Classification of local asymptotics for solutions to heat equations with inverse-square potentials

Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.


Introduction and statement of the main results
We aim to describe the asymptotic behavior near the singularity of solutions to backward evolution equations with inverse square singular potentials of the form (1) u t + ∆u + a(x/|x|) |x| 2 u + f (x, t, u(x, t)) = 0, in R N × (0, T ), where T > 0, N 3, a ∈ L ∞ (S N −1 ) and f : R N × (0, T ) × R → R. Inverse square potentials are related to the well-known classical Hardy's inequality |x| 2 dx, for all u ∈ C ∞ 0 (R N ), N 3, see e.g. [18,20]. Parabolic problems with singular inverse square Hardy potentials arise in the linearization of standard combustion models, see [24]. The properties of the heat operator are strongly affected by the presence of the singular inverse square potential, which, having the same order of homogeneity as the laplacian and failing to belong to the Kato class, cannot be regarded as a lower order term. Hence, singular problems with inverse square potentials represent a borderline case with respect to the classical theory of parabolic equations. Such a criticality makes parabolic equations of type (1) and their elliptic versions quite challenging from the mathematical point of view, thus motivating a large literature which, starting from the pioneering paper by [6], has been devoted to their analysis, see e.g. [18,32] for the parabolic case and [1,29,31] for the elliptic counterpart. In particular, the influence of the Hardy potential in semilinear parabolic problems has been studied in [2], in the case f (x, t, s) = s p , p > 1, and for a(x/|x|) = λ, λ > 0; the analysis carried out in [2] highlighted a deep difference with respect to the classical heat equation (λ = 0), showing that, if λ > 0, there exists a critical exponent p + (λ) such that for p p + (λ), there is no solution even in the weakest sense for any nontrivial initial datum. The present paper is addressed to the problem of describing the behavior of solutions along the directions (λx, λ 2 t) naturally related to the heat operator. Indeed, the unperturbed operator u t + ∆u + a(x/|x|) |x| 2 u is invariant under the action (x, t) → (λx, λ 2 t). Then we are interested in evaluating the asymptotics of u( √ tx, t) as t → 0 + for solutions to (1). Our analysis will show that u( √ tx, t) behaves as a singular self-similar eigenfunction of the Ornstein-Uhlenbeck operator with inverse square potential, multiplied by a power of t related to the corresponding eigenvalue, which can be selected by the limit of a frequency type function associated to the problem.
We consider both linear and subcritical semilinear parabolic equations of type (1). More precisely, we deal with the case f (x, t, s) = h(x, t)s corresponding to the linear problem (2) u t + ∆u + a(x/|x|) with a perturbing potential h satisfying (3) h, h t ∈ L r (0, T ), L N/2 (R N ) for some r > 1, h t ∈ L ∞ loc (0, T ), L N/2 (R N ) , and negligible with respect to the inverse square potential |x| −2 near the singularity in the sense that there exists C h > 0 such that (4) |h(x, t)| C h (1 + |x| −2+ε ) for all t ∈ (0, T ), a.e. x ∈ R N , and for some ε ∈ (0, 2).
We denote as H t ⋆ the dual space of H t and by (Ht) ⋆ ·, · Ht the corresponding duality product.
It will be clear from the parabolic Hardy type inequality of Lemma 2.1 and the Sobolev weighted inequality of Corollary 2.8, that the integral R N f (x, t, u(x, t))φ(x)G(x, t)dx in the above definition is finite for a.e. t ∈ (0, T ), both in the linear case f (x, t, s) = h(x, t)s under assumptions (3)(4) and in the semilinear case f (x, t, s) = ϕ(x, t, s) under condition (5) and for u satisfying (7). We notice that from (12) it follows that v ∈ C 0 ([τ, T ], L), see e.g. [27,Theorem 1.2], where L := L 1 is the completion of C ∞ c (R N ) with respect to the norm v L = R N |v(x)| 2 e −|x| 2 /4 dx 1/2 . Moreover the function is absolutely continuous and 1 2 Ht for a.e. t ∈ (0, T ).
in the sense that, for every φ ∈ H, In particular, if u is a weak solution to (2), whereas, if u is a weak solution to (6), then v(x, t) We give a precise description of the asymptotic behavior at the singularity of solutions to (2) and (6) in terms of the eigenvalues and eigenfunctions of the Ornstein-Uhlenbeck operator with singular inverse square potential (14) L : In order to describe the spectrum of L, we consider the operator −∆ S N −1 − a(θ) on the unit (N − 1)-dimensional sphere S N −1 . For any a ∈ L ∞ S N −1 , −∆ S N −1 − a(θ) admits a diverging sequence of eigenvalues µ 1 (a) < µ 2 (a) · · · µ k (a) · · · , the first of which is simple and can be characterized as (15) µ 1 (a) = min , see [16]. Moreover the quadratic form associated to −∆ − a(x/|x|) is positive definite if and only if see [16,Lemma 2.5]. To each k ∈ N, k 1, we associate a L 2 S N −1 -normalized eigenfunction ψ k of the operator −∆ S N −1 − a(θ) corresponding to the k-th eigenvalue µ k (a), i.e. satisfying In the enumeration µ 1 (a) < µ 2 (a) · · · µ k (a) · · · we repeat each eigenvalue as many times as its multiplicity; thus exactly one eigenfunction ψ k corresponds to each index k ∈ N. We can choose the functions ψ k in such a way that they form an orthonormal basis of L 2 (S N −1 ).
The following proposition describes completely the spectrum of the operator L, thus extending to the anisotropic case the spectral analysis performed in [32, §9.3] in the isotropic case a(θ) ≡ λ; see also [8, §4.2] and [14, §2] for the non singular case.
and µ k (a) is the k-th eigenvalue of the operator −∆ S N −1 −a(θ) on the sphere S N −1 . Each eigenvalue γ m,k has finite multiplicity equal to and a basis of the corresponding eigenspace is ψ j is an eigenfunction of the operator −∆ S N −1 − a(θ) on the sphere S N −1 associated to the j-th eigenvalue µ j (a) as in (17), and P j,n is the polynomial of degree n given by denoting as (s) i , for all s ∈ R, the Pochhammer's symbol (s) i = i−1 j=0 (s + j), (s) 0 = 1.
The following theorems provide a classification of singularity rating of any solution u to (1) based on the limit as t → 0 + of the Almgren type frequency function (see [5,25]), In the linear case f (x, t, u) = h(x, t)u, the behavior of weak solutions to (2) is described by the following theorem.
Theorem 1.5. Let u ≡ 0 be a weak solution to (2) in the sense of Definition 1.1, with h satisfying (3) and (4) and a ∈ L ∞ S N −1 satisfying (16). Then there exist m 0 , k 0 ∈ N, k 0 1, such that where N hu,u is defined in (20) and γ m0,k0 is as in (18). Furthermore, denoting as J 0 the finite set of indices for all τ ∈ (0, 1) there holds where V m,k = V m,k / V m,k L , V m,k are as in (19), An analogous result holds in the semilinear case for solutions to (6) satisfying the further conditions (7) and (8).
The proofs of theorems 1.5 and 1.6 are based on a parabolic Almgren type monotonicity formula combined with blow-up methods. Almgren type frequency functions associated to parabolic equations were first introduced by C.-C. Poon in [25], where unique continuation properties are derived by proving a monotonicity result which is the parabolic counterpart of the monotonicity formula introduced by Almgren in [5] and extended by Garofalo and Lin in [19] to elliptic operators with variable coefficients. A further development in the use of Almgren monotonicity methods to study regularity of solutions to parabolic problems is due to the recent paper [7]. We also mention that an Almgren type monotonicity method combined with blow-up was used in [15] in an elliptic context to study the behavior of solutions to stationary Schrödinger equations with singular electromagnetic potentials. Theorem 1.5 and Theorem 1.6 imply a strong unique continuation property at the singularity, as the following corollary states. Corollary 1.7. Suppose that either u is a weak solution to (2) under the assumptions of Theorem 1.5 or u satisfies (7)(8) and weakly solves (6) under the assumptions of Theorem 1.6. If (28) u(x, t) = O (|x| 2 + t) k as (x, t) → (0, 0) for all k ∈ N, As a byproduct of the proof of Theorems 1.5 and 1.6, we also obtain the following result, which can be regarded as a unique continuation property with respect to time. Proposition 1.8. Suppose that either u is a weak solution to (2) under the assumptions of Theorem 1.5 or u satisfies (7)(8) and weakly solves (6) under the assumptions of Theorem 1.6. If there exists There exists a large literature dealing with strong continuation properties in the parabolic setting. [21] (see too [22]) studies parabolic operators with L N +1 2 time-independent coefficients obtaining a unique continuation property at a fixed time t 0 : the used technique relies on a representation formula for solutions of parabolic equations in terms of eigenvalues of the corresponding elliptic operator and cannot be applied to more general equations with time-dependant coefficients. [26] and [30] use parabolic variants of the Carleman weighted inequalities to obtain a unique continuation property at fixed time t 0 for parabolic operators with time-dependant coefficients. In this direction, it is worth mentioning the work of Chen [8] which contains not only a unique continuation result but also some local asymptotic analysis of solutions to parabolic inequalities with bounded coefficients; the approach is based in recasting equations in terms of parabolic self-similar variables. We also quote [4,9,10,12,13,17] for unique continuation results for parabolic equations with time-dependent potentials by Carleman inequalities and monotonicity methods.
The present paper is organized as follows. In section 2, we state some parabolic Hardy type inequalities and weighted Sobolev embeddings related to equations (2) and (6). In section 3, we completely describe the spectrum of the operator L defined in (14) and prove Proposition 1.4. Section 4 contains an Almgren parabolic monotonicity formula which provides the unique continuation principle stated in Proposition 1.8 and is used in section 5, together with a blow-up method, to prove Theorems 1.5 and 1.6.
Notation. We list below some notation used throughout the paper.
-const denotes some positive constant which may vary from formula to formula.
-dS denotes the volume element on the unit (N − 1)-dimensional sphere S N −1 . -

Parabolic Hardy type inequalities and Weighted Sobolev embeddings
The following lemma provides a Hardy type inequality for parabolic operators. We refer to [25, Proposition 3.1] for a proof.
In the anisotropic version of the above inequality, a crucial role is played by the first eigenvalue µ 1 (a) of the angular operator −∆ S N −1 − a(θ) on the unit sphere S N −1 defined in (15).
The gradient of u can be written in polar coordinates as For all θ ∈ S N −1 , let ϕ θ ∈ C ∞ c ((0, +∞)) be defined by ϕ θ (r) = u(r, θ), and ϕ θ ∈ C ∞ c (R N \ {0}) be the radially symmetric function given by ϕ θ (x) = ϕ θ (|x|). From Lemma 2.1, it follows that where ω N −1 denotes the volume of the unit sphere S N −1 , i.e. ω N −1 = S N −1 dS(θ). On the other hand, from the definition of µ 1 (a) it follows that (31) From (29), (30), and (31), we deduce that The following corollary provides a norm in H t equivalent to · Ht and naturally related to the heat operator with the Hardy potential of equation (1).
Proof. The equality of the two infimum levels follows by the change of variables To prove that they are strictly positive, we argue by contradiction and assume that for every which, by Lemma 2.2, implies that By continuity of the map a → µ 1 (a) with respect to the L ∞ S N −1 -norm, letting ε → 0 the above inequality yields µ 1 (a) + (N −2) 2 4 0, giving rise to a contradiction with (16).
The above results combined with the negligibility assumption (4) on h allow estimating the quadratic form associated to the linearly perturbed equation (2) for small times as follows.
In order to estimate the quadratic form associated to the nonlinearly perturbed equation (6), we derive a Sobolev type embedding in spaces H t . To this purpose, we need the following inequality, whose proof can be found in [11,Lemma 3].
Lemma 2.5. For every u ∈ H, |x|u ∈ L and Corollary 2.6. For every u ∈ H t , there holds From Lemma 2.5 and classical Sobolev embeddings, we can easily deduce the following weighted Sobolev inequality (see also [14]).
The stated inequality follows from classical Sobolev inequalities and Lemma 2.5.
The change of variables u(x) = v(x/ √ t) in Lemma 2.7, yields the following inequality in H t .
Corollary 2.8. For every t > 0, u ∈ H t , and 2 s 2 * , there holds The above Sobolev estimate allows proving the nonlinear counterpart of Corollary 2.4.
Proof. From (5), Hölder's inequality, and Corollary 2.8, we have that, for all u ∈ H t ∩L p+1 (R N ), there holds with C p+1 as in Corollary 2.8. The stated inequality follows from Corollary 2.3 and (33) by choosing t sufficiently small depending on u L p+1 (R N ) .

Spectrum of Ornstein-Uhlenbeck type operators with inverse square potentials
In this section we describe the spectral properties of the operator L defined in (14), extending to anisotropic singular potentials the analysis carried out in [32] for a ≡ λ constant. Following [14], we first prove the following compact embedding.
From Lemma 2.5 and boundedness of u k in H, we deduce that Combining (34), (35), and (36), we obtain that u k → u strongly in L.
From classical spectral theory we deduce the following abstract description of the spectrum of L. (16) holds. Then the spectrum of the operator L defined in (14) consists of a diverging sequence of real eigenvalues with finite multiplicity. Moreover, there exists an orthonormal basis of L whose elements belong to H and are eigenfunctions of L.
Proof. By Corollary 2.3 and the Lax-Milgram Theorem, the bounded linear self-adjoint operator is well defined. Moreover, by Lemma 3.1, T is compact. The result then follows from the Spectral Theorem.
Let us now compute explicitly the eigenvalues of L with the corresponding multiplicities and eigenfunctions by proving Proposition 1.4. Proof of Proposition 1.4. Assume that γ is an eigenvalue of L and g ∈ H\{0} is a corresponding eigenfunction, so that in a weak H-sense. From classical regularity theory for elliptic equations, g ∈ C 1,α loc (R N \ {0}). Hence g can be expanded as Equations (17) and (37) imply that, for every k, and, by the Hardy type inequality of Lemma 2.1, Therefore, w k is a solution of the well known Kummer Confluent Hypergeometric Equation (see [3] and [23]). Then there exist Here M (c, b, t) and, respectively, U (c, b, t) denote the Kummer function (or confluent hypergeometric function) and, respectively, the Tricomi function (or confluent hypergeometric function of the second kind); M (c, b, t) and U (c, b, t) are two linearly independent solutions to the Kummer Confluent Hypergeometric Equation Since N 2 − α k > 1, from the well-known asymptotics of U at 0 (see e.g. [3]), we have that for some const = 0 depending only on N, γ, and α k . On the other hand, M is the sum of the series We notice that M has a finite limit at 0 + , while its behavior at ∞ is singular and depends on the t is a polynomial of degree m in t, which we will denote as P k,m , i.e., If α k 2 + γ ∈ N, then from the well-known asymptotics of M at ∞ (see e.g. [3]) we have that for some const = 0 depending only on N, γ, and α k . Now, let us fix k ∈ N, k 1. From the above description, we have that for some const = 0, and hence for some const = 0. Therefore, condition (40) can be satisfied only for for some const = 0, and hence for some const = 0. Therefore, condition (39) can be satisfied only for solves (38); moreover the function belongs to H, thus providing an eigenfunction of L. We can conclude from the above discussion that if α k 2 + γ ∈ N for all k ∈ N, k 1, then γ is not an eigenvalue of L. On the other hand, if there exist k 0 , m 0 ∈ N, k 0 1, such that and a basis of the corresponding eigenspace is The proof is thereby complete.
and γ m,k = k 2 + m. Hence, in this case we recover the well known fact (see e.g. [8] and [14]) that the eigenvalues of the Ornstein-Uhlenbeck operator −∆ + x 2 · ∇ are the positive half-integer numbers.

By Lemma 3.2, it follows that
is an orthonormal basis of L.
ai ∈ H t and then, since u is a solution to (1) in the sense of of Definition 1.1, for a.e. t ∈ (a i , b i ) we have Therefore, thanks to (i) and (iii), we obtain By the change of variables s = t − a i , we conclude that u i (x, t) = u(x, t + a i ) is a weak solution to (44) in R N ×(0, 2α) in the sense of Definition 1.1. By a further change of variables, we easily obtain that v i (x, t) := u i ( √ tx, t) is a weak solution to (45) in R N × (0, 2α) in the sense of Remark 1.3.
Proof. It follows from Lemma 4.1 and Remark 1.2.
Proof. From Lemma 4.2 and Corollaries 2.4 and 2.9, taking into account that 2α < T , we have that, for all t ∈ (0, 2α), which implies belongs to W 1,1 loc (0, T i ) and its weak derivative is, for a.e. t ∈ (0, T i ), as follows: Proof. Let us first consider case (I), i.e. f (x, t, u) = h(x, t)u, with h(x, t) under conditions (3)(4). We test equation (45) Since, in view of (3-4) and Lemmas 2.1 and 2.5, the integrals in the last two lines of the previous formula are finite for every t ∈ (0, T i ), we conclude that The change of variables u i (x, t) = v i (x/ √ t, t) leads to the conclusion in case (I). Let us now consider case (II), i.e. f (x, t, u) = ϕ(x, t, u) with ϕ satisfying (5) and u satisfying (7)(8). We test equation (45) with (v i ) t (passing through a suitable approximation) and, by Corollary 2.9, we obtain, for all t ∈ (0, T i ), Since in view of hypothesis (5) on ϕ, conditions (7) and (8) on u, and Lemma 2.7 the integrals at the right hand side lines of the previous formula are finite for every t ∈ (0, T i ), we conclude that (v i ) t ∈ L 2 (τ, T i ; L) for all τ ∈ (0, T i ).

The change of variables
Integration by parts yields (these formal computations can be made rigorous through a suitable approximation) thus yielding the conclusion in case (II).
For all i = 1, . . . , k, let us introduce the Almgren type frequency function associated to u i (50) Frequency functions associated to unperturbed parabolic equations of type (1) (i.e. in the case f (x, t, s) ≡ 0) were first studied by C.-C. Poon in [25], where unique continuation properties are derived by proving monotonicity of the quotient in (50). Due to the presence of the perturbing function f (x, t + a i , u(x, t)), the functions N i will not be nondecreasing as in the case treated by Poon; however in both cases (I) and (II), we can prove that their derivatives are integrable perturbations of nonnegative functions wherever the N i 's assume finite values. Moreover our analysis will show that actually the N i 's assume finite values all over (0, 2α).
and ν 2i is as follows: Proof. From Lemma 4.2 and 4.5, it follows that N i ∈ W 1,1 loc (β i , T i ). From (49) we deduce that , which yields the conclusion in view of (47), (48), and Lemma 4.5.
The term ν 2i can estimated as follows.
Proof of Proposition 1.8. It follows immediately from Corollary 4.11.
The following result clarifies the behavior of N (t) as t → 0 + .
Proof. We first observe that N (t) is bounded from below in (0, 2α). Indeed from Corollaries 2.4 and 2.9, we obtain that, for all t ∈ (0, 2α), and hence Let T 0 as in (67). By Schwarz's inequality, ν 1 (t) 0 for a.e. t ∈ (0, T 0 ). Furthermore, from Lemmas 4.7 and 4.8, ν 2 belongs to L 1 (0, T 0 ). In particular, N ′ (t) turns out to be the sum of a nonnegative function and of a L 1 function over (0, T 0 ). Therefore, admits a limit as t → 0 + which is finite in view of (68) and Lemma 4.8.
To prove the claim, we notice that (89) allows passing to the limit in (82). Therefore, in view of (83) and (84) which ensure the vanishing at the limit of the perturbation term, for all φ ∈ H and a.e. t ∈ (0, 1), i.e. w is a weak solution to Testing the difference between (82) and (91) with ( w λn k − w) and integrating with respect to t between τ and 1, we obtain Then, from (83), (85), and (87), we obtain that, for all τ ∈ (0, 1), is a weak solution (in the sense of Definition 1.1) of (93) w t + ∆w + a(x/|x|) |x| 2 w = 0.
In the following lemma, we prove that lim t→0 + t −2γ H(t) is indeed strictly positive.
From (1) and the fact that V m,k (x) is an eigenfuntion of the operator L associated to the eigenvalue γ m,k defined in (18), it follows that Therefore, we have that a.e. and distributionally in (0, √ T 0 ). By integration, we obtain, for all λ,λ ∈ (0,  (41), and an orthonormal basis of E 0 is given by { V m,k : (m, k) ∈ J 0 }. In order to estimate ξ m,k , we distinguish between case (I) and case (II).
We now complete the description of the asymptotics of solutions by combining Lemmas 5.1 and 5.3 and obtaining some convergence of the blowed-up solution continuously as λ → 0 + and not only along subsequences, thus proving Theorems 1.5 and 1.6.
We now prove that the β n,j 's depend neither on the sequence {λ n } n∈N nor on its subsequence {λ n k } k∈N . Let us fix Λ ∈ (0, √ T 0 ) and define u m,i and ξ m,i as in (109-110). By expanding u λ (x, 1) = u(λx, λ 2 ) ∈ L in Fourier series as in (108), from (128) it follows that, for any (m, i) ∈ J 0 , (129) λ −2γ n k u m,i (λ n k ) → In particular the β m,i 's depend neither on the sequence {λ n } n∈N nor on its subsequence {λ n k } k∈N , thus implying that the convergences in (126) and (127) actually hold as λ → 0 + and proving the theorems.
The strong unique continuation property is a direct consequence of Theorems 1.5 and 1.6.
Proof of Corollary 1.7. Let us assume by contradiction that u ≡ 0 in R N × (0, T ) and fix k ∈ N such that k > γ, with γ = γ m0,k0 as in Theorems 1.5 and 1.6. From assumption (28), it follows that, for a.e. (x, t) ∈ R N × (0, 1), On the other hand, from Theorems 1.5 and 1.6, it follows that there exists g ∈ H \ {0} such that g is an eigenfunction of the operator L associated to γ and, for all t ∈ (0, 1) and a.e. x ∈ R N , which, in view of (131), implies g ≡ 0, a contradiction.
Aknowledgements. The authors would like to thank Prof. Susanna Terracini for her interest in their work and for helpful comments and discussions.