Large time behavior of solutions of the heat equation with inverse square potential

Let $L:=-\Delta+V$ be a nonnegative Schr\"odinger operator on $L^2({\bf R}^N)$, where $N\ge 2$ and $V$ is a radially symmetric inverse square potential. In this paper we assume either $L$ is subcritical or null-critical and we establish a method for obtaining the precise description of the large time behavior of $e^{-tL}\varphi$, where $\varphi\in L^2({\bf R}^N,e^{|x|^2/4}\,dx)$.


(Communicated by Hirokazu Ninomiya)
Abstract. Let L := −∆ + V be a nonnegative Schrödinger operator on L 2 (R N ), where N ≥ 2 and V is a radially symmetric inverse square potential. In this paper we assume either L is subcritical or null-critical and we establish a method for obtaining the precise description of the large time behavior of e −tL ϕ, where ϕ ∈ L 2 (R N , e |x| 2 /4 dx).
We say that L : When L is nonnegative, we say that • L is subcritical if, for any W ∈ C 0 (R N ), L − W is nonnegative for all sufficiently small > 0: • L is critical if L is not subcritical. On the other hand, L is said to be supercritical if L is not nonnegative.
Consider the ordinary differential equation under condition (V). Equation (O) has two linearly independent solutions U (a regular solution) andŨ (a singular solution) such that In particular, U ∈ L 2 loc (R N ). Assume that L is nonnegative on L 2 (R N ). Then it follows from [19,Theorem 1.1] that U is positive in (0, ∞) and U (r) ∼ c * v(r) as r → ∞ (1.4) for some positive constant c * , where v(r) := if L is subcritical and λ 2 > λ * , log r if L is subcritical and λ 2 = λ * , (1.5) (See also [27] for the case λ 1 = 0.) We often call U a positive harmonic function for the operator L. When L is critical, following [32], we say that L is positive-critical if U ∈ L 2 (R N ) and that L is null-critical if U ∈ L 2 (R N ). Generally, the behavior of the fundamental solution p = p(x, y, t) corresponding to e −tL can be classified by whether L is either subcritical, null-critical or positive-critical. Indeed, in the case of λ 1 = 0, by [ On the other hand, under condition (V ), the first author of this paper with Kabeya and Ouhabaz recently studied in [19] the Gaussian estimate of the fundamental solution p = p(x, y, t) in the subcritical case and in the critical case with A − (λ 2 ) > −N/2. They proved that holds for all x, y ∈ R N and t > 0, where C is a positive constant (see [19,Theorem 1.3]). For related results, see e.g., [1], [4], [8], [21], [22], [24], [25], [36], [37] and references therein. Only in the subcritical case, the precise description of the large time behavior of e −tL ϕ with ϕ ∈ L 2 (R N , e |x| 2 /4 dx) has been studied in a series of papers [13]- [16] with some additional restrictions such as V ∈ C 1 ([0, ∞)), λ 2 > λ * and the sign of the potential. See also [17].
The purpose of this paper is to establish a method for obtaining the precise description of the large time behavior of e −tL ϕ with ϕ ∈ L 2 (R N , e |x| 2 /4 dx) in the subcritical case and in the null-critical case with A − (λ 2 ) > −N/2, under condition (V). In particular, we show that the solution u of (1.1) behaves as a suitable multiple of if L is subcritical and λ > λ * , which are self-similar solutions of However, due to the fact that v sing (t) ∈ H 1 (R N ) for any t > 0, the arguments in [13]- [16] are not applicable to the critical case. In this paper we study the large time behavior of the function |x| −A e −tL ϕ, instead of e −tL ϕ, with and overcome the difficulty arising from the fact that v sing (t) ∈ H 1 (R N ). As far as we know, this paper is the first one treating the precise large time behavior of e −tL ϕ in the critical case.
(c) Let T > 0 and be a sufficiently small positive constant. Define Then there exists a positive constant C such that In case (S * ) we have: Let N ≥ 2 and assume condition (V ). Let L satisfy (S * ). Let u = u(|x|, t) be a radially symmetric solution of (1.1) such that ϕ ∈ L 2 (R N , e |x| 2 /4 dx).
(i) Let w be as in Theorem 1.1 and K a compact set in R N \ {0}. Then there exists a positive constant C 1 such that where m(ϕ) is as in (1.11). (ii) Let u * , U 2 , F 2 and G 2 be as in Theorem 1.2 with d = 2. Then Furthermore, for any T > 0 and any sufficiently small > 0, there exists a positive constant C 2 such that The function w defined by (1.9) satisfies  and study the large time behavior of w = w(ξ, s) by developing the arguments in a series of papers [11]- [16]. The function ψ d defined by (1.8) is the first eigenfunction of the eigenvalue problem and the corresponding eigenvalue is 0 (see Lemma 2.5). We show that w behaves like a suitable multiple of ψ d as s → ∞. Furthermore, combining the radially symmetry of u with the behavior of w, we prove Theorems 1.1-1.3. The eigenfunction ψ d corresponds to v reg in the subcritical case and v sing in the null-critical case, respectively. In the null-critical case, v reg is transformed by (1.9) into 4 . Hereψ d is the first eigenfunction of the eigenvalue problem In the null-critical case with λ 2 > λ * , we see that 0 < d < 2 and . This justifies that the operator L d has two positive eigenfunctions ψ d andψ d .
The case of d = 0 is on borderline where L V is null-critical and it is not treated in this paper. Indeed, it seems difficult to apply the arguments of this paper to the case of d = 0 since ρ d (ξ) ∼ ξ −1 as ξ → 0 and ρ d ∈ L 1 (R + ).

Nonradial solutions.
We discuss the large time behavior of solutions of (1.1) without the radially symmetry of the solutions.
Let ∆ S N −1 be the Laplace-Beltrami operator on S N −1 . Let {ω k } ∞ k=0 be the eigenvalues of Then ω k = k(N +k −2) for k = 0, 1, 2, . . . . Let {Q k,i } k i=1 and k be the orthonormal system and the dimension of the eigenspace corresponding to ω k , respectively. In particular, Q 0,1 ≡ |S N −1 | −1/2 . For any ϕ ∈ L 2 (R N , e |x| 2 /4 dx), we can find radially symmetric functions {φ k,i } ⊂ L 2 (R N , e |x| 2 /4 dx) such that (see [12] and [13]). Define L k := −∆ + V k (|x|) and V k (r) := V (r) + ω k r −2 . Then for any t > 0. Therefore the behavior of e −tL ϕ is described by a series of the radially symmetric solutions e −tL k φ k,i . (See Section 5.) Furthermore, V k satisfies condition (V) with λ 1 and λ 2 replaced by λ 1 + ω k and λ 2 + ω k , respectively. In particular, L k is subcritical if k ≥ 1. Therefore, applying our results in Section 1.1, we can obtain the precise description of the large time behavior of e −tL ϕ.
As an application of the above argument, we obtain the following result.
The above argument also enables us to obtain the higher order asymptotic expansions of e −tL ϕ. Furthermore, similarly to [11]- [16], it is useful for the study the large time behavior of the hot spots of e −tL ϕ. (See [18].) The rest of this paper is organized as follows. In Section 2 we formulate the definition of the solution of (1.1) and prove some preliminary lemmas. In Section 3 we obtain a priori estimates of radially symmetric solutions of (1.1) by using the comparison principle. In Section 4 we obtain the precise description of the large time behavior of radially symmetric solutions of (1.1) and complete the proofs of Theorems 1.1-1.3. In Section 5, by the argument in Section 1.2 we apply Theorems 1.1-1.3 to prove Theorem 1.4 and Corollary 1.1.

Preliminaries.
We formulate the definition of the solution of (1.1) and obtain some properties related to the operator L. For positive functions f and g defined in (0, R) for some R > 0, we write Similarly, for positive functions f and g defined in (R, ∞) for some R > 0, we write By the letter C we denote generic positive constants and they may have different values also within the same line.

Definition of the solution.
Assume condition (V ) and let L := −∆ + V be nonnegative. In this subsection we consider the Cauchy problem where Problem (P) possesses a unique solution u * such that and we often denote by e −tL * ϕ * the unique solution u * . Since U ∈ C 2 (R N \ {0}) and U > 0 in R N \ {0}, applying the parabolic regularity theorems (see e.g., [20, Chapter IV]) to (P), we see that Proof. By the same argument as in the proof of [12, Lemma 2.1] we obtain assertion (i). We prove assertion (ii). It follows that Then, multiplying (P) by u * e |x| 2 4(1+τ ) and integrating it in for t > 0. Thus assertion (ii) follows. (The proof of assertion (ii) is somewhat formal, however it is justified by use of approximate solutions.) Furthermore, we have: Proof. Let j ∈ {0, 1, 2, . . . } and set v j = ∂ j t u * . By (2.1) it suffices to prove the continuity of v j at (0, t) ∈ R N × (0, ∞). Since v j is radially symmetric, v j satisfies implies thatν is an A 2 weight in a neighborhood of 0 ∈ R N +k . By Lemma 2.1 (i), applying the regularity theorems for parabolic equations with A 2 weight (see e.g., [3] and [11]), we see thatṽ j is continuous at (0, t) ∈ R N +k × (0, ∞). This means that ∂ j t u * is continuous at (0, t) ∈ R N × (0, ∞). Thus Lemma 2.2 follows. We formulate the definition of the solution of (1.1). See also [24] and [25].
Then we say that u is a solution of (1.1) if u * is a solution of (P).
for x ∈ R N and t > 0. On the other hand, by (1.4) and (1.5) we have for t ≥ T (see also (3.7)). These imply (2.4) and Lemma 2.3 follows.

Preliminary lemmas.
We prove a lemma on the decay of U as r → ∞. .2) and (1.5), respectively. In cases (S) and (C) there exists δ > 0 such that It follows from (1.4) and (V) (ii) that if L is subcritical and λ 2 > λ * , can be defined for any r > 0 and satisfies On the other hand, v ± satisfy (2.8) and are linearly independent. Therefore, applying the standard theory for ordinary differential equations, we can find a, b ∈ R such thatṽ(r) = av + (r) + bv − (r) in (0, ∞), that is Assume that L is subcritical. By (1.4), (2.7) and (2. Since Q = (N − 2) 2 + 4λ 2 > 0, by (2.6) we can find δ > 0 such that This implies (2.5) in the subcritical case. Next we assume that L is critical. By (1.4) and (2.7) we see that a = 0 and This together with (2.6) implies that as r → ∞, and (2.5) holds with δ = θ. Thus Lemma 2.4 follows.
At the end of this section we state the following lemma on eigenvalue problem (E).
Proof. We leave the proof to the reader since it is proved by the same argument as in [23, Lemma 2.1].

3.
A priori estimates of radial solutions. Let T > 0 and > 0. Define In this section we prove the following proposition.
(i) There exists C 1 > 0 such that For the proof, we construct supersolutions of problem (P) in D (T ).
Then, for any T > 0 and any sufficiently small > 0, there exists a function W * = W * (x, t) such that Since ζ is monotone decreasing, by (3.5) we have  These imply that Then it follows that 0 ≤ F (x) ≤ C|x| 2 for x ∈ R N . Taking a sufficiently small > 0 if necessary, we obtain This together with (3.6) implies (3.4). Thus Lemma 3.1 follows.
Here c * is as in (1.4), λ := λ 1 − λ 2 and Furthermore, similarly to Lemma 2.4, we see that Then the function F j N given in Proposition 3.1 satisfies SinceṼ (ξ, s) = e s V λ2 (e s 2 ξ), it follows from (V) (ii) that for ξ ∈ (e −θ * s , ∞) and s > 0. Let δ be as in Lemma 2.4. Then, taking a sufficiently small θ > 0 if necessary, we have We prepare some lemmas on estimates of w. for all sufficiently large s > 0.
On the other hand, by Lemma 4.1 (i) we see that (4.7) holds with γ = −d/4. Without loss of generality, we can find j ∈ {0, 1, 2, . . . } such that Next we study the large time behavior ofŵ and prove the following proposition.       for all sufficiently large s > 0. Furthermore, similarly to (4.19), by Lemmas 2.5, 4.3 and 4.4 we obtain for all sufficiently large s > 0. Therefore we deduce from (4.30) and (4.31) that for all sufficiently large s > 0. Since dθ * < dθ < 1 (see (4.6)), by (4.32) we have for all sufficiently large s > 0 and t > 0 with s = log(1 + t).
On the other hand, by (4.28) we apply Proposition 3.1 with D = d/4 and D = 0 to obtain sup 0≤r≤(1+t) for all sufficiently large t > 0. Combining (4.40) with (4.4), we see that for all sufficiently large t > 0. Therefore, by (4.39) and (4.41) we obtain On the other hand, since u * is a radial solution of problem (P), we have for all sufficiently large s > 0. Repeating this argument, we can findθ > 1 such that for all sufficiently large s > 0, instead of (4.32). This implies that for all sufficiently large s > 0. Thus (4.36) holds. Therefore the proof of Proposition 4.2 is complete.
On the other hand, since p(x, y, t) = p(y, x, t), we have for y ∈ R N and τ > 0. Then, applying Theorem 1.4 and letting τ → +0, we obtain the desired results. Thus Corollary 1.1 follows.