Boundedness of Solutions to a Parabolic-Elliptic Keller–Segel Equation in ℝ2 with Critical Mass

Toshitaka Nagai; Tetsuya Yamada

doi:10.1515/ans-2017-6025

Article Open Access

Boundedness of Solutions to a Parabolic-Elliptic Keller–Segel Equation in ℝ² with Critical Mass

Toshitaka Nagai and Tetsuya Yamada

Published/Copyright: June 21, 2017

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From the journal Advanced Nonlinear Studies Volume 18 Issue 2

Abstract

We consider the Cauchy problem for a parabolic-elliptic system in ℝ2, called the parabolic-elliptic Keller–Segel equation, which appears in various fields in biology and physics. In the critical mass case where the total mass of the initial data is 8⁢π, the unboundedness of nonnegative solutions to the Cauchy problem was shown by Blanchet, Carrillo and Masmoudi [7] under some conditions on the initial data, on the other hand, conditions for boundedness were given by Blanchet, Carlen and Carrillo [6] and López-Gómez, Nagai and Yamada [23]. In this paper, we investigate further the boundedness of nonnegative solutions.

Keywords: Parabolic-Elliptic System; Keller–Segel Equation; Chemotaxis System; Critical Mass, Boundedness of Solutions

MSC 2010: 35B45; 35K15; 35K55

1 Introduction

In this paper, we study the boundedness of nonnegative solutions to the Cauchy problem for the parabolic-elliptic system

(CPψ) { ∂ t ⁡ u = Δ ⁢ u - ∇ ⋅ ( u ⁢ ∇ ⁡ ψ ) , t > 0 , x ∈ ℝ 2 , - Δ ⁢ ψ = u , t > 0 , x ∈ ℝ 2 , u ⁢ ( 0 , x ) = u 0 ⁢ ( x ) , x ∈ ℝ 2 ,

where ψ⁢(t,x) is defined by the convolution product of the Newtonian potential N⁢(x) and u⁢(t,x), namely,

ψ ⁢ ( t , x ) := ( N ∗ u ) ⁢ ( t , x ) = ∫ ℝ 2 N ⁢ ( x - y ) ⁢ u ⁢ ( t , y ) ⁢ 𝑑 y , N ⁢ ( x ) = 1 2 ⁢ π ⁢ log ⁡ 1 | x | .

This model is a simplified version of a chemotaxis system derived from the original Keller–Segel model [19] (see also Childress and Percus [12] and Hillen and Painter [16]), which is called the parabolic-elliptic Keller–Segel equation, and can be also regarded as a model of self-attracting particles in ℝ2 (see Biler and Nadzieja [5] and Wolansky [38]). It is known that the decay property u⁢(t)⁢log⁡(1+|x|)∈L1⁢(ℝ2) is necessary and sufficient for ψ⁢(t)∈Lloc1⁢(ℝ2), and that if we assume the decay condition u0⁢log⁡(1+|x|)∈L1⁢(ℝ2) for the nonnegative initial data u0∈L1⁢(ℝ2), then u⁢(t)⁢log⁡(1+|x|)∈L1⁢(ℝ2) for t>0. In this paper, as the decay condition u0⁢log⁡(1+|x|)∈L1⁢(ℝ2) is not assumed, we consider the following Cauchy problem in place of (CPψ):

(CP) { ∂ t ⁡ u = Δ ⁢ u - ∇ ⋅ ( u ⁢ ( ∇ ⁡ N ∗ u ) ) , t > 0 , x ∈ ℝ 2 , u ⁢ ( 0 , x ) = u 0 ⁢ ( x ) , x ∈ ℝ 2 ,

where

( ∇ ⁡ N ∗ u ) ⁢ ( t , x ) := ∫ ℝ 2 ∇ ⁡ N ⁢ ( x - y ) ⁢ u ⁢ ( t , y ) ⁢ 𝑑 y , ∇ ⁡ N ⁢ ( x ) = - 1 2 ⁢ π ⁢ x | x | 2 .

One of the most important properties for (CP) is the mass conservation, namely,

∫ ℝ 2 u ⁢ ( t , x ) ⁢ 𝑑 x = ∫ ℝ 2 u 0 ⁢ ( x ) ⁢ 𝑑 x , t > 0 ,

and the qualitative behavior of nonnegative solutions to (CP) depends on the initial mass of the system. It is known that in the subcritical case ∫ℝ2u0⁢𝑑x<8⁢π the nonnegative solutions exist globally in time and decay to zero as time goes to infinity (see, e.g., [4, 8, 9, 10, 30, 29, 31]), whereas in the supercritical case ∫ℝ2u0⁢𝑑x>8⁢π, there are blowing-up solutions in finite time (see, e.g., [3, 9, 21]). Some related results to finite-time blow-up are found in, e.g., [15, 17, 18, 25, 26, 28, 36, 37], and the references therein.

In this paper, we focus attention on the critical case ∫ℝ2u0⁢𝑑x=8⁢π. The global existence of nonnegative solutions with critical mass initial data was obtained by Biler, Karch, Laurençot, and Nadzieja [4] for radially symmetric solutions, and for non-radially symmetric solutions by Blanchet, Carrillo and Masmoudi [7] under the decay condition

(1.1) ∫ ℝ 2 | x | 2 ⁢ u 0 ⁢ ( x ) ⁢ 𝑑 x < ∞ ,

and by Nagai and Ogawa [32] under the condition u0⁢log⁡(1+|x|)∈L1⁢(ℝ2). It was also shown in [7] that the global nonnegative solution u under assumptions (1.1) is unbounded, more precisely,

lim t → ∞ ⁡ u ⁢ ( t , x ) = 8 ⁢ π ⁢ δ x 0 ⁢ ( x ) in the sense of measures ,

where x0=∫ℝ2x⁢u0⁢(x)⁢𝑑x/(8⁢π) is the center of mass of u0 and δx0⁢(x) stands for the Dirac delta function at x0. The existence of another type of unbounded solution was established by Naito and Senba [35]: there is a radially symmetric nonnegative initial data u0 with ∫ℝ2|x|2⁢u0⁢(x)⁢𝑑x=∞ for which the nonnegative solution u satisfies

0 < lim inf t → ∞ ⁡ ∥ u ⁢ ( t ) ∥ L ∞ < ∞ , lim sup t → ∞ ⁡ ∥ u ⁢ ( t ) ∥ L ∞ = ∞ .

By the two results on the unbounded solutions, the infinite second moment of initial data is not sufficient for the boundedness of nonnegative solutions.

Problem (CP) admits stationary solutions θb,x0 (b>0, x0∈ℝ2) given by

θ b , x 0 ⁢ ( x ) = 8 ⁢ b ( | x - x 0 | 2 + b ) 2 , x ∈ ℝ 2 ,

which belong to L1⁢(ℝ2)∩L∞⁢(ℝ2) and satisfy

∫ ℝ 2 θ b , x 0 ⁢ 𝑑 x = 8 ⁢ π , ∫ ℝ 2 | x | 2 ⁢ θ b , x 0 ⁢ ( x ) ⁢ 𝑑 x = ∞ .

When x0 is the origin, we denote θb,0 by θb for simplicity, namely,

(1.2) θ b ⁢ ( x ) = 8 ⁢ b ( | x | 2 + b ) 2 , x ∈ ℝ 2 .

The following result on boundedness was obtained by Blanchet, Carlen and Carrillo [6, Theorem 1.6]: let u0∈L1⁢(ℝ2) be a nonnegative initial data with ∫ℝ2u0⁢𝑑x=8⁢π satisfying u0⁢log⁡u0,u0⁢log⁡(e+|x|2)∈L1⁢(ℝ2). Assume

(1.3) ℋ b ⁢ [ u 0 ] := ∫ ℝ 2 ( u 0 - θ b ) 2 ⁢ θ b - 1 / 2 ⁢ 𝑑 x < ∞ for some ⁢ b > 0 .

Then supt≥τ⁡∥u⁢(t)∥Lp<∞ for any τ>0 and any 1<p<∞. The functional ℋb was introduced in [6] to study the large-time behavior of nonnegative solutions to (CPψ) for the critical mass case. The next result on boundedness was given by López-Gómez, Nagai and Yamada [23, Theorem 4.1]: let u0∈L1⁢(ℝ2)∩L∞⁢(ℝ2) be a nonnegative initial data with ∫ℝ2u0⁢𝑑x=8⁢π satisfying

(1.4) lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R u 0 ♯ ⁢ 𝑑 x ) > 0 .

Then there is b>0 such that ∥u⁢(t)∥Lp≤∥θb∥Lp on [0,∞) for any 1≤p≤∞. Here, u0♯ is the symmetric rearrangement of u0 (see Section 2.2 for the definition). As θb,x0♯=θb for every x0∈ℝ2 and θb⁢(x)=O⁢(|x|-4) (|x|→∞), it is apparent that condition (1.4) is satisfied for u0=θb,x0. We remark that if (1.4) is satisfied, then ∫ℝ2|x|2⁢u0⁢(x)⁢𝑑x=∞ (see [23, Lemma 4.1]), and that if ℋb⁢[u0]<∞, then there exists h∈L1⁢(ℝ2) such that

(1.5) u 0 = θ b + h , lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R | h | ⁢ 𝑑 x ) = 0 ,

and (1.4) is satisfied (see [23, Lemma 4.3]).

For related studies about (CP) in the critical mass case, we refer to, e.g., [11, 24, 34], and the references therein.

In this paper, we first show that property (1.4) is inherited by the nonnegative solutions of (CP).

Theorem 1.1.

For a nonnegative initial data u0∈L1⁢(R2) with ∫R2u0⁢𝑑x=8⁢π, let u be the (unique) nonnegative mild solution of (CP) and let Tm be the maximal existence time of u. If (1.4) is satisfied, then

(1.6) lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R u ♯ ⁢ ( t , x ) ⁢ 𝑑 x ) > 0 , 0 < t < T m ,

where u♯ is the symmetric rearrangement of u with respect to x. Hence, Tm=∞ and for any τ>0 there exists b>0 such that for all 1≤p≤∞,

(1.7) ∥ u ⁢ ( t ) ∥ L p ≤ ∥ θ b ∥ L p , t ≥ τ .

The definition of mild solutions and their properties are stated in Section 2.1.

We next consider whether Theorem 1.1 is valid under the following condition in place of condition (1.4):

(1.8) lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R u 0 ⁢ 𝑑 x ) > 0 .

Concerning the relation between (1.4) and (1.8), we have that (1.4) implies (1.8) because of

∫ | x | > R f ♯ ⁢ 𝑑 x ≤ ∫ | x | > R | f | ⁢ 𝑑 x for all ⁢ f ∈ L 1 ⁢ ( ℝ 2 )

(see (4.2)). If the statement

(1.9) lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ⁢ 𝑑 x ) > 0 ⟹ lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ♯ ⁢ 𝑑 x ) > 0

for nonnegative functions f∈L1⁢(ℝ2) is true, then condition (1.4) in Theorem 1.1 is replaced by condition (1.8) for boundedness. However, the statement is false in general and a counter example is given in Example 5.2.

For the boundedness of radially symmetric nonnegative solutions, u0♯ in (1.4) can be replaced by u0. As the equation in (CP) is invariant under rotations, the uniqueness of mild solutions ensures that the mild solution u⁢(t,x) of (CP) is radially symmetric in x if the initial data u0 is radially symmetric.

Theorem 1.2.

Let u0∈L1⁢(R2) be a nonnegative initial data with ∫R2u0⁢𝑑x=8⁢π satisfying condition (1.8). If u0 is radially symmetric, then the radially symmetric nonnegative mild solution u of (CP) exists globally in time and

(1.10) sup t > τ ⁡ ∥ u ⁢ ( t ) ∥ L ∞ < ∞ for any ⁢ τ > 0 .

If u0∈L∞⁢(R2) in addition, then (1.10) is valid for τ=0.

We study sufficient conditions for (1.4). Taking into account (1.5) and θb⁢(x)=O⁢(|x|-4) (|x|→∞), we see that if a nonnegative initial data u0∈L1⁢(ℝ2) with ∫ℝ2u0⁢𝑑x=8⁢π satisfies u0⁢(x)≥C⁢|x|-q (|x|≥1) for some 2<q<4 and a constant C>0, then

ℋ b ⁢ [ u 0 ] = ∞ for all ⁢ b > 0 .

For such initial data, we give a comparison test to show the boundedness of the nonnegative solutions.

Theorem 1.3 (Comparison test).

For a nonnegative initial data u0∈L1⁢(R2), let u be the nonnegative solution of (CP). Assume that there is a nonnegative function f∈L1⁢(R2) such that

u 0 ⁢ ( x ) ≥ f ⁢ ( x ) for ⁢ | x | ≫ 1 , lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ♯ ⁢ 𝑑 x ) > 0 .

Then (1.4) holds. Hence, ∥u⁢(t)∥L∞ is bounded on [τ,∞) for any τ>0, provided ∫R2u0⁢𝑑x=8⁢π.

The following is a consequence of Theorem 1.3.

Corollary 1.4.

Let u0∈L1⁢(R2) be a nonnegative initial data with ∫R2u0⁢𝑑x=8⁢π satisfying

(1.11) u 0 ⁢ ( x ) ≥ C ⁢ | x | - 4 , | x | ≫ 1 ,

where C is a positive constant. Then ∥u⁢(t)∥L∞ is bounded on [τ,∞) for any τ>0.

Remark 1.5.

If (1.11) in Corollary 1.4 is replaced by

u 0 ⁢ ( x ) ≤ C ⁢ | x | - q , | x | ≫ 1 ,

where q>4, then ∥u⁢(t)∥L∞→∞ as t→∞ because (1.1) is satisfied.

We consider statement (1.9) again. As mentioned above, (1.9) is false in general. We give some conditions in the following theorem under which (1.9) is true. We denote by |A| the Lebesgue measure of a measurable set A⊂ℝ2.

Theorem 1.6.

Let f∈L1⁢(R2) be a nonnegative function satisfying

f ⁢ ( x ) → 0 as | x | → ∞ ,
μ ⁢ ( t ) := | { x ∈ ℝ 2 : f ⁢ ( x ) > t } | → ∞ as t → + 0 .

Define r⁢(t)=sup⁡{|x|:f⁢(x)≥t} for t>0 and assume that

(1.12) lim inf t → + 0 ⁡ μ ⁢ ( t ) r ⁢ ( t ) 2 > 0 .

Then (1.9) holds.

This paper is organized as follows: In Section 2, we recall the definition of mild solutions to (CP) and their properties, and collect some fundamentals of rearrangements and related results. In Section 3, we prove Theorems 1.1 and 1.2. In Section 4, we begin with useful lemmas related to (1.4), and then prove Theorem 1.3, Corollary 1.4 and Theorem 1.6. In Section 5, we give an application of Theorem 1.6 in Example 5.1 and a counter example to statement (1.9) in Example 5.2.

2 Preliminaries

Throughout this paper, for 1≤p≤∞, Lp⁢(ℝd) stands for the usual Lebesgue space on ℝd with norm ∥⋅∥Lp, and in the case of d=2, for simplicity, we write

L p := L p ⁢ ( ℝ 2 ) , ∥ ⋅ ∥ p := ∥ ⋅ ∥ L p .

We denote by ℤ+ the set of all integers n≥0, and for α=(α1,α2,…,αd)∈ℤ+d, we set

∂ x α := ∂ 1 α 1 ⁡ ∂ 2 α 2 ⁡ ⋯ ⁢ ∂ d α d , ∂ j := ∂ ∂ ⁡ x j .

We denote by Wm,p:=Wm,p⁢(ℝ2) the usual Sobolev spaces.

2.1 Mild Solutions

The following definition introduces the concept of mild solutions for the Cauchy problem (CP).

Definition 2.1.

Given u0∈L1 and T∈(0,∞), a function u:[0,T]×ℝ2→ℝ is said to be a mild solution of (CP) in [0,T] if

u ∈ C ⁢ ( [ 0 , T ] ; L 1 ) ∩ C ⁢ ( ( 0 , T ] ; L 4 / 3 ) , sup0<t<T⁡(t1/4⁢∥u⁢(t)∥4/3)<∞,
u satisfies the integral equation

(2.1) u ⁢ ( t ) = e t ⁢ Δ ⁢ u 0 - ∫ 0 t ∇ ⋅ e ( t - s ) ⁢ Δ ⁢ ( u ⁢ ( s ) ⁢ ( ∇ ⁡ N ∗ u ) ⁢ ( s ) ) ⁢ 𝑑 s , 0 < t < T ,

where et⁢Δ is the heat semigroup

( e t ⁢ Δ ⁢ f ) ⁢ ( x ) = ∫ ℝ 2 G ⁢ ( t , x - y ) ⁢ f ⁢ ( y ) ⁢ 𝑑 y , G ⁢ ( t , x ) = 1 4 ⁢ π ⁢ t ⁢ exp ⁡ ( - | x | 2 4 ⁢ t ) .

A function u:[0,∞)×ℝ2→ℝ is said to be a global mild solution of (CP) with initial data u0 if u is a mild solution of (CP) in [0,T] for all T∈(0,∞).

The integral in (2.1) is well-defined by the Lp-Lq estimates for the heat semigroup et⁢Δ in ℝ2,

(2.2) ∥ ∂ t m ⁡ ∂ x n ⁡ e t ⁢ Δ ⁢ f ∥ p ≤ C ⁢ t - 1 / q + 1 / p - m - n / 2 ⁢ ∥ f ∥ q for all ⁢ f ∈ L q ,

where 1≤q≤p≤∞ and m,n∈ℤ+, and the fact that, for every 4/3≤q<2,

(2.3) ∥ f ⁢ ( ∇ ⁡ N ∗ g ) ∥ 2 ⁢ q / ( 4 - q ) ≤ C q ⁢ ∥ f ∥ q ⁢ ∥ g ∥ q for all ⁢ f , g ∈ L q ,

where Cq is a positive constant depending only on q. Estimate (2.3) follows easily by combining the Hölder inequality with the next inequality of Hardy–Littlewood–Sobolev in ℝ2: for every 1<q<2,

∥ 1 | x | ∗ g ∥ 2 ⁢ q / ( 2 - q ) ≤ C q ⁢ ∥ g ∥ q for all ⁢ g ∈ L q ,

where Cq>0 is a constant depending only on q.

The next result was established in [30] by adapting methods used for the vorticity equation in ℝ2.

Proposition 2.2.

Suppose u0∈L1. Then there exists T=T⁢(u0)∈(0,∞) such that the Cauchy problem (CP) has a unique mild solution u in [0,T]. Moreover, u satisfies the following properties:

u ⁢ ( t ) → u 0 in L 1 as t → 0 .
For every 1 ≤ p ≤ ∞ , there holds u ∈ C ⁢ ( ( 0 , T ] ; L p ) and sup 0 < t < T ⁡ ( t 1 - 1 / p ⁢ ∥ u ⁢ ( t ) ∥ p ) < ∞ .
For every ℓ ∈ ℤ + , α∈ℤ+2 and 1<p<∞, there holds ∂tℓ⁡∂xα⁡u∈C⁢((0,T];Lp).
u is a classical solution of ∂ t ⁡ u = Δ ⁢ u - ∇ ⋅ ( u ⁢ ( ∇ ⁡ N ∗ u ) ) in ( 0 , T ) × ℝ 2 .
∫ ℝ 2 u ⁢ ( t , x ) ⁢ 𝑑 x = ∫ ℝ 2 u 0 ⁢ 𝑑 x for all 0 < t < T .
If u 0 ≥ 0 and u 0 ≢ 0 , then u ⁢ ( t , x ) > 0 for all ( t , x ) ∈ ( 0 , T ) × ℝ 2 .
If u 0 ⁢ log ⁡ ( 1 + | x | ) ∈ L 1 , then u ⁢ ( t ) ⁢ log ⁡ ( 1 + | x | ) ∈ L 1 for all 0 < t < T .

Remark 2.3.

For every u0∈L1 with u0⁢log⁡(1+|x|)∈L1, the Cauchy problem (CPψ) admits a solution (u,ψ) because, by u⁢(t)∈L∞ and u⁢(t)⁢log⁡(1+|x|)∈L1, the function ψ⁢(t):=(N∗u)⁢(t) is well-defined in Lloc1. Moreover, since u⁢(t) is smooth in x, ψ⁢(t) is also smooth in x and -Δ⁢ψ=u.

We mention the behavior of solutions at the maximal existence time of the solution, which is used later. For the proof of the proposition, see, e.g., [7, 30].

Proposition 2.4.

Let u be a nonnegative mild solution of (CP) and let Tm be the maximal existence time of the solution u. If Tm<∞, then

lim sup t → T m ⁡ ∫ ℝ 2 ( 1 + u ⁢ ( t , x ) ) ⁢ log ⁡ ( 1 + u ⁢ ( t , x ) ) ⁢ 𝑑 x = ∞ .

2.2 Rearrangements

For a measurable function f:ℝd→ℝ and t∈ℝ, we use the following notation for simplicity:

{ f > t } = { x ∈ ℝ d : f ( x ) > t } , | f > t | = | { x ∈ ℝ d : f ( x ) > t } | ,

where |A| is the Lebesgue measure of a Lebesgue measurable set A in ℝd.

Let f:ℝd→ℝ be a measurable function. We assume that f vanishes at infinity in the sense that ||f|>t|<∞ for all t>0. The distribution function μf of f is defined by

μ f ( t ) = | | f | > t | , t ≥ 0 ,

and the decreasing rearrangement f∗ of f, the generalized inverse of μf, is defined by

f ∗ ⁢ ( s ) = inf ⁡ { t ≥ 0 : μ f ⁢ ( t ) ≤ s } , s ≥ 0 .

The function f♯:ℝd→ℝ, called the symmetric rearrangement or the Schwarz symmetrization of f, is defined by

f ♯ ⁢ ( x ) = f ∗ ⁢ ( c d ⁢ | x | d ) ,

where cd is the volume of the unit ball in ℝd.

Some basic properties about rearrangements are as follows:

| | f | > t | = | f ♯ > t | = | f ∗ > t | for t>0.
f ∗ is non-increasing and right-continuous on [0,∞).
f ∗ ⁢ ( 0 ) = ∥ f ∥ L ∞ ⁢ ( ℝ d ) , f∗⁢(∞)=0.
If f is bounded and continuous on ℝd, then f∗ and f♯ are bounded and continuous on [0,∞) and ℝd, respectively.
If |f|≤|g|, then f∗≤g∗ and f♯≤g♯.
( c ⁢ f ) ∗ = | c | ⁢ f ∗ , (c⁢f)♯=|c|⁢f♯ for c∈ℝ.
( f + g ) ∗ ⁢ ( s 1 + s 2 ) ≤ f ∗ ⁢ ( s 1 ) + g ∗ ⁢ ( s 2 ) for s1,s2>0.
For s>0 and t>0 satisfying s=|f>t|,

∫ 0 s f ∗ ⁢ 𝑑 σ = ∫ f > t f ⁢ 𝑑 x .

The following properties are well known.

Proposition 2.5.

For every Borel measurable function Φ from ℝ to [0,∞),

∫ ℝ d Φ ⁢ ( | f | ) ⁢ 𝑑 x = ∫ ℝ d Φ ⁢ ( f ♯ ) ⁢ 𝑑 x = ∫ 0 ∞ Φ ⁢ ( f ∗ ) ⁢ 𝑑 s .
Let f , g : ℝ d → ℝ be integrable on ℝ d . If ∫ 0 s f ∗ ⁢ 𝑑 σ ≤ ∫ 0 s g ∗ ⁢ 𝑑 σ for all s > 0 , then

∫ ℝ d Φ ⁢ ( | f | ) ⁢ 𝑑 x ≤ ∫ ℝ d Φ ⁢ ( | g | ) ⁢ 𝑑 x

for all convex functions Φ : [ 0 , ∞ ) → [ 0 , ∞ ) with Φ ⁢ ( 0 ) = 0 .
(The Hardy–Littlewood inequality) Let 1≤p,q≤∞, 1/p+1/q=1. For f∈Lp⁢(ℝd) and g∈Lq⁢(ℝd),

∫ ℝ d | f | ⁢ | g | ⁢ 𝑑 x ≤ ∫ ℝ d f ♯ ⁢ g ♯ ⁢ 𝑑 x = ∫ 0 ∞ f ∗ ⁢ g ∗ ⁢ 𝑑 s .
(Contraction property) Let 1≤p≤∞. For f,g∈Lp⁢(ℝd),

∥ f ∗ - g ∗ ∥ L p ⁢ ( 0 , ∞ ) ≤ ∥ f - g ∥ L p ⁢ ( ℝ d ) , ∥ f ♯ - g ♯ ∥ L p ⁢ ( ℝ d ) ≤ ∥ f - g ∥ L p ⁢ ( ℝ d ) .
(The Pólya-Szegö inequality) Let f∈W1,p⁢(ℝd) (1≤p≤∞). Then f♯∈W1,p⁢(ℝd) and

∥ ∇ ⁡ f ♯ ∥ L p ⁢ ( ℝ d ) ≤ ∥ ∇ ⁡ f ∥ L p ⁢ ( ℝ d ) .

For the properties on rearrangements mentioned above, we refer to, for example, [2, 22, 27, 33].

We now incorporate the following lemma [30, Lemma 5.1].

Lemma 2.6.

Let v⁢(t,x) be a smooth function on (0,T)×R2, satisfying v⁢(t)∈L1∩L∞, such that v is radially symmetric in x and satisfies

∂ t ⁡ v = Δ ⁢ v - ∇ ⋅ ( v ⁢ ( ∇ ⁡ N ∗ v ) ) in ⁢ ( 0 , T ) × ℝ 2 ,

where

( ∇ ⁡ N ∗ v ) ⁢ ( t , x ) := - 1 2 ⁢ π ⁢ ∫ ℝ 2 x - y | x - y | 2 ⁢ v ⁢ ( t , y ) ⁢ 𝑑 y .

Then the functions φ⁢(t,s)=v⁢(t,x), s=π⁢|x|2 and Φ⁢(t,s):=∫0sφ⁢(t,σ)⁢𝑑σ satisfy

∫ ℝ 2 v ⁢ ( t , x ) ⁢ 𝑑 x = ∫ 0 ∞ φ ⁢ ( t , s ) ⁢ 𝑑 s , ⁢ 0 < t < T ,

∂ t ⁡ Φ ⁢ ( t , s ) = 4 ⁢ π ⁢ s ⁢ ∂ s 2 ⁡ Φ ⁢ ( t , s ) + Φ ⁢ ( t , s ) ⁢ ∂ s ⁡ Φ ⁢ ( t , s ) , ⁢ 0 < t < T , s > 0 .

Let u be a nonnegative mild solution of (CP) on [0,T) with nonnegative initial data u0∈L1. For the decreasing rearrangement u∗ of the solution u with respect to the space variable x, define the function H⁢(t,s) by

H ⁢ ( t , s ) = ∫ 0 s u ∗ ⁢ ( t , σ ) ⁢ 𝑑 σ , 0 < t < T , s ≥ 0 .

Then we have the following proposition [13, 14, 30].

Proposition 2.7.

For almost all t∈(0,T),

∂ t ⁡ H - 4 ⁢ π ⁢ s ⁢ ∂ s 2 ⁡ H - H ⁢ ∂ s ⁡ H ≤ 0 , a.a. ⁢ s > 0 .

We give the following comparison principle for later uses (see [23, Proposition 3.3]).

Proposition 2.8.

For nonnegative initial data u0,v0∈L1, let u⁢(t,x) and v⁢(t,x) be nonnegative solutions of (CP) on [0,T) corresponding to the initial data u0 and v0, respectively. Assume that v0 and v are radially symmetric in x and

v ⁢ ( t , x ) = φ ⁢ ( t , s ) , v 0 ⁢ ( x ) = φ 0 ⁢ ( s ) , s = π ⁢ | x | 2 .

∫ 0 s u 0 ∗ ⁢ 𝑑 σ ≤ ∫ 0 s φ 0 ⁢ 𝑑 σ for all ⁢ s > 0 ,

then

∫ 0 s u ∗ ⁢ ( t , σ ) ⁢ 𝑑 σ ≤ ∫ 0 s φ ⁢ ( t , σ ) ⁢ 𝑑 σ for all ⁢ s > 0 , 0 < t < T .

3 Proofs of Theorems 1.1 and 1.2

In order to prove Theorems 1.1 and 1.2, we begin with the following lemma, which is a slight modification of [23, Lemma 4.2] with the decreasing rearrangement f∗ of f replaced by f0 defined in Lemma 3.1. The proof of Lemma 3.1 is similar to that of [23, Lemma 4.2], and is omitted.

Lemma 3.1.

Let f∈L1 be a nonnegative radially symmetric function satisfying ∫R2f⁢𝑑x=8⁢π. Define f0 by f⁢(x)=f0⁢(s), s=π⁢|x|2.

Assume that

lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ⁢ 𝑑 x ) = 1 π ⁢ lim inf s → ∞ ⁡ ( s ⁢ ∫ s ∞ f 0 ⁢ 𝑑 σ ) > 0 .

Then there exist b 0 > 0 and s 0 > 0 such that

(3.1) ∫ 0 s f 0 ⁢ 𝑑 σ < ∫ 0 s θ b 0 ∗ ⁢ 𝑑 σ for all ⁢ s ≥ s 0 .
If f ∈ L 1 ∩ L ∞ and ( 3.1 ) is satisfied, then there exists b ∈ ( 0 , b 0 ) such that

∫ 0 s f 0 ⁢ 𝑑 σ < ∫ 0 s θ b ∗ ⁢ 𝑑 σ for all ⁢ s > 0 .

We remark that as θb defined by (1.2) is radially symmetric and decreasing in |x|, we have the equalities θb⁢(x)=θb♯⁢(x)=θb∗⁢(π⁢|x|2) and

θ b ∗ ⁢ ( s ) = 8 ⁢ π 2 ⁢ b ( s + π ⁢ b ) 2 , ∫ 0 s θ b ∗ ⁢ 𝑑 σ = 8 ⁢ π ⁢ s s + π ⁢ b .

Proof of Theorem 1.1.

Let Tm be the maximal existence time of u. Recall that u0♯⁢(x)=u0∗⁢(s), s=π⁢|x|2. Applying Lemma 3.1 (i) as f=u0♯ and f0=u0∗ yields that there exist b0>0 and s0>0 such that

(3.2) ∫ 0 s u 0 ∗ ⁢ 𝑑 σ < ∫ 0 s θ b 0 ∗ ⁢ 𝑑 σ for all ⁢ s ≥ s 0 .

Define H0⁢(s), H⁢(t,s) and Θb0⁢(s) by

H 0 ⁢ ( s ) = ∫ 0 s u 0 ∗ ⁢ 𝑑 σ , H ⁢ ( t , s ) = ∫ 0 s u ∗ ⁢ ( t , σ ) ⁢ 𝑑 σ , ⁢ 0 < t < T m , s ≥ 0 ,

Θ b 0 ⁢ ( s ) = ∫ 0 s θ b 0 ∗ ⁢ 𝑑 σ = 8 ⁢ π ⁢ s s + π ⁢ b 0 , ⁢ s ≥ 0 .

Then

(3.3) { 4 ⁢ π ⁢ s ⁢ Θ b 0 ′′ ⁢ ( s ) + Θ b 0 ⁢ ( s ) ⁢ Θ b 0 ′ ⁢ ( s ) = 0 , s > 0 , Θ b 0 ⁢ ( 0 ) = 0 , Θ b 0 ⁢ ( ∞ ) = 8 ⁢ π ,

where =′d/ds, and by Proposition 2.7,

(3.4) { ∂ t ⁡ H - 4 ⁢ π ⁢ s ⁢ ∂ s 2 ⁡ H - H ⁢ ∂ s ⁡ H ≤ 0 , 0 < t < T m , s > 0 , H ⁢ ( t , ∞ ) = 8 ⁢ π , 0 < t < T m .

Here we used the fact that by Proposition 2.5 (i),

H ⁢ ( t , ∞ ) = ∫ ℝ 2 u ⁢ ( t , x ) ⁢ 𝑑 x = ∫ ℝ 2 u 0 ⁢ 𝑑 x = 8 ⁢ π .

We observe that H⁢(t,s0) is continuous in t∈[0,Tm) by Proposition 2.5 (iv) and u∈C⁢([0,Tm);L1). As H⁢(0,s0)=H0⁢(s0)<Θb0⁢(s0) by (3.2), there exists τ0>0 such that

(3.5) H ⁢ ( t , s 0 ) < Θ b 0 ⁢ ( s 0 ) , 0 < t ≤ τ 0 .

By (3.2)–(3.5), the comparison principle on the domain (0,τ0)×(s0,∞) ensures that

H ⁢ ( t , s ) ≤ Θ b 0 ⁢ ( s ) , 0 < t ≤ τ 0 , s ≥ s 0 ,

and thus b0 can be shortened to get

H ⁢ ( t , s ) < Θ b 0 ⁢ ( s ) , 0 < t ≤ τ 0 , s ≥ s 0 .

As u⁢(τ)∈L1∩L∞ (0<τ<Tm), we remark that u♯⁢(τ)∈L1∩L∞ by the basic property (iv) on rearrangements and Proposition 2.5 (i). Applying Lemma 3.1 (ii) as f=u♯⁢(τ) and f0=u∗⁢(τ) yields that for any 0<τ≤τ0 there exists b∈(0,b0) such that

H ⁢ ( τ , s ) < Θ b ⁢ ( s ) , s > 0 .

Hence, by the comparison principle on the domain (τ,Tm)×(0,∞), we obtain that

H ⁢ ( t , s ) ≤ Θ b ⁢ ( s ) , τ ≤ t < T m , s ≥ 0 .

It follows from this inequality and H⁢(t,∞)=Θb⁢(∞)=8⁢π that

s ⁢ ∫ s ∞ u ∗ ⁢ ( t , σ ) ⁢ 𝑑 σ = s ⁢ ( 8 ⁢ π - H ⁢ ( t , s ) ) ≥ s ⁢ ( 8 ⁢ π - Θ b ⁢ ( s ) ) = 8 ⁢ π 2 ⁢ b ⁢ s s + π ⁢ b ,

which proves (1.6).

As u⁢(τ) is in L1∩L∞ for 0<τ<Tm and (1.6) for t=τ is satisfied, taking τ and u⁢(τ) as the initial time and the initial data, we obtain Tm=∞ and (1.7) by the boundedness result of López-Gómez, Nagai and Yamada [23] mentioned in Section 1. ∎

To prove Theorem 1.2, we need the following lemma. This lemma was already proven by Moser’s technique (see, e.g., [1, 20]), but we give another proof.

Lemma 3.2.

For a nonnegative initial data u0∈L1∩L∞, let u be the nonnegative mild solution of (CP) on [0,T) for some 0<T≤∞. Assume that

A := sup 0 < t < T ⁡ ∥ ( ∇ ⁡ N ∗ u ) ⁢ ( t ) ∥ ∞ < ∞ .

Then there exists a positive constant C⁢(∥u0∥1,∥u0∥∞,A) depending only on ∥u0∥1, ∥u0∥∞ and A such that

(3.6) ∥ u ⁢ ( t ) ∥ ∞ ≤ C ⁢ ( ∥ u 0 ∥ 1 , ∥ u 0 ∥ ∞ , A ) , 0 < t < T .

Proof.

Take any t0>0 with 4⁢t0<T and fix it. By the definition of mild solutions, the following integral equation is satisfied: for τ≥0 with t0+τ<T,

u ⁢ ( t 0 + τ ) = e t 0 ⁢ Δ ⁢ u ⁢ ( τ ) - ∫ 0 t 0 ∇ ⋅ e ( t 0 - s ) ⁢ Δ ⁢ ( u ⁢ ( s + τ ) ⁢ ( ∇ ⁡ N ∗ u ) ⁢ ( s + τ ) ) ⁢ 𝑑 s .

Let 1≤q≤p≤∞ and 1/q-1/p<1/2. By the Lp-Lq estimates (2.2) for et⁢Δ and ∥u⁢(t)∥1=∥u0∥1, we obtain that

∥ u ⁢ ( t 0 + τ ) ∥ p ≤ ∥ e t 0 ⁢ Δ ⁢ u ⁢ ( τ ) ∥ p + C ⁢ ∫ 0 t 0 ( t 0 - s ) - 1 / q + 1 / p - 1 / 2 ⁢ ∥ u ⁢ ( s + τ ) ⁢ ( ∇ ⁡ N ∗ u ) ⁢ ( s + τ ) ∥ q ⁢ 𝑑 s

≤ C ⁢ t 0 - 1 + 1 / p ⁢ ∥ u ⁢ ( τ ) ∥ 1 + C ⁢ A ⁢ ∫ 0 t 0 ( t 0 - s ) - 1 / q + 1 / p - 1 / 2 ⁢ ∥ u ⁢ ( s + τ ) ∥ q ⁢ 𝑑 s

(3.7) ≤ C ⁢ B 1 ⁢ ( t 0 , p ) ⁢ ∥ u 0 ∥ 1 + C ⁢ A ⁢ B 2 ⁢ ( t 0 , p , q ) ⁢ sup 0 < s < t 0 ⁡ ∥ u ⁢ ( s + τ ) ∥ q ,

where

B 1 ⁢ ( t 0 , p ) = t 0 - 1 + 1 / p , B 2 ⁢ ( t 0 , p , q ) = t 0 - 1 / q + 1 / p + 1 / 2 ⁢ ∫ 0 1 ( 1 - s ) - 1 / q + 1 / p - 1 / 2 ⁢ 𝑑 s .

Take q=1 and p=4/3 in (3.7). Then

∥ u ⁢ ( t 0 + τ ) ∥ 4 / 3 ≤ C ⁢ B 1 ⁢ ( t 0 , 4 / 3 ) ⁢ ∥ u 0 ∥ 1 + C ⁢ A ⁢ B 2 ⁢ ( t 0 , 4 / 3 , 1 ) ⁢ ∥ u 0 ∥ 1 .

Replacing t0+τ by t in this inequality, we have

(3.8) ∥ u ⁢ ( t ) ∥ 4 / 3 ≤ C t 0 ⁢ ( 1 + A ) ⁢ ∥ u 0 ∥ 1 , t 0 ≤ t < T .

Here, we denote by Ct0 a positive constant depending only on t0. Next, taking q=4/3 and p=3 in (3.7) for τ≥t0 with t0+τ<T, we have

∥ u ⁢ ( t 0 + τ ) ∥ 3 ≤ C ⁢ B 1 ⁢ ( t 0 , 3 ) ⁢ ∥ u 0 ∥ 1 + C ⁢ A ⁢ B 2 ⁢ ( t 0 , 3 , 4 / 3 ) ⁢ sup 0 < s < t 0 ⁡ ∥ u ⁢ ( s + τ ) ∥ 4 / 3 .

Replacing t0+τ by t in this inequality yields that

(3.9) ∥ u ⁢ ( t ) ∥ 3 ≤ C t 0 ⁢ ∥ u 0 ∥ 1 + C t 0 ⁢ A ⁢ sup t 0 < t < T ⁡ ∥ u ⁢ ( t ) ∥ 4 / 3 , 2 ⁢ t 0 ≤ t < T .

Take q=3 and p=∞ in (3.7) for τ≥2⁢t0 with t0+τ<T. Then

∥ u ⁢ ( t 0 + τ ) ∥ ∞ ≤ C ⁢ B 1 ⁢ ( t 0 , ∞ ) ⁢ ∥ u 0 ∥ 1 + C ⁢ A ⁢ B 2 ⁢ ( t 0 , ∞ , 3 ) ⁢ sup 0 < s < t 0 ⁡ ∥ u ⁢ ( s + τ ) ∥ 3 ,

from which it follows that

(3.10) ∥ u ⁢ ( t ) ∥ ∞ ≤ C t 0 ⁢ ∥ u 0 ∥ 1 + C t 0 ⁢ A ⁢ sup 2 ⁢ t 0 < t < T ⁡ ∥ u ⁢ ( t ) ∥ 3 , 3 ⁢ t 0 ≤ t < T .

Summing up (3.8)–(3.10), we obtain

(3.11) ∥ u ⁢ ( t ) ∥ ∞ ≤ C t 0 ⁢ ( 1 + A + A 2 + A 3 ) ⁢ ∥ u 0 ∥ 1 , 3 ⁢ t 0 ≤ t < T .

We estimate ∥u⁢(t)∥∞ for 0<t≤3⁢t0, using the integral equation (2.1). Calculating similar to (3.7) and using

∥ e t ⁢ Δ ⁢ u 0 ∥ p ≤ C ⁢ ∥ u 0 ∥ p for ⁢ 1 ≤ p ≤ ∞

by (2.2), we have

∥ u ⁢ ( t ) ∥ 4 / 3 ≤ C ⁢ ∥ u 0 ∥ 4 / 3 + C t 0 ⁢ A ⁢ ∥ u 0 ∥ 1 , 0 < t ≤ 3 ⁢ t 0 .

Similarly,

∥ u ⁢ ( t ) ∥ 3 ≤ C ⁢ ∥ u 0 ∥ 3 + C t 0 ⁢ A ⁢ sup 0 < t ≤ 3 ⁢ t 0 ⁡ ∥ u ⁢ ( t ) ∥ 4 / 3 , 0 < t ≤ 3 ⁢ t 0 ,

and thus

∥ u ⁢ ( t ) ∥ ∞ ≤ C ⁢ ∥ u 0 ∥ ∞ + C t 0 ⁢ A ⁢ sup 0 < t ≤ 3 ⁢ t 0 ⁡ ∥ u ⁢ ( t ) ∥ 3 , 0 < t ≤ 3 ⁢ t 0 .

Hence,

∥ u ⁢ ( t ) ∥ ∞ ≤ C ⁢ ∥ u 0 ∥ ∞ + C t 0 ⁢ ( A ⁢ ∥ u 0 ∥ 3 + A 2 ⁢ ∥ u 0 ∥ 4 / 3 + A 3 ⁢ ∥ u 0 ∥ 1 ) , 0 < t ≤ 3 ⁢ t 0 .

Combining this estimate for 0<t≤3⁢t0 with (3.11) concludes (3.6). ∎

The next lemma on the Poisson equation is used in the proof of Theorem 1.2.

Lemma 3.3.

Let f be in L1∩Lp for some 1<p≤∞ and put

ψ ⁢ ( x ) = ∫ ℝ 2 ( N ⁢ ( x - y ) - N ⁢ ( y ) ) ⁢ f ⁢ ( y ) ⁢ 𝑑 y , x ∈ ℝ 2 ,

where N⁢(x)=-1/(2⁢π)⁢log⁡|x|. Then the following holds:

ψ ⁢ ( x ) is well-defined for all x ∈ ℝ 2 , and ψ ∈ C loc α ⁢ ( ℝ 2 ) for every 0 < α < 1 .
- Δ ⁢ ψ = f in the sense of distributions in ℝ 2 .
A weak derivative of ψ is given by

(3.12) ∂ x j ⁡ ψ ⁢ ( x ) = ∫ ℝ 2 ( ∂ x j ⁡ N ) ⁢ ( x - y ) ⁢ f ⁢ ( y ) ⁢ 𝑑 y for almost every ⁢ x ∈ ℝ 2 .
If p > 2 , then ψ has a derivative for every x∈ℝ2 given by (3.12), and ψ∈Cloc1+α⁢(ℝ2) for every 0<α<1-2/p.

Proof.

Take any x0∈ℝ2 and R>1. Let φR∈C0∞ be such that 0≤φR≤1 and

φ R ⁢ ( x ) = 1 for ⁢ | x - x 0 | ≤ 2 ⁢ R , φ R ⁢ ( x ) = 0 for ⁢ | x - x 0 | ≥ 3 ⁢ R .

Put f1=φR⁢f and f2=(1-φR)⁢f, and define

ψ i ⁢ ( x ) = - 1 2 ⁢ π ⁢ ∫ ℝ 2 ( log ⁡ | x - y | - log ⁡ | y | ) ⁢ f i ⁢ ( y ) ⁢ 𝑑 y , i = 1 , 2 .

Then ψ=ψ1+ψ2. As the support of f1 is compact and f1∈L1∩Lp, we have

ψ 1 ⁢ ( x ) = - 1 2 ⁢ π ⁢ ∫ ℝ 2 log ⁡ | x - y | ⁢ f 1 ⁢ ( y ) ⁢ 𝑑 y + 1 2 ⁢ π ⁢ ∫ ℝ 2 log ⁡ | y | ⁢ f 1 ⁢ ( y ) ⁢ 𝑑 y ,

and hence, by the book of Lieb and Loss [22, Theorems 6.21 and 10.2], we obtain (i)–(iv) in Lemma 3.3 with ψ and f replaced by ψ1 and f1, respectively. Next, as f2⁢(x)=0 (|x-x0|≤2⁢R) and f2∈L1, we derive that ψ2∈C∞⁢(B3⁢R/2⁢(x0)) and for every α∈ℤ+2 with |α|≥1,

∂ x α ⁡ ψ 2 ⁢ ( x ) = ∫ ℝ 2 ( ∂ x α ⁡ N ) ⁢ ( x - y ) ⁢ f 2 ⁢ ( y ) ⁢ 𝑑 y for all ⁢ x ∈ B 3 ⁢ R / 2 ⁢ ( x 0 ) ,

where BR⁢(x0)={x∈ℝ2:|x-x0|<R}. Hence, ψ2 is harmonic in B3⁢R/2⁢(x0). Therefore, the assertions of the lemma are obtained. ∎

Proof of Theorem 1.2.

Let Tm be the maximal existence time of the solution u. As u0⁢(x) and u⁢(t,x) are radially symmetric in x, we define φ⁢(t,s) and φ0⁢(s) for 0≤t<Tm, s≥0 by

u ⁢ ( t , x ) = φ ⁢ ( t , s ) , u 0 ⁢ ( x ) = φ 0 ⁢ ( s ) , s = π ⁢ | x | 2 .

Put

Φ ⁢ ( t , s ) = ∫ 0 s φ ⁢ ( t , σ ) ⁢ 𝑑 σ , Φ 0 ⁢ ( s ) = ∫ 0 s φ 0 ⁢ 𝑑 σ , s ≥ 0 .

Then, by Lemma 2.6,

{ ∂ t ⁡ Φ ⁢ ( t , s ) = 4 ⁢ π ⁢ s ⁢ ∂ s 2 ⁡ Φ ⁢ ( t , s ) + Φ ⁢ ( t , s ) ⁢ ∂ s ⁡ Φ ⁢ ( t , s ) , 0 < t < T m , s > 0 , Φ ⁢ ( t , 0 ) = 0 , Φ ⁢ ( t , ∞ ) = 8 ⁢ π , 0 < t < T m , Φ ⁢ ( 0 , s ) = Φ 0 ⁢ ( s ) , s ≥ 0 .

As (1.8) is equivalent to

lim inf s → ∞ ⁡ ( s ⁢ ∫ s ∞ φ 0 ⁢ 𝑑 σ ) > 0 ,

duplicating the argument in the proof of Theorem 1.1 with u0∗, u∗ and H replaced by φ0, φ and Φ, respectively, we get that for any 0<τ<Tm there exists b>0 such that

(3.13) Φ ⁢ ( t , s ) ≤ Θ b ⁢ ( s ) = 8 ⁢ π ⁢ s s + π ⁢ b , τ ≤ t < T m , s ≥ 0 .

For τ≤t<Tm, define the function ψ~⁢(t,x) by

ψ ~ ⁢ ( t , x ) = - 1 2 ⁢ π ⁢ ∫ ℝ 2 ( log ⁡ | x - y | - log ⁡ | y | ) ⁢ u ⁢ ( t , y ) ⁢ 𝑑 y , x ∈ ℝ 2 .

As u⁢(t)∈L1∩L∞ for any τ≤t<Tm, by Lemma 3.3, ψ~⁢(t,x) is well-defined for all x∈ℝ2, and it holds that

- Δ ⁢ ψ ~ ⁢ ( t ) = u ⁢ ( t ) in the sense of distributions in ⁢ ℝ 2 ,

ψ ~ ⁢ ( t ) ∈ C loc 1 + α ⁢ ( ℝ 2 ) for ⁢ 0 < α < 1 , ∇ ⁡ ψ ~ ⁢ ( t , x ) = ( ∇ ⁡ N ∗ u ) ⁢ ( t , x ) .

Taking into account u⁢(t)∈C∞⁢(ℝ2), we see that ψ~⁢(t)∈C∞⁢(ℝ2) and

- Δ ⁢ ψ ~ ⁢ ( t , x ) = u ⁢ ( t , x ) for all ⁢ x ∈ ℝ 2 .

As (u,ψ~) is radially symmetric with respect to x, we define u^⁢(t,r) and ψ^⁢(t,r) by

u ⁢ ( t , x ) = u ^ ⁢ ( t , r ) , ψ ~ ⁢ ( t , x ) = ψ ^ ⁢ ( t , r ) , r = | x | .

Then, by -Δ⁢ψ~=u, we have

- 1 r ⁢ ∂ r ⁡ ( r ⁢ ∂ r ⁡ ψ ^ ) = u ^ ,

and hence

- r ⁢ ∂ r ⁡ ψ ^ ⁢ ( t , r ) = ∫ 0 r u ^ ⁢ ( t , η ) ⁢ η ⁢ 𝑑 η = 1 2 ⁢ π ⁢ ∫ 0 π ⁢ r 2 φ ⁢ ( t , σ ) ⁢ 𝑑 σ = 1 2 ⁢ π ⁢ Φ ⁢ ( t , π ⁢ r 2 ) .

From this relation and (3.13) it follows that for τ≤t<Tm and x∈ℝ2,

| ( ∇ ⁡ N ∗ u ) ⁢ ( t , x ) | = | ∇ ⁡ ψ ~ ⁢ ( t , x ) | = | ∂ r ⁡ ψ ^ ⁢ ( t , r ) | = 1 2 ⁢ π ⁢ r ⁢ Φ ⁢ ( t , π ⁢ r 2 ) ≤ 4 ⁢ r r 2 + b ≤ 2 b .

Applying Lemma 3.2 as T=Tm-τ for the solution u⁢(t+τ,x), we have that

(3.14) sup 0 < t < T m - τ ⁡ ∥ u ⁢ ( t + τ ) ∥ ∞ ≤ C ⁢ ( ∥ u ⁢ ( τ ) ∥ 1 , ∥ u ⁢ ( τ ) ∥ ∞ , A ) ,

where

A = sup τ ≤ t < T m ⁡ ∥ ( ∇ ⁡ N ∗ u ) ⁢ ( t ) ∥ ∞ ≤ 2 b .

It follows from (3.14) that lim supt→Tm⁡∥u⁢(t)∥∞<∞. Therefore, Tm=∞ by Proposition 2.4, that is, u exists globally in time, and ∥u⁢(t)∥∞ is bounded on [τ,∞) by (3.14).

If u0∈L∞ in addition, then Φ0⁢(s)≤Θb⁢(s) (s≥0) for some b>0 by Lemma 3.1, and hence (3.13) is valid for τ=0. Therefore, (1.10) is concluded for τ=0. ∎

4 Sufficient Conditions for (1.4)

In this section, we prove Theorems 1.3 and 1.6 and Corollary 1.4. We begin with two lemmas.

Lemma 4.1.

Let f ∈ L 1 and assume that

lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R | f | ⁢ 𝑑 x ) = 0 .

Then

lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ♯ ⁢ 𝑑 x ) = lim s → ∞ ⁡ ( s ⁢ ∫ s ∞ f ∗ ⁢ 𝑑 σ ) = 0 .
Let f , g ∈ L 1 and assume that

lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ♯ ⁢ 𝑑 x ) > 0 , lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R | g | ⁢ 𝑑 x ) = 0 .

Then

(4.1) lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R ( f + g ) ♯ ⁢ 𝑑 x ) > 0 .

Proof.

(i) We observe that by Proposition 2.5 (iii),

∫ | x | < R | f | ⁢ 𝑑 x = ∫ ℝ 2 χ B R ⁢ | f | ⁢ 𝑑 x ≤ ∫ ℝ 2 χ B R ♯ ⁢ f ♯ ⁢ 𝑑 x = ∫ | x | < R f ♯ ⁢ 𝑑 x ,

where χBR is the characteristic function of BR={x∈ℝ2:|x|<R} and we used χBR♯=χBR. Hence, thanks to ∥f∥1=∥f♯∥1 by Proposition 2.5 (i), we have

(4.2) ∫ | x | > R f ♯ ⁢ 𝑑 x ≤ ∫ | x | > R | f | ⁢ 𝑑 x ,

which yields the conclusion.

(ii) Putting h=f+g and applying the basic property (vii) on rearrangements, we observe that by f=h+(-g),

f ∗ ⁢ ( 2 ⁢ s ) ≤ h ∗ ⁢ ( s ) + g ∗ ⁢ ( s ) , s > 0 ,

and hence

∫ s ∞ h ∗ ⁢ ( σ ) ⁢ 𝑑 σ ≥ ∫ s ∞ f ∗ ⁢ ( 2 ⁢ σ ) ⁢ 𝑑 σ - ∫ s ∞ g ∗ ⁢ ( σ ) ⁢ 𝑑 σ = 1 2 ⁢ ∫ 2 ⁢ s ∞ f ∗ ⁢ ( σ ) ⁢ 𝑑 σ - ∫ s ∞ g ∗ ⁢ ( σ ) ⁢ 𝑑 σ .

By (i), we have

lim s → ∞ ⁡ ( s ⁢ ∫ s ∞ g ∗ ⁢ 𝑑 σ ) = 0 .

Therefore,

lim inf s → ∞ ⁡ ( s ⁢ ∫ s ∞ h ∗ ⁢ 𝑑 σ ) ≥ 1 2 ⁢ lim inf s → ∞ ⁡ ( s ⁢ ∫ 2 ⁢ s ∞ f ∗ ⁢ 𝑑 σ ) > 0 ,

which proves (4.1). ∎

Lemma 4.2.

For f∈L1 and R0>0, define f0∈L1 by

f 0 ⁢ ( x ) = 0 for ⁢ | x | ≤ R 0 , f 0 ⁢ ( x ) = f ⁢ ( x ) for ⁢ | x | > R 0 .

Then

f 0 ♯ ⁢ ( x ) ≥ f ♯ ⁢ ( 2 ⁢ x ) , ⁢ | x | ≥ R 0 ,

∫ | x | > R f 0 ♯ ⁢ 𝑑 x ≥ 1 2 ⁢ ∫ | x | > 2 ⁢ R f ♯ ⁢ 𝑑 x , ⁢ R ≥ R 0 .

Proof.

Define f1∈L1 by

f 1 ⁢ ( x ) = f ⁢ ( x ) for ⁢ | x | ≤ R 0 , f 1 ⁢ ( x ) = 0 for ⁢ | x | > R 0 .

Since ||f1|>t|≤πR02 for any t>0, we have

f 1 ∗ ⁢ ( s ) = 0 , s ≥ π ⁢ R 0 2 .

By f=f0+f1, we observe that f∗⁢(2⁢s)≤f0∗⁢(s)+f1∗⁢(s) for s>0 by the basic property (vii) on rearrangements, and thus

f 0 ∗ ⁢ ( s ) ≥ f ∗ ⁢ ( 2 ⁢ s ) , s ≥ π ⁢ R 0 2 ,

from which it follows that

∫ s ∞ f 0 ∗ ⁢ 𝑑 σ ≥ 1 2 ⁢ ∫ 2 ⁢ s ∞ f ∗ ⁢ 𝑑 σ , s ≥ π ⁢ R 0 2 .

Hence, the inequalities on f0♯ are obtained by virtue of f0♯⁢(x)=f0∗⁢(π⁢|x|2). ∎

Proof of Theorem 1.3.

By u0⁢(x)≥f⁢(x) (|x|≫1), there is a number R0>1 such that u0⁢(x)≥f⁢(x) (|x|≥R0). Define f0∈L1 by

f 0 ⁢ ( x ) = 0 for ⁢ | x | < R 0 , f 0 ⁢ ( x ) = f ⁢ ( x ) for ⁢ | x | ≥ R 0 .

As u0≥f0≥0, we have u0♯≥f0♯ by the basic property (v) on rearrangements, and thus, by Lemma 4.2,

∫ | x | > R u 0 ♯ ⁢ 𝑑 x ≥ ∫ | x | > R f 0 ♯ ⁢ 𝑑 x ≥ 1 2 ⁢ ∫ | x | ≥ 2 ⁢ R f ♯ ⁢ 𝑑 x , R ≥ R 0 ,

from which it follows that

lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R u 0 ♯ ⁢ 𝑑 x ) ≥ 1 2 ⁢ lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > 2 ⁢ R f ♯ ⁢ 𝑑 x ) > 0 .

Hence, if ∫ℝ2u0⁢𝑑x=8⁢π, then Theorem 1.1 ensures the boundedness of ∥u⁢(t)∥∞ on [τ,∞) for any τ>0. ∎

Proof of Corollary 1.4.

Taking b>0 such that C=8⁢b, we have that

u 0 ⁢ ( x ) ≥ C | x | 4 ≥ 8 ⁢ b ( | x | 2 + b ) 2 = θ b ⁢ ( x ) , | x | ≥ R 0 ,

for a positive number R0>1. As θb♯=θb and

lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R θ b ⁢ 𝑑 x ) > 0 ,

the conclusion of the corollary is an immediate consequence of Theorem 1.3. ∎

Proof of Theorem 1.6.

We observe that by the definition of decreasing rearrangements,

| f > f ∗ ( s ) | ≤ s ≤ | f ≥ f ∗ ( s ) | for all s > 0 ,

and by |f>t|→∞ (t→+0),

f ∗ ⁢ ( s ) > 0 for all ⁢ s > 0 , f ∗ ⁢ ( s ) → 0 as ⁢ s → ∞ .

We claim that

(4.3) ∫ 0 s f ∗ ⁢ 𝑑 σ ≤ ∫ | x | ≤ r ⁢ ( t ) f ⁢ 𝑑 x for ⁢ t := f ∗ ⁢ ( s ) , s > 0 .

In fact, in the case of s=|f>t|, by the basic property (viii) on rearrangements and {f≥t}⊂{|x|≤r(t)}, we have

∫ 0 s f ∗ ⁢ 𝑑 σ = ∫ f > t f ⁢ 𝑑 x ≤ ∫ | x | ≤ r ⁢ ( t ) f ⁢ 𝑑 x .

In the case of |f>t|<s≤|f≥t|, put s1=|f>t| and s2=|f≥t|. By the basic property (i) on rearrangements, we observe that

s 1 = | f ∗ > t | , s 2 = | f ∗ ≥ t | , s 2 - s 1 = | f = t | = | f ∗ = t | ,

and thus f∗⁢(σ)=t for s1≤σ≤s2. By these equalities, we have that

∫ 0 s f ∗ ⁢ 𝑑 σ = ∫ 0 s 1 f ∗ ⁢ 𝑑 σ + ( s - s 1 ) ⁢ t

≤ ∫ f > t f ⁢ 𝑑 x + ( s 2 - s 1 ) ⁢ t = ∫ f ≥ t f ⁢ 𝑑 x ≤ ∫ | x | ≤ r ⁢ ( t ) f ⁢ 𝑑 x .

Hence, claim (4.3) is obtained.

As ∫0∞f∗⁢𝑑s=∫ℝ2f⁢𝑑x and μ(t):=|f>t|≤s for t=f∗⁢(s), it follows from (4.3) that

s ⁢ ∫ s ∞ f ∗ ⁢ 𝑑 σ ≥ μ ⁢ ( t ) ⁢ ∫ | x | > r ⁢ ( t ) f ⁢ 𝑑 x = μ ⁢ ( t ) r ⁢ ( t ) 2 ⋅ r ⁢ ( t ) 2 ⁢ ∫ | x | > r ⁢ ( t ) f ⁢ 𝑑 x .

Hence, as t=f∗⁢(s)→0 (s→∞) and r⁢(t)→∞ (t→+0), by (1.12) we have that

lim inf s → ∞ ⁡ ( s ⁢ ∫ s ∞ f ∗ ⁢ 𝑑 σ ) ≥ C ⁢ lim inf t → 0 ⁡ ( r ⁢ ( t ) 2 ⁢ ∫ | x | > r ⁢ ( t ) f ⁢ 𝑑 x )

for a constant C>0. Therefore, this inequality concludes that (1.9) holds. ∎

5 Examples

Example 5.1.

For given a1>0 and a2>0, define {rn}n=0∞ by

r 0 = 0 , r 1 = a 1 , r 2 ⁢ n = r 2 ⁢ n - 1 + a 2 , r 2 ⁢ n + 1 = r 2 ⁢ n + a 1 .

Put

A = ⋃ n = 1 ∞ A 2 ⁢ n - 1 , A n = { x ∈ ℝ 2 : r n ≤ | x | ≤ r n + 1 }

and define f⁢(x) by

(5.1) f ⁢ ( x ) = χ A ⁢ ( x ) ⁢ | x | - 4 , x ∈ ℝ 2 ,

where χA is the characteristic function of A. Then it holds that

(5.2) lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ⁢ 𝑑 x ) > 0 ,

(5.3) lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ♯ ⁢ 𝑑 x ) > 0 .

Let us prove (5.2). Observing that by r2⁢n=r2⁢n-1+a2 and r2⁢n+1=r2⁢n+a1,

∫ r 2 ⁢ n ≤ | x | ≤ r 2 ⁢ n + 1 | x | - 4 ⁢ 𝑑 x = ( r 2 ⁢ n + 2 r 2 ⁢ n ) 2 ⁢ 2 ⁢ a 1 ⁢ r 2 ⁢ n + a 1 2 2 ⁢ a 2 ⁢ r 2 ⁢ n + 1 + a 2 2 ⁢ ∫ r 2 ⁢ n + 1 ≤ | x | ≤ r 2 ⁢ n + 2 | x | - 4 ⁢ 𝑑 x

and

lim n → ∞ ⁡ ( r 2 ⁢ n + 2 r 2 ⁢ n ) 2 ⁢ 2 ⁢ a 1 ⁢ r 2 ⁢ n + a 1 2 2 ⁢ a 2 ⁢ r 2 ⁢ n + 1 + a 2 2 = a 1 a 2 ,

we see that there is n0∈ℕ such that

∫ r 2 ⁢ n ≤ | x | ≤ r 2 ⁢ n + 1 | x | - 4 ⁢ 𝑑 x ≤ 2 ⁢ a 1 a 2 ⁢ ∫ r 2 ⁢ n + 1 ≤ | x | ≤ r 2 ⁢ n + 2 | x | - 4 ⁢ 𝑑 x for all ⁢ n ≥ n 0 .

Then for r2⁢n-1≤R≤r2⁢n+1 (n≥n0),

∫ | x | > R | x | - 4 ⁢ 𝑑 x ≤ ∫ | x | > R χ A ⁢ ( x ) ⁢ | x | - 4 ⁢ 𝑑 x + ∫ | x | ≥ r 2 ⁢ n ( 1 - χ A ⁢ ( x ) ) ⁢ | x | - 4 ⁢ 𝑑 x

= ∫ | x | > R f ⁢ 𝑑 x + ∑ k = n ∞ ∫ r 2 ⁢ k ≤ | x | ≤ r 2 ⁢ k + 1 | x | - 4 ⁢ 𝑑 x

(5.4) ≤ ( 1 + 2 ⁢ a 1 a 2 ) ⁢ ∫ | x | > R f ⁢ 𝑑 x .

Hence, (5.2) follows from (5.4).

Next, we prove (5.3) by applying Theorem 1.6. For (r2⁢n)-4≤t<(r2⁢n-1)-4, we have

μ ( t ) := | f > t | = π ∑ k = 1 n - 1 ( ( r 2 ⁢ k ) 2 - ( r 2 ⁢ k - 1 ) 2 ) + π ( t - 1 / 2 - ( r 2 ⁢ n - 1 ) 2 ) ,

and for (r2⁢n+1)-4≤t<(r2⁢n)-4,

| f > t | = π ∑ k = 1 n ( ( r 2 ⁢ k ) 2 - ( r 2 ⁢ k - 1 ) 2 ) .

Put

s 2 ⁢ n = | f > ( r 2 ⁢ n ) - 4 | = π ∑ k = 1 n ( ( r 2 ⁢ k ) 2 - ( r 2 ⁢ k - 1 ) 2 ) .

Then

(5.5) μ ⁢ ( t ) = { s 2 ⁢ ( n - 1 ) + π ⁢ ( t - 1 / 2 - ( r 2 ⁢ n - 1 ) 2 ) if ⁢ ( r 2 ⁢ n ) - 4 ≤ t < ( r 2 ⁢ n - 1 ) - 4 , s 2 ⁢ n if ⁢ ( r 2 ⁢ n + 1 ) - 4 ≤ t < ( r 2 ⁢ n ) - 4 .

We next observe that by r2⁢k=r2⁢k-1+a2 and r2⁢k-1=(a1+a2)⁢(k-1)+a1,

s 2 ⁢ n = π ⁢ ∑ k = 1 n { ( r 2 ⁢ k ) 2 - ( r 2 ⁢ k - 1 ) 2 } = π ⁢ ∑ k = 1 n ( 2 ⁢ a 2 ⁢ r 2 ⁢ k - 1 + a 2 2 )

(5.6) = π ⁢ ∑ k = 1 n { 2 ⁢ a 2 ⁢ ( a 1 + a 2 ) ⁢ ( k - 1 ) + 2 ⁢ a 1 ⁢ a 2 + a 2 2 } ≥ a 2 ⁢ ( a 1 + a 2 ) ⁢ π ⁢ n 2 .

By (5.5) and (5.6),

μ ⁢ ( t ) ≥ s 2 ⁢ ( n - 1 ) ≥ a 2 ⁢ ( a 1 + a 2 ) ⁢ π ⁢ ( n - 1 ) 2 , ( r 2 ⁢ n + 1 ) - 4 ≤ t < ( r 2 ⁢ n - 1 ) - 4 .

As r⁢(t)=sup⁡{|x|:f⁢(x)≥t}, we observe that

r ⁢ ( t ) ≤ r 2 ⁢ n if ⁢ ( r 2 ⁢ n ) - 4 ≤ t < ( r 2 ⁢ n - 1 ) - 4 , r ⁢ ( t ) = r 2 ⁢ n if ⁢ ( r 2 ⁢ n + 1 ) - 4 ≤ t < ( r 2 ⁢ n ) - 4 .

Hence, by r2⁢n=(a1+a2)⁢n, we have that for (r2⁢n+1)-4≤t<(r2⁢n-1)-4,

μ ⁢ ( t ) r ⁢ ( t ) 2 ≥ a 2 ⁢ ( a 1 + a 2 ) ⁢ π ⁢ ( n - 1 ) 2 ( a 1 + a 2 ) 2 ⁢ n 2 ,

which yields that

lim inf t → + 0 ⁡ μ ⁢ ( t ) r ⁢ ( t ) 2 ≥ a 2 ⁢ π a 1 + a 2 .

Therefore, by Theorem 1.6, inequality (5.3) is established.

With the function f defined by (5.1), we construct two kinds of nonnegative initial data u0∈L1∩L∞. We denote by u the nonnegative solution of (CP) corresponding to the initial data u0.

Case 1: Radial case. Define u0∈L1∩L∞ by u0=c⁢f, where c is taken as ∫ℝ2u0⁢𝑑x=8⁢π. Then u0 is radially symmetric and

0 ≤ u 0 ≤ C ⁢ | x | - 4 for ⁢ | x | ≥ 1 , lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R u 0 ⁢ 𝑑 x ) > 0 .

By Theorem 1.2, ∥u⁢(t)∥∞ is bounded on [0,∞).

We remark that ℋb⁢[u0]=∞ for all b>0, where ℋb is the functional given by (1.3). In fact, we observe that by u0⁢(x)=0 for x∉A,

ℋ b ⁢ [ u 0 ] ≥ ∫ ℝ 2 / A θ b 1 / 2 ⁢ 𝑑 x

= 2 ⁢ 2 ⁢ b ⁢ ∑ k = 0 ∞ ∫ r 2 ⁢ k ≤ | x | ≤ r 2 ⁢ k + 1 1 | x | 2 + b ⁢ 𝑑 x

= 4 ⁢ π ⁢ 2 ⁢ b ⁢ ∑ k = 0 ∞ ∫ r 2 ⁢ k r 2 ⁢ k + 1 r r 2 + b ⁢ 𝑑 r

= 2 ⁢ π ⁢ 2 ⁢ b ⁢ ∑ k = 0 ∞ log ⁡ ( ( r 2 ⁢ k + 1 ) 2 + b ( r 2 ⁢ k ) 2 + b ) .

As r2⁢k+1=r2⁢k+a1 and r2⁢k=(a1+a2)⁢k, we have

( r 2 ⁢ k + 1 ) 2 + b ( r 2 ⁢ k ) 2 + b = 1 + 2 ⁢ a 1 ⁢ r 2 ⁢ k + a 1 2 ( r 2 ⁢ k ) 2 + b ≥ 1 + 2 ⁢ a 1 ⁢ ( a 1 + a 2 ) ⁢ k ( a 1 + a 2 ) 2 ⁢ k 2 + b ,

and hence, by log⁡(1+s)≥s/2 (0≤s≤1),

ℋ b ⁢ [ u 0 ] ≥ 2 ⁢ π ⁢ 2 ⁢ b ⁢ a 1 ⁢ ( a 1 + a 2 ) ⁢ ∑ k = 2 ∞ k ( a 1 + a 2 ) 2 ⁢ k 2 + b = ∞ .

Case 2: Non-radial case. Let cf be the same as in Case 1 and take g∈L1∩L∞ satisfying

(5.7) c ⁢ f + g ≥ 0 , ∫ ℝ 2 g ⁢ 𝑑 x = 0 , lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R | g | ⁢ 𝑑 x ) = 0 .

Define u0∈L1∩L∞ by u0=c⁢f+g. We observe that by (5.3) and (c⁢f)♯=c⁢f♯,

lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R ( c ⁢ f ) ♯ ⁢ 𝑑 x ) > 0 .

Hence, by Lemma 4.1 (ii),

lim inf R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R u 0 ♯ ⁢ 𝑑 x ) > 0 .

Therefore, ∥u⁢(t)∥∞ is bounded on [0,∞).

We remark that if |x|2⁢g,|g|∈L1 in addition to (5.7), then ℋb⁢[u0]=∞ for all b>0. In fact, observing that

( u 0 - θ b ) 2 ⁢ θ b - 1 / 2 = ( c ⁢ f - θ b ) 2 ⁢ θ b - 1 / 2 + g ⁢ θ b - 1 / 2 + 2 ⁢ ( c ⁢ f - u 0 )

≥ ( c ⁢ f - θ b ) 2 ⁢ θ b - 1 / 2 - | g | ⁢ | x | 2 + b 2 ⁢ 2 ⁢ b - 2 ⁢ | g | ,

we have that

ℋ b ⁢ [ u 0 ] ≥ ℋ b ⁢ [ c ⁢ f ] - 1 2 ⁢ 2 ⁢ b ⁢ ∫ ℝ 2 | g ⁢ ( x ) | ⁢ ( | x | 2 + b ) ⁢ 𝑑 x - 2 ⁢ ∫ ℝ 2 | g | ⁢ 𝑑 x .

Hence, as ℋb⁢[c⁢f]=∞, we conclude ℋb⁢[u0]=∞.

Example 5.2.

Let a⁢(t) and b⁢(t) be such that

a ⁢ ( t ) = t - α , b ⁢ ( t ) = t - β for ⁢ 0 < t ≤ 1 , 0 < β < α < 1 , α + β < 1 ,

and define the function f⁢(x) (x=(x1,x2)∈ℝ2) by

f ⁢ ( x ) = { 1 if ⁢ x 1 2 a ⁢ ( 1 ) 2 + x 2 2 b ⁢ ( 1 ) 2 ≤ 1 , t if ⁢ x 1 2 a ⁢ ( t ) 2 + x 2 2 b ⁢ ( t ) 2 = 1 , 0 < t < 1 .

Define the parameter regions of (α,β) as follows:

P 1 : 0 < β < α , 1 2 ≤ α + β < 1 ,

P 2 : 0 < β < α , α + β < 1 2 , ⁢ 3 ⁢ α + β ≥ 1 ,

P 3 : 0 < β < α , 3 α + β < 1 .

Put

A = lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ⁢ 𝑑 x ) , A ♯ = lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ♯ ⁢ 𝑑 x ) .

Then it holds that

(5.8) ( α , β ) ∈ P 1 ⟹ A > 0 , A ♯ > 0 ,

(5.9) ( α , β ) ∈ P 2 ⟹ A > 0 , A ♯ = 0 ,

(5.10) ( α , β ) ∈ P 3 ⟹ A = A ♯ = 0 .

Statement (1.9) in Section 1 is false for (α,β)∈P2.

In what follows, we prove (5.8)–(5.10). As it is apparent that

{ f > t } = { ∅ , t ≥ 1 , { x ∈ ℝ 2 : x 1 2 a ⁢ ( t ) 2 + x 2 2 b ⁢ ( t ) 2 < 1 } , 0 < t < 1 ,

we have that

μ ( t ) := | f > t | = { 0 , t ≥ 1 , π ⁢ a ⁢ ( t ) ⁢ b ⁢ ( t ) , 0 < t < 1 ,

r ⁢ ( t ) := sup ⁡ { | x | : f ⁢ ( x ) ≥ t } = a ⁢ ( t ) = t - α , ⁢ 0 < t < 1 ,

μ ⁢ ( t ) r ⁢ ( t ) 2 = π ⁢ b ⁢ ( t ) a ⁢ ( t ) = π ⁢ t α - β → 0 ⁢ ⁢ as ⁢ t → 0 .

We also see that

∫ ℝ 2 f ⁢ ( x ) ⁢ 𝑑 x = π ⁢ ∫ 0 1 a ⁢ ( t ) ⁢ b ⁢ ( t ) ⁢ 𝑑 t < ∞ .

Let f⁢(x)=t∈(0,1). Then, by b⁢(t)≤|x|≤a⁢(t),

| x | - 1 / β ≤ f ⁢ ( x ) ≤ | x | - 1 / α , | x | > 1 .

As f∗ on (π,∞) equals the inverse function of μ on (0,1), we have

f ∗ ⁢ ( s ) = 1 for ⁢ 0 ≤ s ≤ π , f ∗ ⁢ ( s ) = ( s π ) - 1 / ( α + β ) for ⁢ s > π ,

and thus, for s>π,

s ⁢ ∫ s ∞ f ∗ ⁢ 𝑑 σ = π 1 / ( α + β ) ⁢ α + β 1 - ( α + β ) ⁢ s 2 - 1 / ( α + β ) .

Hence, we derive that

(5.11) lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ♯ ⁢ 𝑑 x ) = lim s → ∞ ⁡ ( s ⁢ ∫ s ∞ f ∗ ⁢ 𝑑 σ ) = { ∞ , α + β > 1 / 2 , π 2 , α + β = 1 / 2 , 0 , α + β < 1 / 2 .

It follows from (4.2) and (5.11) that

(5.12) α + β > 1 2 ⟹ lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ⁢ 𝑑 x ) = ∞ .

We next study

(5.13) lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ⁢ 𝑑 x ) for ⁢ 0 < β < α , α + β ≤ 1 2 .

For this aim, by {f>s}=∅ (s≥1), we first note that for 0<t<1,

∫ | x | > a ⁢ ( t ) f ⁢ ( x ) ⁢ 𝑑 x = ∫ 0 1 ( ∫ | x | > a ⁢ ( t ) χ { f > s } ⁢ ( x ) ⁢ 𝑑 x ) ⁢ 𝑑 s = ∫ 0 1 g ⁢ ( s , t ) ⁢ 𝑑 s ,

where χ{f>s} is the characteristic function of {f>s} and

g ( s , t ) = ∫ | x | > a ⁢ ( t ) χ { f > s } ( x ) d x = | { | x | > a ( t ) } ∩ { f > s } | .

Define γ⁢(t) for 0<t<1 by b⁢(γ⁢(t))=a⁢(t). Then

0 < γ ⁢ ( t ) = t α / β < t .

We observe that

(5.14) g ⁢ ( s , t ) = { 0 , t ≤ s ≤ 1 , π ⁢ { a ⁢ ( s ) ⁢ b ⁢ ( s ) - a ⁢ ( t ) 2 } , 0 < s ≤ γ ⁢ ( t ) .

Indeed, (5.14) follows from the following:

{ | x | > a ( t ) } ∩ { f > s } = ∅ if t ≤ s ≤ 1 ,

{ | x | < a ( t ) } ⊂ { f > s } if 0 < s ≤ γ ( t ) .

In the case of γ⁢(t)<s<t<1, to describe g⁢(s,t) by a⁢(s) and b⁢(s), we consider the intersection point (ξ⁢(s,t),η⁢(s,t)) in {x1≥0,x2≥0} of the two curves Γ1⁢(s), Γ2⁢(t):

Γ 1 ⁢ ( s ) : x 1 2 a ⁢ ( s ) 2 + x 2 2 b ⁢ ( s ) 2 = 1 , Γ 2 ⁢ ( t ) : x 1 2 + x 2 2 = a ⁢ ( t ) 2 .

The point (ξ⁢(s,t),η⁢(s,t)) is given by

(5.15) ξ ⁢ ( s , t ) = a ⁢ ( s ) ⁢ a ⁢ ( t ) 2 - b ⁢ ( s ) 2 a ⁢ ( s ) 2 - b ⁢ ( s ) 2 = t - α ⁢ s 2 ⁢ β - t 2 ⁢ α s 2 ⁢ β - s 2 ⁢ α ,

(5.16) η ⁢ ( s , t ) = b ⁢ ( s ) ⁢ 1 - ξ ⁢ ( s , t ) 2 a ⁢ ( s ) 2 = a ⁢ ( t ) 2 - ξ ⁢ ( s , t ) 2

and

0 < ξ ⁢ ( s , t ) < a ⁢ ( t ) for ⁢ γ ⁢ ( t ) < s < t , ξ ⁢ ( γ ⁢ ( t ) , t ) = 0 , ξ ⁢ ( t , t ) = a ⁢ ( t ) .

We remark that there are no intersection points of the two curves for 0<s<γ⁢(t). Then g⁢(s,t) is expressed by the following: for γ⁢(t)<s<t<1,

(5.17) g ⁢ ( s , t ) = 4 ⁢ { ∫ ξ ⁢ ( s , t ) a ⁢ ( s ) b ⁢ ( s ) ⁢ 1 - x 1 2 a ⁢ ( s ) 2 ⁢ 𝑑 x 1 - ∫ ξ ⁢ ( s , t ) a ⁢ ( t ) a ⁢ ( t ) 2 - x 1 2 ⁢ 𝑑 x 1 } .

It follows from (5.14) and (5.17) that

lim s → t ± 0 ⁡ g ⁢ ( s , t ) = 0 ,

(5.18) lim s → γ ⁢ ( t ) ± 0 ⁡ g ⁢ ( s , t ) = π ⁢ { a ⁢ ( γ ⁢ ( t ) ) ⁢ b ⁢ ( γ ⁢ ( t ) ) - a ⁢ ( t ) 2 } = π ⁢ a ⁢ ( t ) ⁢ { a ⁢ ( γ ⁢ ( t ) ) - a ⁢ ( t ) } .

Here we used ξ⁢(γ⁢(t),t)=0, ξ⁢(t,t)=a⁢(t) and b⁢(γ⁢(t))=a⁢(t).

We claim that

(5.19) lim t → 0 ⁡ ( a ⁢ ( t ) 2 ⁢ ∫ | x | > a ⁢ ( t ) f ⁢ 𝑑 x ) = 2 ⁢ lim t → 0 ⁡ { a ⁢ ( t ) 4 ⁢ ∫ γ ⁢ ( t ) t ( π 2 - arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) ⁢ 𝑑 s } ,

provided that the limit on the right-hand side of (5.19) exists in the extended real values. To prove the claim, we first note that by g⁢(s,t)=0 (0<t≤s≤1),

(5.20) ∫ | x | > a ⁢ ( t ) f ⁢ ( x ) ⁢ 𝑑 x = ∫ 0 γ ⁢ ( t ) g ⁢ ( s , t ) ⁢ 𝑑 s + ∫ γ ⁢ ( t ) t g ⁢ ( s , t ) ⁢ 𝑑 s .

By (5.14),

(5.21) ∂ ⁡ g ∂ ⁡ t ⁢ ( s , t ) = - 2 ⁢ π ⁢ a ⁢ ( t ) ⁢ a ′ ⁢ ( t ) for ⁢ 0 < s < γ ⁢ ( t ) .

For γ⁢(t)<s<t, differentiating (5.17) with respect to t, we have that by (5.16),

∂ ⁡ g ∂ ⁡ t ⁢ ( s , t ) = - 4 ⁢ { b ⁢ ( s ) ⁢ 1 - ξ ⁢ ( s , t ) 2 a ⁢ ( s ) 2 - a ⁢ ( t ) 2 - ξ ⁢ ( s , t ) 2 } ⁢ ∂ ⁡ ξ ∂ ⁡ t ⁢ ( s , t ) - 4 ⁢ ∫ ξ ⁢ ( s , t ) a ⁢ ( t ) ∂ ∂ ⁡ t ⁢ a ⁢ ( t ) 2 - x 1 2 ⁢ 𝑑 x 1

= - 4 ⁢ ∫ ξ ⁢ ( s , t ) a ⁢ ( t ) a ⁢ ( t ) ⁢ a ′ ⁢ ( t ) a ⁢ ( t ) 2 - x 1 2 ⁢ 𝑑 x 1 ,

from which it follows that

(5.22) ∂ ⁡ g ∂ ⁡ t ⁢ ( s , t ) = - 4 ⁢ a ⁢ ( t ) ⁢ a ′ ⁢ ( t ) ⁢ ( π 2 - arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) .

Hence, differentiating (5.20) with respect to t∈(0,1), by (5.18), (5.21) and (5.22), we derive that

d d ⁢ t ⁢ ∫ | x | > a ⁢ ( t ) f ⁢ ( x ) ⁢ 𝑑 x = ∫ 0 γ ⁢ ( t ) ∂ ⁡ g ∂ ⁡ t ⁢ ( s , t ) ⁢ 𝑑 s + ∫ γ ⁢ ( t ) t ∂ ⁡ g ∂ ⁡ t ⁢ ( s , t ) ⁢ 𝑑 s

= - 2 ⁢ π ⁢ a ⁢ ( t ) ⁢ a ′ ⁢ ( t ) ⁢ γ ⁢ ( t ) - 4 ⁢ a ⁢ ( t ) ⁢ a ′ ⁢ ( t ) ⁢ ∫ γ ⁢ ( t ) t ( π 2 - arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) ⁢ 𝑑 s .

Therefore, applying L’Hôpital’s rule and observing that by β<1/4,

lim t → 0 ⁡ a ⁢ ( t ) 4 ⁢ γ ⁢ ( t ) = lim t → 0 ⁡ t - 4 ⁢ α + α / β = 0 ,

we obtain claim (5.19).

By claim (5.19), it is shown that

(5.23) lim t → 0 ⁡ ( a ⁢ ( t ) 2 ⁢ ∫ | x | > a ⁢ ( t ) f ⁢ 𝑑 x ) = 1 2 ⁢ ∫ 0 1 ( 1 - s 2 ⁢ α ) - 1 / 2 ⁢ s - β ⁢ 𝑑 s ⁢ lim t → 0 ⁡ t - 3 ⁢ α - β + 1 .

Thanks to (5.23) and a⁢(t)=t-α, the question on (5.13) is concluded as follows:

(5.24) lim R → ∞ ⁡ ( R 2 ⁢ ∫ | x | > R f ⁢ 𝑑 x ) = { ∞ , 3 ⁢ α + β > 1 , c , 3 ⁢ α + β = 1 , 0 , 3 ⁢ α + β < 1 ,

where

c = 1 2 ⁢ ∫ 0 1 ( 1 - s 2 ⁢ α ) - 1 / 2 ⁢ s - β ⁢ 𝑑 s .

To prove (5.23) by (5.19), we recall γ⁢(t)=tα/β by b⁢(γ⁢(t))=a⁢(t), and observe that by ξ⁢(t,t)=a⁢(t) and ξ⁢(γ⁢(t),t)=0,

d d ⁢ t ⁢ ∫ γ ⁢ ( t ) t ( π 2 - arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) ⁢ 𝑑 s

= ( π 2 - arcsin ⁡ ξ ⁢ ( t , t ) a ⁢ ( t ) ) - ( π 2 - arcsin ⁡ ξ ⁢ ( γ ⁢ ( t ) , t ) a ⁢ ( t ) ) ⁢ γ ′ ⁢ ( t ) - ∫ γ ⁢ ( t ) t ∂ ∂ ⁡ t ⁢ ( arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) ⁢ 𝑑 s

= - π 2 ⁢ γ ′ ⁢ ( t ) - ∫ γ ⁢ ( t ) t ∂ ∂ ⁡ t ⁢ ( arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) ⁢ 𝑑 s .

By (5.15), we have ξ⁢(s,t)/a⁢(t)=(s2⁢β-t2⁢α)1/2⁢(s2⁢β-s2⁢α)-1/2, and thus

∂ ∂ ⁡ t ⁢ ( arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) = - α ⁢ t 2 ⁢ α - 1 ⁢ ( t 2 ⁢ α - s 2 ⁢ α ) - 1 / 2 ⁢ ( s 2 ⁢ β - t 2 ⁢ α ) - 1 / 2 .

Hence,

d d ⁢ t ⁢ ∫ γ ⁢ ( t ) t ( π 2 - arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) ⁢ 𝑑 s = - π ⁢ α 2 ⁢ β ⁢ t α / β - 1 + α ⁢ t 2 ⁢ α - 1 ⁢ ∫ γ ⁢ ( t ) t ( t 2 ⁢ α - s 2 ⁢ α ) - 1 / 2 ⁢ ( s 2 ⁢ β - t 2 ⁢ α ) - 1 / 2 ⁢ 𝑑 s ,

from which it follows that

(5.25) 1 ( a ⁢ ( t ) - 4 ) ′ ⁢ d d ⁢ t ⁢ ∫ γ ⁢ ( t ) t ( π 2 - arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) ⁢ 𝑑 s = - π 8 ⁢ β ⁢ t α / β - 4 ⁢ α + 1 4 ⁢ t - 2 ⁢ α ⁢ ∫ γ ⁢ ( t ) t ( t 2 ⁢ α - s 2 ⁢ α ) - 1 / 2 ⁢ ( s 2 ⁢ β - t 2 ⁢ α ) - 1 / 2 ⁢ 𝑑 s .

As α/β-4⁢α>0 because of β<1/4, it is apparent that

(5.26) lim t → 0 ⁡ π 8 ⁢ β ⁢ t α / β - 4 ⁢ α = 0 .

Next, putting s=t⁢σ and γ1⁢(t)=tα/β-1, we have that

t - 2 ⁢ α ⁢ ∫ γ ⁢ ( t ) t ( t 2 ⁢ α - s 2 ⁢ α ) - 1 / 2 ⁢ ( s 2 ⁢ β - t 2 ⁢ α ) - 1 / 2 ⁢ 𝑑 s = t - 3 ⁢ α - β + 1 ⁢ ∫ γ 1 ⁢ ( t ) 1 ( 1 - σ 2 ⁢ α ) - 1 / 2 ⁢ ( σ 2 ⁢ β - t 2 ⁢ ( α - β ) ) - 1 / 2 ⁢ 𝑑 σ

(5.27) = t - 3 ⁢ α - β + 1 ⁢ I ⁢ ( t ) ,

where

I ⁢ ( t ) = ∫ γ 1 ⁢ ( t ) 1 φ ⁢ ( s , t ) ⁢ 𝑑 s , φ ⁢ ( s , t ) := ( 1 - s 2 ⁢ α ) - 1 / 2 ⁢ ( s 2 ⁢ β - t 2 ⁢ ( α - β ) ) - 1 / 2 .

We claim that

(5.28) lim t → 0 ⁡ I ⁢ ( t ) = ∫ 0 1 ( 1 - s 2 ⁢ α ) - 1 / 2 ⁢ s - β ⁢ 𝑑 s .

In fact, we first divide the integral into two parts:

I ⁢ ( t ) = ( ∫ γ 1 ⁢ ( t ) 2 ⁢ γ 1 ⁢ ( t ) + ∫ 2 ⁢ γ 1 ⁢ ( t ) 1 ) ⁢ φ ⁢ ( s , t ) ⁢ d ⁢ s .

Let γ1⁢(t)<s<2⁢γ1⁢(t)<1/2. Then φ⁢(s,t)≤(1-2-2⁢α)-1/2⁢(s2⁢β-t2⁢(α-β))-1/2, and, putting s=γ1⁢(t)⁢σ=tα/β-1⁢σ, we obtain

∫ γ 1 ⁢ ( t ) 2 ⁢ γ 1 ⁢ ( t ) ( s 2 ⁢ β - t 2 ⁢ ( α - β ) ) - 1 / 2 ⁢ 𝑑 s = t - α + β + α / β - 1 ⁢ ∫ 1 2 ( σ 2 ⁢ β - 1 ) - 1 / 2 ⁢ 𝑑 σ .

As -α+β+α/β-1=(α-β)⁢(1-β)/β>0, we derive that

0 ≤ ∫ γ 1 ⁢ ( t ) 2 ⁢ γ 1 ⁢ ( t ) φ ⁢ ( s , t ) ⁢ 𝑑 s ≤ C ⁢ t - α + β + α / β - 1 → 0 as ⁢ t → 0 .

Let 2⁢γ1⁢(t)<s<1. Then tα-β≤(s/2)β, and hence

φ ⁢ ( s , t ) ≤ ( 1 - 2 - 2 ⁢ β ) - 1 / 2 ⁢ ( 1 - s 2 ⁢ α ) - 1 / 2 ⁢ s - β .

Putting A⁢(t)=(2⁢γ1⁢(t),1)⊂ℝ, we have that

φ ⁢ ( s , t ) ⁢ χ A ⁢ ( t ) ⁢ ( s ) ≤ ( 1 - 2 - 2 ⁢ β ) - 1 / 2 ⁢ ( 1 - s 2 ⁢ α ) - 1 / 2 ⁢ s - β , 0 < s < 1 ,

where χA⁢(t) is the characteristic function of A⁢(t), and

lim t → 0 ⁡ φ ⁢ ( s , t ) ⁢ χ A ⁢ ( t ) ⁢ ( s ) = ( 1 - s 2 ⁢ α ) - 1 / 2 ⁢ s - β , 0 < s < 1 .

As the function (1-s2⁢α)-1/2⁢s-β is integrable on (0,1), Lebesgue’s convergence theorem ensures that

∫ 2 ⁢ γ 1 ⁢ ( t ) 1 φ ⁢ ( s , t ) ⁢ 𝑑 s = ∫ 0 1 φ ⁢ ( s , t ) ⁢ χ A ⁢ ( t ) ⁢ ( s ) ⁢ 𝑑 s → ∫ 0 1 ( 1 - s 2 ⁢ α ) - 1 / 2 ⁢ s - β ⁢ 𝑑 s as ⁢ t → 0 .

Hence, claim (5.28) is derived. Summing up (5.25)–(5.28) and applying L’Hôpital’s rule, we arrive at

lim t → 0 ⁡ { a ⁢ ( t ) 4 ⁢ ∫ γ ⁢ ( t ) t ( π 2 - arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) ⁢ 𝑑 s } = lim t → 0 ⁡ 1 ( a ⁢ ( t ) - 4 ) ′ ⁢ d d ⁢ t ⁢ ∫ γ ⁢ ( t ) t ( π 2 - arcsin ⁡ ξ ⁢ ( s , t ) a ⁢ ( t ) ) ⁢ 𝑑 s

(5.29) = 1 4 ⁢ ∫ 0 1 ( 1 - s 2 ⁢ α ) - 1 / 2 ⁢ s - β ⁢ 𝑑 s ⁢ lim t → 0 ⁡ t - 3 ⁢ α - β + 1 .

Therefore, (5.23) follows from (5.19) and (5.29).

In conclusion, (5.8) follows from (5.11), (5.12) and (5.24), and (5.9) and (5.10) from (5.11) and (5.24).

Communicated by Julian Lopez Gomez

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: JP15KT0019

Funding statement: This work was supported by JSPS KAKENHI, Grant Number JP15KT0019.

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Received: 2017-05-09

Revised: 2017-05-25

Accepted: 2017-05-25

Published Online: 2017-06-21

Published in Print: 2018-04-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2017-6025

Keywords for this article

Parabolic-Elliptic System; Keller–Segel Equation; Chemotaxis System; Critical Mass, Boundedness of Solutions

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