solves ∂ t v ǫ − ∆ v ǫ + e − ℓ ( ǫ ) | v ǫ | p − v ǫ = 0 in B ǫ × (0 , ∞ ) v ǫ = 0 in ∂B ǫ × (0 , ∞ ) v ǫ ( x,
0) = ∞ δ in B ǫ , (4.9)where the initial condition is to be understood in the sense lim k →∞ kδ . We put w ǫ ( x, t ) = ǫ / ( p − e − ℓ ( ǫ ) / ( p − v ǫ ( ǫx, ǫ t ) . Then w ǫ = w is independent of ǫ and solves ∂ t w − ∆ w + | w | p − w = 0 in B × (0 , ∞ ) w = 0 in ∂B × (0 , ∞ ) w ( x,
0) = ∞ δ in B . (4.10)8Therefore u (0 , ≥ v ǫ (0 ,
1) = ǫ − / ( p − e ℓ ( ǫ ) / ( p − w (0 , ǫ − ) . (4.11)The longtime behaviour is given in [7] where it is provedlim τ →∞ e λ τ w (0 , τ ) = κφ (0) . In this formula φ is the first eigenfunction of − ∆ in W , ( B ), λ the corresponding eigenvalueand κ >
0. Thus u (0 , ≥ δǫ − / ( p − e ℓ ( ǫ ) / ( p − e λ ǫ − φ (0) , (4.12)for some δ >
0, if ǫ is small enough. If we assumelim ǫ → (cid:18) p − ǫ − + ℓ ( ǫ ) p − − λ ǫ − (cid:19) = ∞ , (4.13)it implies u (0 ,
1) = ∞ = ⇒ u (0 , t ) = ∞ ∀ t > . (4.14)Moreover, the unit ball B can be replaced by any ball B R and λ by λ R = R − λ . Therefore thesufficient condition for a Razor blade is that it exists some c > ǫ → (cid:0) ℓ ( ǫ ) − cǫ − (cid:1) = ∞ . (4.15)An equivalent condition is lim inf ǫ → ǫ ℓ ( ǫ ) > . (4.16)The general case. If p > β > β > N ( p − −
2, and we write e − ℓ ( x ) = | x | β e − ℓ ( x ) − β ln | x | . For
R > x ˜ ℓ ( x ) := ℓ ( x ) + β ln | x | is positive, increasing and satisfies the sameblow-up condition (4.2 ) as ℓ . Clearly u k is bounded from below on B R × (0 , ∞ ) by the solution˜ u := ˜ u k of ∂ t ˜ u − ∆˜ u + | x | β e − ˜ ℓ ( x ) | ˜ u | p − ˜ u = 0 in B R × (0 , ∞ )˜ u = 0 in ∂B R × (0 , ∞ )˜ u ( x,
0) = kδ in B R . (4.17)Therefore, for 0 < ǫ < R , ˜ u ∞ is bounded from below on B ǫ × (0 , ∞ ) by the solution v ǫ of ∂ t v ǫ − ∆ v ǫ + | x | β e − ˜ ℓ ( ǫ ) | v ǫ | p − v ǫ = 0 in B ǫ × (0 , ∞ ) v ǫ = 0 in ∂B ǫ × (0 , ∞ ) v ǫ ( x,
0) = ∞ δ in B ǫ . (4.18)If we set w ǫ ( x, t ) = ǫ (2+ β ) / ( p − e − ℓ ( ǫ ) / ( p − v ǫ ( ǫx, ǫ t ) , then w ǫ = w is independent of ǫ and ∂ t w − ∆ w + | x | β | w | p − w = 0 in B × (0 , ∞ ) w = 0 in ∂B × (0 , ∞ ) w ( x,
0) = ∞ δ in B . (4.19)9By a straightforward adaptation of the result of [7], there still holdslim τ →∞ e λ τ w (0 , τ ) = κφ (0)for some κ >
0. The remaining of the proof is the same as in case 1 < p < /N . Step 2: Asymptotic behaviour.
A key observation is that, for any τ > ǫ > Z ǫ u ∞ ( x, τ ) dx = ∞ . (4.20)We give the proof in the case 1 < p < /N , the general case being similar. By step 1 Z B ǫ u ( x, τ ) dx ≥ Z B ǫ v ǫ ( x, τ ) dx = ǫ − / ( p − N e ℓ ( ǫ ) / ( p − Z B w ( y, ǫ − τ ) dy. (4.21)If we fix τ and use [7], there exists ǫ such that w ( y, ǫ − τ ) ≥ − κe − λ ǫ − τ φ ( y ) for ǫ ≤ ǫ and y ∈ B . Therefore Z B ǫ u ( x, τ ) dx ≥ cǫ − / ( p − N e ℓ ( ǫ ) / ( p − − λ ǫ − τ , (4.22)for some constant c >
0. If τ is small enough, the right-hand side of (4.22 ) tends to infinity as ǫ →
0, so does the left-hand side. This implies (4.20 ). For any k > ǫ >
0, there exists m = m ( ǫ ) > Z B ǫ min { u ( x, τ ) , m } dx = k, thus, if we set φ m = min { u ( x, τ ) , m } χ Bǫ , then u is bounded from below on R N × ( τ, ∞ ) by thesolution v = v ǫ,k of (cid:26) ∂ t v − ∆ v + e − ℓ ( x ) | v | p − v = 0 in R N × ( τ, ∞ ) v ( x, τ ) = φ m ( x ) in R N . (4.23)When ǫ → φ m ( . ) → kδ weakly in M ( R N ). By standard approximation property, v ( ǫ, k ) → v ,k which is a solution of (cid:26) ∂ t v − ∆ v + e − ℓ ( x ) | v | p − v = 0 in R N × ( τ, ∞ ) v ( ., τ ) = kδ in R N . (4.24)By uniqueness, v ,k ( x, t ) = u k ( x, t − τ ). Letting k → ∞ yields to u ∞ ( x, t + τ ) ≥ u ∞ ( x, t ) ∀ ( x, t ) ∈ Q T . (4.25)This implies that t u ∞ ( x, t ) is increasing for every x ∈ R N . Because u ( x, t ) ≤ U ( x ), it isstraightforward that lim x →∞ u ( x, t ) = ˜ U ( x ) exists in R N \ { } . Step 3: Identification of the limit. If ζ ∈ C ∞ ( R N \ { } ), there holds Z T +1 T Z R N (cid:16) − u ( x, t )∆ ζ ( x ) + e − ℓ ( x ) u p ( x, t ) ζ ( x ) (cid:17) dx dt = Z R N ( u ( x, T ) − u ( x, T + 1)) ζ ( x ) dx. By Lebesgue’s theorem Z R N (cid:16) − ˜ U ( x )∆ ζ ( x ) + e − ℓ ( x ) ˜ U p ( x ) ζ ( x ) (cid:17) dx = 0 , (4.26)0and, from (4.20 ), Z ǫ ˜ U ( x ) dx = ∞ , (4.27)for any ǫ >
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