Existence and nonexistence of solutions to the Hardy parabolic equation
aa r X i v : . [ m a t h . A P ] F e b EXISTENCE AND NONEXISTENCE OF SOLUTIONS TOTHE HARDY PARABOLIC EQUATION
KOTARO HISA AND MIKOŁAJ SIERŻĘGA
Abstract.
In this paper, we obtain necessary conditions and sufficient conditions on the initialdata for the local-in-time solvability of the Cauchy problem ∂ t u + ( − ∆) θ u = | x | − γ u p , x ∈ R N , t > , u (0) = µ in R N , where N ≥ , < θ ≤ , p > , γ > and µ is a nonnegative Radon measure on R N . Usingthese conditions, we attempt to identify the optimal strength of the singularity of µ for theexistence of solutions to this problem. Introduction
Consider nonnegative solutions to the Cauchy problem for the Hardy parabolic equation(1.1) ( ∂ t u + ( − ∆) θ u = | x | − γ u p , x ∈ R N , t > ,u (0) = µ in R N , where N ≥ , < θ ≤ , p > , γ > and µ is a nonnegative Radon measure in R N . Here ( − ∆) θ/ denotes the fractional power of the Laplace operator − ∆ in R N . If γ = 0 , this equationis the Fujita-type equation. Throughout this paper we assume < γ < min { θ, N } . In the case of < θ < , we assume the additional condition(1.2) γ < θ ( p − . In the case of θ = 2 , (1.2) is not necessary. In this paper, we attempt to give necessaryconditions and sufficient conditions for the local-in-time solvability of the Cauchy problem (1.1)and to identify the optimal strength of the singularity of µ for the existence of local-in-timesolutions to (1.1). The potential term | x | − γ promotes blow-up of solutions to (1.1) at the origin,on the other hand, its effect is hardly noticeable far from the origin. For this reason, far from theorigin, the optimal singularity of µ is expected to be same as that of the Fujita-type equation,while at the origin, it is expected to be weaker than that of the Fujita-type equation. However,to our knowledge, there seem to be no results describing this prediction.We recall the local-in-time solvability of the Fujita-type equation and the optimal singularityof its initial data. Let us consider nonnegative solutions to the semilinear parabolic equation(1.3) ∂ t v + ( − ∆) θ v = v p , x ∈ R N , t > , v (0) = ν in R N , where N ≥ , < θ ≤ , p > and ν is a nonnegative Radon measure in R N . The local-in-time solvability of Cauchy problem (1.3) has been studied in many papers (see e.g. [1, 3, 6, 10, Mathematics Subject Classification.
Primary 35K58; Secondary 35A01, 35K15, 35K67, 35R11.
Key words and phrases.
Hardy parabolic equation, Fractional Laplacian, Solvability.The first author was supported by JSPS KAKENHI Grant Number JP19H05599.The second author was supported by NCN grant 2017/26/D/ST1/00614.
13, 15–17, 20, 24–26, 28, 31, 32] and references therein). Among them, Baras–Pierre [3] obtainednecessary conditions for the local-in-time solvability in the case of θ = 2 . Subsequently, the firstauthor of this paper and Ishige [13] obtained a generalization to the case of < θ ≤ . Precisely,the following results have already been obtained.(a) Let < T < ∞ . Assume that problem (1.3) possesses a nonnegative solution. Then ν must satisfy the following: – If < p < p , then sup x ∈ R N ν ( B ( x, < ∞ ; – If p = p , then sup x ∈ R N ν ( B ( x, σ )) < c ∗ h log (cid:16) e + T θ σ (cid:17)i − Nθ for all < σ < T θ ; – If p > p , then sup x ∈ R N ν ( B ( x, σ )) < c ∗ σ N − θp − for all < σ < T θ .Here p := 1 + θ/N and c ∗ is a constant depending only on N , θ and p .Then, we can find a large constant C ∗ > with the following property: Set(1.4) Ψ( x ) := | x | − N (cid:20) log (cid:18) e + 1 | x | (cid:19)(cid:21) − Nθ − if p = p , | x | − θp − , if p > p . (b) Problem (1.3) possesses no local-in-time solutions if ν is a nonnegative measurable func-tion in R N satisfying ν ( x ) ≥ C ∗ Ψ( x ) in a neighborhood of the origin.On the other hand, Kozono–Yamazaki [20], Robinson and the second author of this paper [24] andthe first author of this paper and Ishige [13] obtained sufficient conditions for the local-in-timesolvability. These results proved that there exists a small constant c ∗ > such that if ν satisfies ≤ ν ( x ) ≤ c ∗ Ψ( x ) in R N , then problem (1.3) possesses a local-in-time solution. By combiningthese conditions, we see that Ψ( x ) is the optimal singularity of ν for the existence of local-in-time solutions to problem (1.3). We are interested in finding similar necessary conditions andsufficient conditions for the local-in-time solvability of the Cauchy problem (1.1) and identifyingthe optimal singularity of the initial data.Now let us return to the Cauchy problem (1.1). This problem has been studied in [2, 4, 5, 7–9,12, 21–23, 29, 30]. Among them, in the case of θ = 2 , Ben Slimene–Tayachi–Weissler [5] provedthat the Cauchy problem (1.1) is locally well-posed in L q ( R N ) with a suitable q ≥ . Moreover,in the case of p > − γ ) /N , they proved that if µ satisfies ≤ µ ( x ) ≤ c | x | − − γp − in R N for sufficiently small c > , then (1.1) possesses a global-in-time solution. However, there seemto be no results on necessary conditions such as assertion (a). Further still, it seems that in thecase of < θ < no results covering the sufficient conditions are available.In order to state our main results, we introduce some notation and formulate the definitionof solutions to (1.1). For x ∈ R N and r > , let B ( x, r ) := { y ∈ R N : | x − y | < r } and | B ( x, r ) | be the volume of B ( x, r ) . Furthermore, for f ∈ L loc ( R N ) , we set − Z B ( x,r ) f ( y ) dy := 1 | B ( x, r ) | Z B ( x,r ) f ( y ) dy. Let G = G ( x, t ) be the fundamental solution to(1.5) ∂ t v + ( − ∆) θ v = 0 in R N × (0 , ∞ ) , where < θ ≤ . HE HARDY PARABOLIC EQUATION 3
Definition 1.1.
Let u be a nonnegative measurable function in R N × (0 , T ) , where < T ≤ ∞ .We say that u is a solution to (1.1) in R N × [0 , T ) if u satisfies (1.6) ∞ > u ( x, t ) = Z R N G ( x − y, t ) dµ ( y ) + Z t Z R N G ( x − y, t − s ) | y | − γ u ( y, s ) p dyds for almost all x ∈ R N and < t < T . In what follows, define p γ := 1 + θ − γN for N ≥ , < θ ≤ and γ ≥ .Now we are ready to state the main results of this paper. In Theorem 1.1 and 1.2 we obtainnecessary conditions for the local-in-time solvability of the Cauchy problem (1.1). Theorem 1.1.
Let u be a solution to (1.1) in R N × [0 , T ) , where < T < ∞ . Then, there existsa constant C > depending only on N , θ , p and γ , such that (1.7) sup z ∈ R N (cid:18) − Z B ( z,σ ) | x | γp − dx (cid:19) − µ ( B ( z, σ )) ≤ C σ N − θp − for all < σ < T θ . In particular, in the case of p = p γ , there exists a constant C ′ > dependingonly on N , θ and γ , such that (1.8) µ ( B (0 , σ )) ≤ C ′ " log e + T θ σ ! − Nθ − γ for all < σ < T /θ . Since the effect of the singularity | x | − γ is hardly noticeable away from the origin, the solutionto (1.1) is expected to behave like that to (1.3). Theorem 1.2 shows a necessary condition forthe local-in-time solvability of the Cauchy problem (1.1) far from the origin in the case of p = p (compare with (a) as above). Theorem 1.2.
Let z ∈ R N and p = p . Let u be a solution to (1.1) in R N × [0 , T ) , where < T < ∞ . Assume that | z | > T /θ . Then, there exists a constant C > depending only on N , θ and γ , such that µ ( B ( z, σ )) ≤ C | z | γp − " log e + T θ σ ! − Nθ for all < σ < T θ . From Theorems 1.1 and 1.2, we have the following remarks.
Remark 1.1.
There exists a constant C ∗ > with the following property: (i) Problem (1.1) possesses no local-in-time solution if µ is a nonnegative measurable func-tion in R N satisfying µ ( x ) ≥ C ∗ | x | − N (cid:20) log (cid:18) e + 1 | x | (cid:19)(cid:21) − Nθ − γ − if p = p γ ,C ∗ | x | − θ − γp − , if p > p γ , in the neighborhood of the origin. K. HISA AND M. SIERŻĘGA (ii)
Let z ∈ R N \ { } . Problem (1.1) possesses no local-in-time solution if µ is a nonnegativemeasurable function in R N satisfying µ ( x ) ≥ C ∗ | z | γp − Ψ( x ) in the neighborhood of z . Remark 1.2.
Let µ = δ z in R N , where δ z is the Dirac measure concentrated at z ∈ R N . Then,the following holds: • If p γ ≤ p and z = 0 , problem (1.1) possesses no local-in-time solution; • If p ≤ p and z ∈ R N , problem (1.1) possesses no local-in-time solution. The proof of (1.7) is based on [14, Theorem 1.1] and [19, Proposition 1]. In this proof, weconsider the weak solution to (1.1) and give an upper bound for µ by substituting a suitable testfunction. On the other hand, the proofs of (1.8) and Theorem 1.2 are based on [13, Lemma 3.2],which proved a necessary condition in the case of γ = 0 . Let u be a solution (in the sense ofDefinition 1.1) to (1.1) in R N × [0 , T ) and z ∈ R N . Following [13], we employ an iterationargument to get a lower estimate related to Z R N G ( x, t ) u ( x + z, t ) dx, and we prove (1.8) and Theorem 1.2. In particular, in the proof of Theorem 1.2, in order to applyan argument used previously for the Fujita-type equation [13], we have to estimate the potentialterm | x | − γ properly, and assumption T /θ < | z | plays an important role in it. Moreover, it isimportant to estimate the integral Z R N G ( y, t ) | y + z | γp − dy from above for t > , and the assumption (1.2) guarantees that this value is finite (see (2.2)below).We give a sufficient condition for the local-in-time solvability of problem (1.1). Theorem 1.3.
Let p > p γ , α > , r > and T > . Assume that r > satisfies (1.9) r ∈ (cid:18) N ( p − θ − γ − ǫ, N ( p − θ − γ (cid:19) for sufficiently small ǫ > . Then there exists a constant C > , depending only on N , θ , p , γ and r , such that, if µ is a nonnegative measurable function in R N satisfying (1.10) sup z ∈ R N (cid:18) − Z B ( z,σ ) µ ( y ) r dy (cid:19) r ≤ C σ − θ − γp − for all < σ < T /θ , then problem (1.1) possesses a solution in R N × [0 , T ) . As a corollary of Theorem 1.3, we have
Corollary 1.1.
Let p > p γ . There exists a constant c ∗ > such that if µ satisfies (1.11) ≤ µ ( x ) ≤ c ∗ | x | − θ − γp − in R N , then problem (1.1) possesses a local-in-time solution. HE HARDY PARABOLIC EQUATION 5
By Corollary 1.1, we see that the singularity of µ as in (1.11) is the optimal one at the originin the case of p > p γ . However, in other cases, the optimal singularity has not been obtained.The rest of this paper is organized as follows. In Section 2 we collect some properties ofthe fundamental solution G and prepare some preliminary lemmas. In Section 3 we proveTheorem 1.1. In Section 4 we prove Theorem 1.2. In Section 5 we prove Theorem 1.3.2. Preliminaries
In this section, we collect some properties of the fundamental solution G to (1.5) and preparepreliminary lemmas. In what follows the letter C denotes a generic positive constant dependingonly on N , θ , p and γ .Let N ≥ and < θ ≤ . The fundamental solution G to (1.5) is a positive and smoothfunction in R N × (0 , ∞ ) and has the following properties: G ( x, t ) = t − Nθ G (cid:16) t − θ x, (cid:17) , (2.1) C − (1 + | x | ) − N − θ ≤ G ( x, ≤ C (1 + | x | ) − N − θ if < θ < , (2.2) G ( · , is radially symmetric and G ( x, ≤ G ( y, if | x | ≥ | y | , (2.3) G ( x, t ) = Z R N G ( x − y, t − s ) G ( y, s ) dy, (2.4) Z R N G ( x, t ) dx = 1 , (2.5)for all x , y ∈ R N and < s < t . For any φ ∈ L loc ( R N ) , we identify φ with the Radon measure φ dx . For any Radon measure µ in R N , we define [ S ( t ) µ ]( x ) := Z R N G ( x − y, t ) dµ ( y ) , x ∈ R N , t > . Furthermore, we have the following lemmas.
Lemma 2.1.
There exists a constant
C > such that k S ( t ) µ k L ∞ ( R N ) ≤ Ct − Nθ sup z ∈ R N µ ( B ( x, t θ )) for any Radon measure µ in R N and t > .Proof. This lemma was proved in [13, Lemma 2.1]. (cid:3)
Lemma 2.2.
Let µ be a nonnegative Radon measure in R N and < T ≤ ∞ . Assume thatthere exists a supersolution v to (1.1) in R N × [0 , T ) . Then, there exists a solution to (1.1) in R N × [0 , T ) .Proof. Set u := S ( t ) µ . Define u k ( k = 2 , , · · · ) inductively by(2.6) u k ( t ) := S ( t ) µ + Z t S ( t − s ) | · | − γ u k − ( s ) p . Let v be a supersolution to (1.1) in R N × [0 , T ) , where < T ≤ ∞ . Then, it follows inductivelythat ≤ u ( x, t ) ≤ u ( x, t ) ≤ · · · ≤ u k ( x, t ) ≤ · · · ≤ v ( x, t ) < ∞ for almost all x ∈ R N and t ∈ (0 , T ) . This implies that u ( x, t ) := lim k →∞ u k ( x, t ) K. HISA AND M. SIERŻĘGA is well-defined for almost all x ∈ R N and t ∈ (0 , T ) . Furthermore, by (2.6), we see that u satisfies(1.6) for almost all x ∈ R N and t ∈ (0 , T ) . Thus, Lemma 2.2 follows. (cid:3) Proof of Theorem 1.1
In this section, we prove Theorem 1.1. First, we prove (1.7). The proof is based on theargument in [14, Theorem 1.1] and [19, Proposition 1]. Therefore, we note the following remark.
Remark 3.1.
Let < T < ∞ . If u satisfies (1.6) , then u also satisfies (3.1) Z T Z R N ( u ( − ϕ t + ( − ∆) θ ϕ ) − | x | − γ u p ϕ ) dxdt = Z R N ϕ (0) dµ for all ϕ ∈ C ∞ ( R N × [0 , T ]) with ϕ ( T ) = 0 . The proof of (1.7) relies on substituting a suitable test function into (3.1).
Proof of (1.7) . Let u be a solution to (1.1) in R N × [0 , T ) , where < T < ∞ . For z ∈ R N and σ ∈ (0 , T /θ ) , let ζ : R N × [0 , ∞ ) → [0 , be a C ∞ -function which satisfies ζ ( x, t ) = ( in B ( z, − /θ σ ) × [0 , − σ θ ] , outside B ( z, σ ) × [0 , σ θ ) . By substituting ϕ ( x, t ) = ζ ( x, t ) s as a test function in (3.1), where s is an integer and satisfies s > p/ ( p − , we get − s Z σ θ Z B ( z,σ ) uζ t ζ s − dxdt + Z σ θ Z R N u ( − ∆) θ ζ s dxdt = Z σ θ Z B ( z,σ ) | x | − γ u p ζ s dxdt + Z B ( z,σ ) ζ (0) s dµ. (3.2)It follows from (3.2) and the Young inequality that Z σ θ Z B ( z,σ ) | x | − γ u p ζ s dxdt + Z B ( z,σ ) ζ (0) s dµ = − s Z σ θ Z B ( z,σ ) uζ t ζ s − dxdt + Z σ θ Z R N u ( − ∆) θ ζ s dxdt ≤ − s Z σ θ Z B ( z,σ ) uζ t ζ s − dxdt + s Z σ θ Z B ( z,σ ) uζ s − ( − ∆) θ ζ dxdt ≤ C Z σ θ Z B ( z,σ ) u | ζ t | ζ s − dxdt + C Z σ θ Z B ( z,σ ) uζ s − | ( − ∆) θ ζ | dxdt ≤ Z σ θ Z B ( z,σ ) | x | − γ u p ζ s dxdt + C Z σ θ Z B ( z,σ ) | x | γp − | ζ t | pp − ζ s − pp − dxdt + C Z σ θ Z B ( z,σ ) | x | γp − | ( − ∆) θ ζ | pp − ζ s − pp − dxdt. HE HARDY PARABOLIC EQUATION 7
Here, we also used the inequality ( − ∆) θ/ ζ s ≤ sζ s − ( − ∆) θ/ ζ (see [18] for details). Since s > p/ ( p − , we have Z B ( z,σ ) ζ (0) s dµ ≤ C Z σ θ Z B ( z,σ ) | x | γp − | ζ t | pp − dxdt + C Z σ θ Z B ( z,σ ) | x | γp − | ( − ∆) θ ζ | pp − dxdt. (3.3)Now, we choose in (3.3) the function ζ ( x, t ) = ψ ( σ − θ t ) ξ ( σ − x ) , where ψ : [0 , ∞ ) → [0 , is asmooth function which satisfies ψ ( t ) = 1 on [0 , − ] , ψ ( t ) = 0 outside [0 , and ξ : R N → [0 , is a smooth function which satisfies ξ ( x ) = 1 in B ( z, − θ ) , ξ ( x ) = 0 in B ( z, . Since the functions ψ and ξ can be chosen such that(3.4) | ∂ t ψ ( σ − θ t ) | ≤ Cσ − θ and | ( − ∆) θ ξ ( σ − x ) | ≤ Cσ − θ , (3.3) with (3.4) yields µ ( B ( z, − θ σ )) ≤ Cσ − θp − Z B ( z,σ ) | x | γp − dx for all z ∈ R N and < σ < T θ . Then, we can find a positive constant m depending only N , θ , p and γ , such that sup z ∈ R N (cid:18) − Z B ( z,σ ) | x | γp − dx (cid:19) − µ ( B ( z, σ )) ≤ m sup z ∈ R N (cid:18) − Z B ( z,σ ) | x | γp − dx (cid:19) − µ ( B ( z, − θ σ )) ≤ Cmσ N − θp − for all < σ < T θ . Therefore, we obtain the desired estimate and the proof of (1.7) iscomplete. (cid:3) In order to prove (1.8) and complete the proof of Theorem 1.1, we prepare the followinglemma, which has been obtained in [13, Lemma 3.2].
Lemma 3.1.
Let u be a solution to (1.1) in R N × [0 , T ) , where < T < ∞ . Let z ∈ R N and ρ > with (2 ρ ) θ < T . Then, there exists a constant c ∗ > depending only on N such that (3.5) u ( x + z, (2 ρ ) θ ) ≥ c ∗ G ( x, ρ θ ) µ ( B ( z, ρ )) for almost all x ∈ R N . Now we are ready to prove (1.8). The proof is based on the argument in [13, Lemma 3.3].
Proof of (1.8) . We assume p = p γ . Let ν > be a sufficiently small constant and ρ be such that < ρ < ( νT ) θ . Set v ( x, t ) := u ( x, t + (2 ρ ) θ ) for almost all x ∈ R N and t ∈ (0 , T − (2 ρ ) θ ) . Since u is a solutionto (1.1) in R N × [0 , T ) , it follows from (1.6) that for < τ < T − (2 ρ ) θ ,(3.6) v ( x, t ) = Z R N G ( x − y, t − τ ) v ( y, τ ) dy + Z tτ Z R N G ( x − y, t − s ) | y | − γ v ( y, s ) p dyds K. HISA AND M. SIERŻĘGA holds for almost all x ∈ R N , t ∈ ( τ, T − (2 ρ ) θ ) . In the case of < θ < , by (2.1) and (2.2) wehave G ( x − t, τ ) ≥ Cτ − Nθ (cid:18) | x | + | y | τ θ (cid:19) − N − θ ≥ Cτ − Nθ (cid:18) | y | τ θ (cid:19) − N − θ ≥ CG ( y, τ ) for all x ∈ R N with | x | < τ θ , y ∈ R N and τ > . This time (3.6) with t = 2 τ gives(3.7) Z R N G ( y, τ ) v ( y, τ ) dy < ∞ for almost all τ ∈ (0 , [ T − (2 ρ ) θ ] / . On the other hand, in the case of θ = 2 , we have G ( x − y, τ ) ≥ (8 πt ) − N exp (cid:18) − | x | + 2 | y | τ (cid:19) ≥ C (4 πt ) − N exp (cid:18) − | y | τ (cid:19) = CG ( y, τ ) for all x ∈ R N with | x | < τ θ , y ∈ R N and τ > . Then (3.6) with t = 3 τ yields(3.8) Z R N G ( y, τ ) v ( y, τ ) dy < ∞ for almost all τ ∈ (0 , [ T − (2 ρ ) θ ] / . Furthermore, by (2.4), (3.5) and (3.6) with τ = 0 we have v ( x, t ) − Z t Z R N G ( x − y, t − s ) | y | − γ v ( y, s ) p dyds ≥ c ∗ µ ( B (0 , ρ )) Z R N G ( x − y, t ) G ( y, ρ θ ) dy = c ∗ µ ( B (0 , ρ )) G ( x, t + ρ θ ) (3.9)for almost all x ∈ R N and < t < T − (2 ρ ) θ , where c ∗ is the constant in Lemma 3.1. Set w ( t ) := Z R N G ( x, t ) v ( x, t ) dx. By (3.7) and (3.8), we see that w ( t ) < ∞ for almost all t ∈ (0 , [ T − (2 ρ ) θ ] / . Then, it followsfrom (2.4) and (3.9) that ∞ > w ( t ) ≥ c ∗ µ ( B (0 , ρ )) Z R N G ( x, t + ρ θ ) G ( x, t ) dx + Z R N Z t Z R N G ( x − y, t − s ) G ( x, t ) | y | − γ v ( y, s ) p dydsdx ≥ c ∗ µ ( B (0 , ρ )) G (0 , t + ρ θ ) + Z tρ θ Z R N G ( y, t − s ) | y | − γ v ( y, s ) p dyds (3.10)for almost all ρ θ < t < [ T − (2 ρ ) θ ] / . Now it follows from (2.1) and (2.3) that G ( y, t − s ) = (2 t − s ) − Nθ G y (2 t − s ) θ , ! ≥ (cid:16) s t (cid:17) Nθ s − Nθ G (cid:18) ys θ , (cid:19) = (cid:16) s t (cid:17) Nθ G ( y.s ) (3.11) HE HARDY PARABOLIC EQUATION 9 for y ∈ R N and < s < t . By (2.1), (3.10) and (3.11), we obtain ∞ > w ( t ) ≥ c ∗ µ ( B (0 , ρ )) G (0 , t + ρ θ ) + Z tρ θ (cid:16) s t (cid:17) Nθ Z R N G ( y, s ) | y | − γ v ( y, s ) p dyds ≥ Cµ ( B (0 , ρ )) t − Nθ + Z tρ θ (cid:16) s t (cid:17) Nθ Z R N G ( y, s ) | y | − γ v ( y, s ) p dyds (3.12)for almost all ρ θ < t < [ T − (2 ρ ) θ ] / . By the Hölder inequality, we have(3.13) Z R N G ( y, s ) | y | − γ v ( y, s ) p dy ≥ (cid:18)Z R N G ( y, s ) | y | γp − dy (cid:19) − ( p − w ( s ) p . In the case of θ = 2 , by direct computations we obtain(3.14) Z R N G ( y, s ) | y | γp − dy ≤ Cs γ p − . On the other hand, in the case of < θ < , by virtue of assumption (1.2) we see that Z R N G ( y, s ) | y | γp − dy ≤ Cs − Nθ Z R N (cid:18) | y | s θ (cid:19) − N − θ | y | γp − dy = Cs γθ ( p − Z R N (1 + | y | ) − N − θ | y | γp − dy ≤ Cs γθ ( p − . (3.15)By (3.12), (3.13), (3.14) and (3.15), we get(3.16) ∞ > w ( t ) ≥ c µ ( B (0 , ρ )) t − Nθ + c t − Nθ Z tρ θ s Nθ − γθ w ( s ) p ds for almost all ρ θ < t < [ T − (2 ρ ) θ ] / , where c > and c > are constants depending only on N , θ , p and γ .For k = 1 , , · · · , we define the sequence { a k } inductively as(3.17) a := c , a k +1 := c a pk p − p k − k = 1 , , · · · ) . Furthermore, set(3.18) f k ( t ) := a k µ ( B (0 , ρ )) p k − t − Nθ (cid:18) log tρ θ (cid:19) pk − − p − , k = 1 , , · · · . We claim that(3.19) w ( t ) ≥ f k ( t ) , k = 1 , , · · · , for almost all ρ θ < t < [ T − (2 ρ ) θ ] / . By (3.16), we see that (3.19) holds for k = 1 . We assumethat (3.19) holds with some k ∈ { , , · · · } . Then, due to (3.16), we infer that w ( t ) ≥ c t − Nθ Z tρ θ s Nθ − γθ f k ( s ) p ds = c t − Nθ Z tρ θ s Nθ − γθ a k µ ( B (0 , ρ )) p k − s − Nθ (cid:18) log sρ θ (cid:19) pk − − p − p ds = c a pk µ ( B (0 , ρ )) p k t − Nθ Z tρ θ s − (cid:18) log sρ θ (cid:19) pk − pp − ds = c a pk p − p k − µ ( B (0 , ρ )) p k t − Nθ (cid:18) log tρ θ (cid:19) pk − p − = a k +1 µ ( B (0 , ρ )) p k t − Nθ (cid:18) log tρ θ (cid:19) pk − p − = f k +1 ( t ) for almost all ρ θ < t < [ T − (2 ρ ) θ ] / . Therefore, we conclude that that (3.19) holds for all k = 1 , , · · · .Next, we claim that there exists a constant β > such that(3.20) a k ≥ β p k , k = 1 , , · · · . Set b k := − p − k log a k . We prove that there exists a constant C > such that b k ≤ C . By (3.17),we see that − log a k +1 = − p log a k + log (cid:20) c p − p k − (cid:21) . This implies that(3.21) b k +1 − b k = p − k − log (cid:20) c p − p k − (cid:21) ≤ Cp − k − ( k + 1) , k = 1 , , · · · for some constant C > . By (3.21) we see that b k +1 = b + k X j =1 ( b j +1 − b j ) ≤ b + C k X j =1 p − j − ( j + 1) ≤ C for k = 1 , , · · · . This implies (3.20). Taking a sufficiently small ν if necessary, by (3.18), (3.19)and (3.20) we see that ∞ > w ( t ) ≥ f k +1 ( t ) ≤ " β p µ ( B (0 , ρ )) (cid:18) log tρ θ (cid:19) p − p k t − Nθ (cid:18) log tρ θ (cid:19) − p − ≤ " β p µ ( B (0 , ρ )) (cid:18) log T ρ θ (cid:19) p − p k t − Nθ (cid:18) log tρ θ (cid:19) − p − , k = 1 , , · · · for almost all T / < t < T / . Then it follows that β p µ ( B (0 , ρ )) (cid:20) log T ρ θ (cid:21) p − ≤ , HE HARDY PARABOLIC EQUATION 11 which in turn implies that(3.22) µ ( B (0 , ρ )) ≤ C (cid:20) log T ρ θ (cid:21) − p − ≤ C (cid:20) log Tρ θ (cid:21) − p − ≤ C (cid:20) log (cid:18) e + Tρ θ (cid:19)(cid:21) − p − for < ρ < ( νT ) /θ . By (1.7) there exists a constant C ∗ > such that µ ( B (0 , ρ )) ≤ C ∗ for ( νT ) /θ ≤ ρ < T /θ . Since p = p γ , we see that(3.23) (cid:20) log (cid:18) e + Tρ θ (cid:19)(cid:21) − p − ≥ h log (cid:16) e + ν − θ (cid:17)i − p − = C ≥ CC ∗ µ ( B (0 , ρ )) for ( νT ) /θ ≤ ρ < T /θ . Combining (3.22) and (3.23), we obtain µ ( B (0 , ρ )) ≤ C (cid:20) log (cid:18) e + Tσ θ (cid:19)(cid:21) − p − for all < σ < T /θ . Therefore, we obtain the desired result and the proof of Theorem 1.1 iscomplete. (cid:3) Proof of Theorem 1.2
In this section, we prove Theorem 1.2. The proof is based on the argument found in [13,Lemma 3.3]. The following lemma is the key to the proof.
Lemma 4.1.
Assume the same conditions as in Theorem . Then there exists a constant
C > depending only on N , θ and γ such that (4.1) Z R N G ( y, s ) | y + z | γp − dy ≤ Cρ − N Z B ( z,ρ ) | y | γp − dy for almost all ρ θ < s < T / .Proof. The proof is divided into two steps.
We prove that Z R N G ( y, s ) | y + z | γp − dy ≤ C Z B (0 ,s θ ) G ( y, s ) | y + z | γp − dy for almost all ρ θ < s < T / . For this purpose, it is sufficient to show that(4.2) Z B (0 ,s θ ) c G ( y, s ) | y + z | γp − dy ≤ C Z B (0 ,s θ ) G ( y, s ) | y + z | γp − dy for almost all ρ θ < s < T / . First, we give an upper estimate to the integral on B (0 , s /θ ) c . Byvirtue of (1.2), we see that Z B (0 ,s θ ) c G ( y, s ) | y + z | γp − dy = Z B (0 ,s θ ) c G ( y, s ) | y | γp − (cid:18) | y + z || y | (cid:19) γp − dy ≤ Z B (0 ,s θ ) c G ( y, s ) | y | γp − (cid:18) | z || y | (cid:19) γp − dy ≤ C (cid:18) | z | s θ (cid:19) γp − Z B (0 ,s θ ) c G ( y, s ) | y | γp − dy ≤ C ( | z | + s θ ) γp − (4.3)for almost all ρ θ < s < T / .Second, we give a lower estimate to the integral on B (0 , s /θ ) . Since | z | > s /θ > | y | for y ∈ B (0 , s /θ ) , we see that Z B (0 ,s θ ) G ( y, s ) | y + z | γp − dy ≥ Z B (0 ,s θ ) G ( y, s )( | z | − | y | ) γp − dy ≥ ( | z | − s θ ) γp − Z B (0 ,s θ ) G ( y, s ) dy ≥ C ( | z | − s θ ) γp − (4.4)for almost all < s < T / .Combining (4.3) and (4.4), we obtain Z B (0 ,s θ ) c G ( y, s ) | y + z | γp − dy ≤ C | z | + s θ | z | − s θ ! γp − Z B (0 ,s θ ) G ( y, s ) | y + z | γp − dy (4.5)for almost all ρ θ < s < T / . Since ρ θ < s < T / and | z | > T /θ , we have | z | + s θ | z | − s θ ≤ | z | + ( T / θ | z | − ( T / θ ≤ | z | + | z | / θ | z | − | z | / θ ≤ C. This together with (4.5) yields (4.2).
We give an upper estimate to the integral on B (0 , s /θ ) . By (2.1) and (2.3), we see HE HARDY PARABOLIC EQUATION 13 that Z B (0 ,s θ ) G ( y, s ) | y + z | γp − dy ≤ C Z B (0 ,ρ ) G ( y, ρ θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s θ ρ y + z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) γp − dy ≤ C Z B (0 ,ρ ) G ( y, ρ θ ) s θ ρ | y | + | z | ! γp − dy ≤ C ( | z | + s θ ) γp − Z B (0 ,ρ ) G ( y, ρ θ ) | y + z | γp − | y + z | − γp − dy ≤ C | z | + s θ | z | − ρ ! γp − G (0 , ρ θ ) Z B (0 ,ρ ) | y + z | γp − dy ≤ C | z | + s θ | z | − ρ ! γp − ρ − N Z B ( z,ρ ) | y | γp − dy (4.6)for almost all ρ θ < t < T / . Since ρ θ < s < T / and | z | > T /θ , we have | z | + s θ | z | − ρ ≤ | z | + ( T / θ | z | − ( T / θ ≤ | z | + | z | / θ | z | − | z | / θ ≤ C. By combining (4.2) and (4.6), we obtain (4.1). (cid:3)
Proof of Theorem 1.2.
We assume p = p . Let ν > be a sufficiently small constant. Let ρ besuch that < ρ < ( νT ) θ . Set v ( x, t ) := u ( x + z, t + (2 ρ ) θ ) for almost all x ∈ R N and t ∈ (0 , T − (2 ρ ) θ ) . Since u is asolution to (1.1) in R N × [0 , T ) , it follows from (1.6) that v ( x, t ) = Z R N G ( x − y, t ) v ( y, dy + Z t Z R N G ( x − y, t − s ) | y + z | − γ v ( y, s ) p dyds holds for almost all x ∈ R N and t ∈ (0 , T − (2 ρ ) θ ) . Then, we see that Z R N G ( x, t ) v ( x, t ) dy < ∞ holds for almost all < t < [ T − (2 ρ ) θ ] / . Set w ( t ) := Z R N G ( x, t ) v ( x, t ) dx for almost all < t < [ T − (2 ρ ) θ ] / . By an argument similar to that used in the proof of (1.8),we obtain ∞ > w ( t ) ≥ c µ ( B ( z, ρ )) t − Nθ + c t − Nθ Z tρ θ s Nθ (cid:18)Z R N G ( y, s ) | y + z | γp − dy (cid:19) − ( p − w ( s ) p ds (4.7) for almost all ρ θ < t < [ T − (2 ρ ) θ ] / . By applying Lemma 4.1 to (4.7), we have ∞ > w ( t ) ≥ c µ ( B ( z, ρ )) t − Nθ + c t − Nθ (cid:18) ρ − N Z B ( z,ρ ) | y | γp − dy (cid:19) − ( p − Z tρ θ s Nθ w ( s ) p ds for almost all ρ θ < t < [ T − (2 ρ ) θ ] / . Again, by an iteration scheme similar to the one used inthe proof of (1.8), we have ∞ > w ( t ) ≥ " β p (cid:18) ρ − N Z B ( z,ρ ) | y | γp − dy (cid:19) − µ ( B ( z, ρ )) (cid:18) log tρ θ (cid:19) p − p k × t − Nθ ρ − N Z B ( z,ρ ) | y | γp − dy (cid:18) log tρ θ (cid:19) − p − ≥ " β p (cid:18) ρ − N Z B ( z,ρ ) | y | γp − dy (cid:19) − µ ( B ( z, ρ )) (cid:18) log T ρ θ (cid:19) p − p k × t − Nθ ρ − N Z B ( z,ρ ) | y | γp − dy (cid:18) log tρ θ (cid:19) − p − , k = 1 , , · · · for almost all T / < t < T / , where β > is a constant. Then it follows that β p (cid:18) ρ − N Z B ( z,ρ ) | y | γp − dy (cid:19) − µ ( B ( z, ρ )) (cid:20) log (cid:18) e + T ρ θ (cid:19)(cid:21) p − ≤ for all < ρ < ( νT ) /θ . Similarly to the proof of (1.8), we obtain (cid:18) − Z B ( z,σ ) | y | γp − dy (cid:19) − µ ( B ( z, σ )) ≤ C (cid:20) log (cid:18) e + Tσ θ (cid:19)(cid:21) − p − for all z ∈ R N with | z | > T /θ and < σ < T /θ . This is the desired estimate and the proof ofTheorem 1.2 is complete. (cid:3) Proof of Theorem 1.3
In this section, we prove Theorem 1.3. To simplify the notation, we set V ( x ) := | x | − γ . Proof of Theorem 1.3.
By Lemma 2.2, it is sufficient to construct a supersolution to (1.1). Fur-thermore, it is sufficient to consider the case of T = 1 . Indeed, for any solution u to (1.1) in R N × [0 , T ) , where < T < ∞ , we see that u λ ( x, t ) := λ ( θ − γ ) / ( p − u ( λx, λ θ t ) with λ = T /θ isalso a solution to (1.1) in R N × [0 , . Let α > be sufficiently close to N/γ . Set ρ ( t ) := t − γθ − θ − γθ p − ( p − α ′ ) and W ( t ) := S ( t ) µ + ρ ( t )( S ( t ) µ r ) rα ′ , where α ′ = α/ ( α − . We will show that W ( t ) is a supersolution to (1.1) in R N × [0 , . Since p > p γ , α > is sufficiently close to N/γ and (1.9) holds, we see that(5.1) − θ − γθ p − (cid:18) p − α ′ (cid:19) > HE HARDY PARABOLIC EQUATION 15 and(5.2) θ − γθ p (cid:18) − p − (cid:18) p − α ′ (cid:19)(cid:19) − θ − γθ α ′ > . Since G satisfies (2.4) and (2.5), by the Hölder inequality and the Jensen inequality, we have S ( t − s ) V ( S ( s ) µ ) p ≤ ( S ( t − s ) V α ) α ( S ( t − s )( S ( s ) µ ) pα ′ ) α ′ ≤ k S ( t − s ) V α k α L ∞ ( R N ) ( k S ( s ) µ k pα ′ − L ∞ ( R N ) S ( t ) µ ) α ′ ≤ k S ( t − s ) V α k α L ∞ ( R N ) k S ( s ) µ r k r ( p − α ′ ) L ∞ ( R N ) ( S ( t ) µ r ) rα ′ . (5.3)Since < α < N/γ , by (1.10) and Lemma 2.1 we have(5.4) k S ( t − s ) V α k α L ∞ ( R N ) ≤ C ( t − s ) − γθ and(5.5) k S ( s ) µ r k r L ∞ ( R N ) ≤ CC s − θ − γθ p − for almost all < s < t . Then by (5.3), (5.4) and (5.5) we have S ( t − s ) V ( S ( s ) µ ) p ≤ CC p − α ′ ( t − s ) − γθ s − θ − γθ p − ( p − α ′ )( S ( t ) µ r ) rα ′ for almost all < s < t . By (5.1), the right hand side is integrable with respect to s . Then weobtain Z t S ( t − s ) V ( S ( s ) µ ) p ds ≤ CC p − α ′ t − γθ − θ − γθ p − ( p − α ′ )( S ( t ) µ r ) rα ′ = CC p − α ′ ρ ( t )( S ( t ) µ r ) rα ′ (5.6)for almost all < t < . Similarly to (5.3), we see that S ( t − s ) V ( ρ ( s )( S ( s ) µ r ) rα ′ ) p ≤ ρ ( s ) p ( S ( t − s ) V α ) α ( S ( t − s )( S ( s ) µ r ) pr ) α ′ ≤ Cρ ( s ) p ( t − s ) − γθ ( S ( t − s )( S ( s ) µ r ) p ) rα ′ ≤ Cρ ( s ) p ( t − s ) − γθ k S ( s ) µ r k p − rα ′ L ∞ ( R N ) ( S ( t ) µ r ) rα ′ ≤ CC p − α ′ ( t − s ) − γθ s θ − γθ p ( − p − ( p − α ′ )) − θ − γθ α ′ ( S ( t ) µ r ) rα ′ for almost all < s < t . By (5.2) we have Z t S ( t − s ) V ( ρ ( s )( S ( s ) µ r ) rα ′ ) p ds ≤ CC p − α ′ t − γθ + θ − γθ p ( − p − ( p − α ′ )) − θ − γθ α ′ ( S ( t ) µ r ) rα ′ = CC p − α ′ ρ ( t )( S ( t ) µ r ) rα ′ (5.7) for almost all < t < . Combining (5.6) and (5.7), we see that S ( t ) µ + Z t S ( t − s ) V ( W ( s )) p ds ≤ S ( t ) µ + 2 p − Z t S ( t − s ) V ( S ( s ) µ ) p ds + 2 p − Z t S ( t − s ) V ( ρ ( s )( S ( s ) µ r ) rα ′ ) p ds ≤ S ( t ) µ + C ( C p − α ′ + C p − α ′ ) ρ ( t )( S ( t ) µ r ) rα ′ for almost all < t < . Taking a sufficiently small constant C > if necessary, W ( t ) is asupersolution to (1.1) in R N × [0 , . Thus, the proof of Theorem 1.3 is complete. (cid:3) Acknowledgments.
The first author of this paper is grateful to Professor K. Ishige for math-ematical discussions and proofreading of the manuscript.
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Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
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