UNCONDITIONAL UNIQUENESS AND NON-UNIQUENESS FOR HARDY-H´ENON PARABOLIC EQUATIONS

. We study the problems of uniqueness for Hardy-H´enon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (H´enon type) in the nonlinear term. To deal with the Hardy-H´enon type nonlinearities, we employ weighted Lorentz spaces as solution spaces. We prove unconditional uniqueness and non-uniqueness, and we establish uniqueness criterion for Hardy-H´enon parabolic equations in the weighted Lorentz spaces. The results extend the previous works on the Fujita equation and Hardy equations in Lebesgue spaces.


Introduction and main results
1.1. Introduction and our setting. We consider the Cauchy problem of the Hardy-Hénon parabolic equation where T > 0, d ∈ N, γ ∈ R, α > 1, q ∈ [1, ∞], r ∈ (0, ∞] and s ∈ R. Here, ∂ t := ∂ ∂t is the time derivative, ∆ := d j=1 is the Laplace operator on R d , u = u(t, x) is an unknown complex-valued function on (0, T ) × R d , u 0 = u 0 (x) is a prescribed complex-valued function on R d , and L q,r s (R d ) is the weighted Lorentz space (see Definition 2.3), which includes the Lebesgue space L q (R d ) = L q,q 0 (R d ) as a special case r = q and s = 0. The equation (1.1) in the case γ = 0 is the Fujita equation, which has been extensively studied in various directions. The equation (1.1) with γ < 0 is known as a Hardy parabolic equation, while that with γ > 0 is known as a Hénon parabolic equation. The corresponding stationary problem to (1.1), that is, − ∆U = |x| γ |U | α−1 U, (1.2) was proposed by Hénon as a model to study the rotating stellar systems (see [19]), and has also been extensively studied in the mathematical context, especially in the fields of nonlinear analysis and variational methods (see [14] for example).
In this paper we study the problem on unconditional uniqueness and non-uniqueness for (1.1) in weighted Lorentz spaces L q,r s (R d ). Here, unconditional uniqueness means uniqueness of the solution to (1.1) in the sense of the integral form u(t) = e t∆ u 0 + t 0 e (t−τ )∆ (| · | γ |u(τ )| α−1 u(τ )) dτ (1.3) in L ∞ (0, T ; L q,r s (R d )) or C([0, T ]; L q,r s (R d )), where T > 0 and {e t∆ } t>0 is the heat semigroup. We say that non-uniqueness holds for (1.1) if unconditional uniqueness fails. In contrast, we say that conditional uniqueness holds if uniqueness of the solution to (1.1) holds in the entire space with some auxiliary function spaces. In addition, we also study uniqueness criterion which is a necessary and sufficient condition on the Duhamel term (i.e. the second term in the right-hand side of (1.3)) for uniqueness to hold.
Let us here state previous works on uniqueness for (1.1). For (1.1) with γ ≤ 0, the problem on uniqueness has been well studied (see [3, 4, 7-9, 18, 27, 29, 36, 38, 42, 43] for example). In the study of unconditional uniqueness for (1.1) in Lebesgue spaces L q (R d ) or Lorentz spaces L q,r (R d ), the following two critical exponents are known to be important. The first one is the so-called scale-critical exponent q c given by and we say that the problem (1.1) is scale-critical if q = q c , scale-subcritical if q > q c , and scale-supercritical if q < q c . The second one is the critical exponent Q c given by Q c = Q c (d, γ, α) := dα d + γ , (1.5) which is related to well-definedness of the Duhamel term in (1.3) in L q,r (R d ). In fact, the nonlinear term |x| γ |u| α−1 u ∈ L 1 loc (R d ) for any u ∈ L q,r (R d ) if and only if "q > Q c " or "q = Q c and r ≤ α". In the case γ = 0, unconditional uniqueness for (1.1) in C([0, T ]; L q (R d )) was proved in the double subcritical case q > max{q c , Q c } by Weissler [42] and in the single critical cases q = Q c > q c and q = q c > Q c by Brezis and Cazenave [7]. In the double critical case q = q c = Q c , non-uniqueness was proved for some initial data u 0 ∈ L q (R d ) by Terraneo [38], and then, for any initial data u 0 ∈ L q (R d ) by Matos and Terraneo [27]. In [38], uniqueness criterion was also obtained in the double critical case. In the scale-supercritical case q < q c , non-uniqueness for (1.1) was proved for initial data u 0 = 0 by Haraux and Weissler [18]. Uniqueness and non-uniqueness have also been studied for heat equations with exponential nonlinearities (see [21,23] and references therein). In the Hardy case − min{2, d} < γ < 0, similar results were obtained by [4,36], where the Lorentz spaces L q,r (R d ) is used to study unconditional uniqueness in the critical case q = Q c in [36]. In contrast, the Hénon case γ > 0 has not been well studied. This is due to the difficulty of treating the increasing potential |x| γ in the nonlinear term at infinity. To overcome this difficulty, the weighted spaces are effective, and recently, conditional uniqueness was obtained in L q s (R d ) = L q,q s (R d ) in [10]; however, unconditional uniqueness and non-uniqueness are completely open. The main purpose of this paper is to prove unconditional uniqueness, non-uniqueness and uniqueness criterion for (1.1) with all γ > − min{2, d}, including the Hénon case, in L q,r s (R d ). is the Hardy-Sobolev exponent. Table 1 and Table 2 summarize the previous results on uniqueness for (1.1) with γ ≤ 0.
1.2. Statement of the results. To describe our results, let us give some definitions and notation. For T ∈ (0, ∞] and a quasi-normed space X , we denote by L ∞ (0, T ; X) the space of functions u : (0, T ) → X such that u L ∞ (0,T ;X) := sup t∈(0,T ) u(t) X < ∞, and by C([0, T ]; X) the space of functions u ∈ L ∞ (0, T ; X) such that u(t) X is continuous in t ∈ [0, T ]. The space L q,r s (R d ) is defined as the completion of C ∞ 0 (R d ) with respect to · L q,r s (see Definition 2.3). Definition 1.1. Let T > 0 and X = L q,r s (R d ) or L q,r s (R d ). We say that a function u = u(t, x) on (0, T ) × R d is a mild solution to (1.1) with initial data u 0 ∈ X in C([0, T ]; X) (L ∞ (0, T ; X) resp.) if u belongs to C([0, T ]; X) (L ∞ (0, T ; X) resp.) and satisfies the integral equation (1.3) for almost everywhere (t, x) ∈ (0, T ) × R d .
We define two critical cases in the framework of L q,r s (R d ) in a similar manner to q c and Q c , respectively. The equation (1.1) is invariant under the following scale transformation: u λ (t, x) := λ 2+γ α−1 u(λ 2 t, λx), λ > 0. More precisely, if u is a solution to (1.1), then so is u λ with the rescaled initial data λ 2+γ α−1 u 0 (λx). Moreover, we calculate qc ) u 0 L q,r s , λ > 0. Hence, if q and s satisfy s d then u λ (0) L q,r s = u 0 L q,r s for any λ > 0, i.e., the norm u λ (0) L q,r s is invariant with respect to λ. Therefore, we say that the problem (1.1) is scale-critical if s d + 1 q = 1 qc , scale-subcritical if s d + 1 q < 1 qc , and scale-supercritical if s d + 1 q > 1 qc . Another critical case is when the following holds: This is related to local integrability of the nonlinear term |x| γ |u| α−1 u. In fact, which is often referred to the Serrin exponent (see [34,35] and also [16]). The exponents α * , q c and Q c are related as follows: In our results on unconditional uniqueness below, we assume that (1.10) Our results on unconditional uniqueness are the following: . Let T > 0, and let d, γ, α, q, r, s be as in (1.10). Assume either (1) or (2): . Theorem 1.3 (Scale-critical case). Let T > 0, and let d, γ, α, q, r, s be as in (1.10). Assume d ≥ 3, q = ∞, and either (1) or (2): Qc . Then unconditional uniqueness holds for (1.1) in C([0, T ]; L q,r s (R d )). Remark 1.4. In Theorem 1.2 (1), the condition "r ≤ α if q = α" comes from the restriction on parameters in linear estimates. More precisely, the condition is due to the restriction r 1 = 1 for linear estimates with q 1 = 1 in Proposition 3.1 (see (3.4) and also Lemma 4.1 (ii)).
Next, we consider the following two cases where the unconditional uniqueness is not obtained in the above theorems: r > α in the single critical case I; r > α * − 1 in the double critical case.
In the single critical case I, the condition r ≤ α naturally appears from the viewpoint of well-definedness of mild solutions to (1.1) as seen in (1.8). On the other hand, when r > α, we can define mild solutions to (1.1) with the auxiliary condition and we know that conditional uniqueness holds (see [10,Theorem 1.13]). We are interested in the questions whether unconditional uniqueness holds in this case or whether the conditional uniqueness can be improved. Unfortunately, we do not know if unconditional uniqueness holds in this case. However, we can give the following sufficient condition for uniqueness to hold which improves the conditional uniqueness [10, Theorem 1.13]. Proposition 1.5. Let T > 0, and let d, γ, α, q, r, s be as in (1.10). Assume that α < α * , q = ∞, α < r ≤ ∞, and s d + 1 Here, r is the Hölder conjugate of r , i.e., 1 = 1 r + 1 r . In the double critical case, we prove the result on non-uniqueness for (1.1) if α * − 1 < r ≤ ∞. More precisely, we have the following: . By Theorem 1.3 (2) and Theorem 1.6, we reveal that the exponent r = α * − 1 is a threshold of dividing unconditional uniqueness and non-uniqueness for (1.1) in the double critical case. In Theorem 1.6, one of two different solutions is regular and the other is singular at x = 0 (see Section 5), and this threshold comes essentially from the logarithmic rate of the singularity at x = 0 of the singular solution (see Theorem 5.5). In addition, we give the following uniqueness criterion. Theorem 1.7. Let T > 0, and let d, γ, α, q, r, s be as in (1.10) (1.12) . Remark 1.8. The exponent r = α * − 1 in (1.12) of Theorem 1.7 is optimal for the same reason as above (see Theorem 5.4).
In the scale-supercritical case, we have the following result on non-uniqueness for (1.1). Here, we define the exponents α F and α HS by which are often referred to as the Fujita exponent (see [32,33]) and the critical Hardy-Sobolev exponent (see [26]).
Then the equation (1.1) has a global positive solution in C([0, ∞); L q,r s (R d )) with initial data 0. To visually understand our above results, we give Figure 2 for the case γ < 0 and min{ 1 qc , 1 Qc } < 1 qc .
Herein, we compare our results with previous ones. Our results generalize the previous works [4,7,18,27,36,38,42], since s can be taken as s = 0 if γ ≤ 0 in our results. More precisely, our results on unconditional uniqueness (Theorem 1.  . The figure shows the domain of (α, s) for d ≥ 3 and q > 1. Here, α 0 := min{1, 1 + γ d }, α F , α * , α HS are given in Figure 1, s c , S c are given in (1.14), and s * := d − 2 − d q . Table 3 and Table 4  To easily compare our results with the previous work [10] which includes the Hénon case γ > 0, we can rewrite our results by using the two critical exponents on s: 14) The exponents s c and S c correspond to q c and Q c in the case without weights, respectively. In fact, we can see that . The results in [10] show local well-posedness, including the conditional uniqueness, for (1.1) if s ≤ s c and non-existence of positive mild solution to (1.1) for some initial data u 0 ≥ 0 if s > s c . However, unconditional uniqueness and non-uniqueness are not mentioned in [10]. Our results are summarized in Figure 3.
This paper is organized as follows. In Section 2, we summarize the definitions and fundamental lemmas on Lorentz spaces and weighted Lorentz spaces. In Section 3, we establish the two kinds of weighted linear estimates. In Subsection 3.1, we extend the usual L q 1 -L q 2 estimates to the weighted Lorentz spaces, which are fundamental tools in this paper. In Subsection 3.2, we prove a certain space-time estimate in the weighted Lorentz spaces. We call it the weighted Meyer inequality. This inequality corresponds to a certain endpoint case of the weighted Strichartz estimates, and it is an important tool in studying the scale-critical case. In Section 4, we prove our results on unconditional uniqueness and uniqueness criterion (Theorem 1.2, Theorem 1.3, Proposition 1.5 and Theorem 1.7), based on the weighted linear estimates. In Section 5, we prove our result on non-uniqueness (Theorem 1.6). In Section 6, we discuss the non-uniqueness in the scale-supercritical case and prove Proposition 1.9. In Section 7, we give a remark on the number of solutions in the double critical case, and additional results on the critical singular case γ = − min{2, d} and the exterior problem on domains not containing the origin.

Weighted Lorentz spaces
In this paper we use the symbols a b and b a for a, b ≥ 0 which mean that there exists a constant C > 0 such that a ≤ Cb. The symbol a ∼ b means that a b and b a happen simultaneously. Let Ω be a domain in R d . We denote by C ∞ 0 (Ω) the set of all C ∞ -functions having compact support in Ω, and by L 0 (Ω) the set of all Lebesgue measurable functions on Ω. We define the distribution function d f of a function f by where |A| denotes the Lebesgue measure of a set A. Definition 2.1. For 0 < q, r ≤ ∞, the Lorentz space L q,r (Ω) is defined by where f * is the decreasing rearrangement of f given by We refer to [15] for the properties of the distribution function, the decreasing rearrangement and the Lorentz space.
(i) The weighted Lebesgue space L q s (Ω) is defined by L q s (Ω) := f ∈ L 0 (Ω) ; f L q s < ∞ endowed with a quasi-norm The space L q s (Ω) is defined as the completion of C ∞ 0 (Ω) with respect to · L q s (Ω) . (ii) The weighted Lorentz space L q,r s (Ω) is defined by L q,r s (Ω) := f ∈ L 0 (Ω) ; f L q,r s (Ω) < ∞ endowed with a quasi-norm f L q,r s (Ω) := | · | s f L q,r (Ω) . The space L q,r s (Ω) is defined as the completion of C ∞ 0 (Ω) with respect to · L q,r s (Ω) . Only when Ω = R d , we omit Ω and we write · L q,r s = · L q,r s (R d ) for simplicity.
Remark 2.4. There are several ways to define weighted Lorentz spaces. For example, the definitions in [11,13,24] are different from ours.
Lemma 2.6 (Generalized Hölder's inequality). Let 0 < q, q 1 , q 2 < ∞ and 0 < r, r 1 , r 2 ≤ ∞. Then the following assertions hold: then there exists a constant C > 0 such that f g L q,r ≤ C f L q 1 ,r 1 g L q 2 ,r 2 for any f ∈ L q 1 ,r 1 (R d ) and g ∈ L q 2 ,r 2 (R d ). (ii) There exists a constant C > 0 such that f g L q,r ≤ C f L q,r g L ∞ for any f ∈ L q,r (R d ) and g ∈ L ∞ (R d ).
Lemma 2.7 (Generalized Young's inequality). Let 1 < q, q 1 , q 2 < ∞ and 0 < r, r 1 , r 2 ≤ ∞. Then the following assertions hold: then there exists a constant C > 0 such that then there exists a constant C > 0 such that Lemmas 2.6 and 2.7 are originally proved by O'Neil [31] (see also Yap [44] for Lorentz spaces with second exponents less than one). Lemma

Linear estimates
In this section, we summarize linear estimates for the heat semigroup in the weighted Lorentz spaces.

3.1.
Smoothing and time decay estimates in weighted spaces. Let {e t∆ } t>0 be the heat semigroup whose element is defined by Here, * denotes the convolution operator and S (R d ) denotes the space of tempered distributions on R d . We use the notation · X→Y for the operator norm from a quasi-normed space X to another one Y , i.e., for an operator T from X into Y . In this subsection, we prove the following: Remark 3.2. The estimate (3.1) can be also obtained for 0 < q 2 ≤ 1. More precisely, let d ∈ N, 1 ≤ q 1 ≤ ∞, 0 < q 2 ≤ 1, 0 < r 1 , r 2 ≤ ∞ and s 1 , s 2 ∈ R, and assume (3.2)-(3.7) with the additional condition Then we have (3.1) for any t > 0. The additional condition (3.8) is required due to use of the embedding On the other hand, we can also prove the necessity of (3.2)-(3.7), but we do not know if (3.8) is necessary. The proof is similar to that of Proposition 3.1, and we omit it. In the proofs of the nonlinear estimates (Lemmas 4.1, 4.2, 5.2 and 5.9), we do not use the case 0 < q 2 ≤ 1.
Remark 3.3. The estimates (3.1) are known in some particular cases, for example, the case s 2 = 0 in Lebesgue spaces in [4], the case s 2 ≥ 0 in Lorentz spaces in [37], and the case q 1 ≤ q 2 in Lebesgue spaces in [30,39] (see also [10]). Similar estimates are proved in Herz spaces and weak Herz spaces in [30,39]. However, this does not change the results in [36] as the endpoint case is not used in [36].
To reduce (3.1) for e t∆ into that for e ∆ , we give the following lemma.
for any t > 0. Since for t > 0 and x ∈ R d , we have Hence, (3.9) is proved.
Proof of the necessity part of Proposition 3.1. For the condition (3.7), see Remark 2.5 (b).
Step 1: and (3.4). If either of these fails, then and (3.5). Suppose either of these fails, i.e., Hence, it is impossible to obtain (3.1). The case Hence, e t∆ f → 0 in S (R d ) as t → 0. Combining this with the continuity e t∆ f → f in S (R d ) as t → 0, we have f = 0 by uniqueness of the limit. However, this is a contradiction to f = 0. Thus, The proof is based on the translation argument as in [11,37]. In fact, take a non-negative function By making the changes of variables, we have (3.10) The weight | · τ + x 0 | s 2 has the uniform lower bounds with respect to sufficient large τ : is obtained, we deduce from (3.10), (3.11) and positivity of e ∆ f that s 2 ≤ s 1 . Therefore, it is enough to show (3.11). In the case s 1 ≥ 0, we have the uniform upper bound which implies (3.11). In the other case s 1 < 0, the weight has a singularity only at x = x * (τ ) = (−τ, 0, . . . , 0), and is increasing with respect to τ for each x ∈ {x 1 ≥ 0}. Here, we note that | · τ + x 0 | s 1 f ∈ L q 1 ,r 1 for any τ > 0, since the singular points x * (τ ) are not included in supp f for any τ > 0. Hence, and we can use the monotone convergence theorem (Lemma A.2) to obtain This implies (3.11). Thus, the necessity of s 2 ≤ s 1 is proved.
Since f is a positive, radially symmetric and decreasing function, we have for |x| ≥ 1 sufficiently large. If (3.6) fails, then f ∈ L q 2 ,r 2 s 2 (R d ), which implies e ∆ f ∈ L q 2 ,r 2 s 2 (R d ) by (3.13). The proof of the necessity part is finished. Proof of the sufficiency part of Proposition 3.1. By Lemma 3.5, it is enough to prove (3.1) with t = 1: (3.14) We start the proof with the case 1 < q 1 , q 2 < ∞. We first prove (3.14) with the non-endpoint case: From Lemma 3.5 and the embedding L q 1 , We divide the proof into three cases: In the case s 2 ≥ 0, we use the inequality |x| ( 3.16) Then we use Lemma 2.6 (i) and Lemma 2.7 (i) to estimate Here, we note that such p 1 , p 2 , p 3 and p 4 exist if (3.15) and s 2 ≥ 0 hold. Hence, (3.14) is proved in this case.
In the case s 2 < 0 ≤ s 1 , we use Lemma 2.6 (i) to obtain Then it has been proved above that Thus, the case s 2 < 0 ≤ s 1 is also proved.
In the case s 1 < 0, setting g := |x| s 1 |f |, and using the inequality Then we use Lemma 2.6 (i) and Lemma 2.7 (i) to estimate Here, we note that such p 6 , p 7 , p 8 and p 9 exist if (3.14) and s 1 < 0 hold. Thus, the case s 1 < 0 is also proved.
Next, we consider the endpoint cases (3.4), (3.5) or (3.6) with 1 < q 1 , q 2 < ∞. Here, we give only sketch of proofs of single endpoint cases. If two or more endpoints overlap, simply combine them.
As to the case (3.4), i.e., s 1 d + 1 q 1 = 1 and r 1 ≤ 1, we note that s 1 ≥ 0, and the proof is almost the same as the non-endpoint case (3.15) with s 1 ≥ 0. In fact, we can take , p 3 = q 2 and p 4 = 1, and use Lemma 2.7 (iii) (instead of Lemma 2.7 (i)) in (3.18), where f L p 4 ,∞ is replaced by f L 1 and the restriction r 1 ≤ 1 appears.
As to the case (3.5), i.e., s 2 d + 1 q 2 = 0 and r 2 = ∞, we note that s 2 < 0, and the proof is similar to the non-endpoint case (3.15) with s 2 < 0 ≤ s 1 or s 2 ≤ s 1 < 0. For s 2 < 0 ≤ s 1 , we use Lemma 2.6 (ii) to obtain will be given later (see the proof of the case q 2 = ∞ below). For s 2 ≤ s 1 < 0, we also have As to the case (3.6), i.e., s 1 d + 1 q 1 = s 2 d + 1 q 2 and r 1 ≤ r 2 , we can use Lemma 2.7 (iii) to make a similar argument to the non-endpoint case. In fact, when s 2 ≥ 0, this case corresponds to taking p 1 = 1, .17) and (3.18). In particular, in (3.17), Lemma 2.7 (iii) is used and the restriction r 1 ≤ r 2 is required: The case s 2 < 0 is similar, and we may omit it.
The case 1 < q 1 < ∞ and q 2 = ∞ is the estimate (3.14) with Since s 2 ≥ 0, this case is proved in a similar way to (3.17) and (3.18). In fact, we deduce from Lemma 2.7 (ii) and Lemma 2.6 (i) that where 1 ≤ p 10 < d s 2 and d d−s 2 < p 11 ≤ ∞ satisfy 1 = 1 p 10 + 1 p 11 and 1 p 11 = s 1 −s 2 d + 1 q 1 , and where d d−s 2 ≤ p 12 < ∞ and 1 < p 13 ≤ d s 2 satisfy 1 = 1 p 12 + 1 p 13 and 1 p 12 = s 1 d + 1 q 1 . Here, we note that such p 10 , p 11 , p 12 and p 13 exist if 0 ≤ s 2 ≤ s 1 and s 1 d + 1 q 1 < 1. For the case s 1 d + 1 q 1 = 1, the first term can be estimated in the same way as (3.17) (where we take p 8 = d s 2 and p 9 = d d−s 2 ). For the second term, we take p 10 = ∞ and p 11 = 1 and we use Lemma 2.7 (ii) to obtain Thus, the estimate (3.14) is proved in the case q 2 = ∞.
The case q 1 = 1 and 1 < q 2 < ∞ is the estimate (3.14) with The proof is similar to (3.22) and (3.23). Let s 2 d + 1 q 2 > 0. As to the first term, it follows from Lemma 2.6 (i) and Lemma 2.7 (ii) that The second term can be estimated as Here, we note that such p 14 and p 15 exist if s 2 ≤ s 1 ≤ 0 and 0 < s 2 d + 1 q 2 < s 1 d + 1. For the case s 2 d + 1 q 2 = 0, the first term can be estimated in the same way as (3.27) (where we take p 14 = − d s 1 ). For the second term, we we have only to take p 15 = ∞ and r 2 = ∞ and use Young's inequality f * g L ∞ ≤ f L 1 g L ∞ . Thus, the estimate (3.14) is proved in the case q 1 = 1 and 1 < q 2 < ∞. The proof of Proposition 3.1 is finished.

Weighted Meyer inequality.
In this subsection, we shall prove the following proposition, which is a key tool to study unconditional uniqueness and uniqueness criterion in the scale-critical case and the construction of a singular solution in the double critical case. Proposition 3.6. Let T ∈ (0, ∞], and let d ≥ 3, 1 ≤ q 1 ≤ ∞, 0 < q 2 < ∞, 0 < r 1 ≤ ∞ and s 1 ,  for any t ∈ (0, T ) and f ∈ L ∞ (0, T ; L q 1 ,r 1 s 1 (R d )). The case s 1 = s 2 = 0 is known as Meyer's inequality and is proved by Meyer [28] (see also [38]).
Proof. We shall prove only the case q 1 > 1 and s 1 d + 1 q 1 < 1, since the proofs of the other cases are similar. By the argument in [28], it suffices to prove that where we define and we may assume that sup without loss of generality. Let λ ∈ (0, ∞) be arbitrarily fixed. For τ ∈ (0, ∞), which is to be determined later, we divide g into two parts: Let p 0 and p 1 be such that Then, by Proposition 3.1 and Remark 3.2, we have Now, the definition of the Lorentz norms yields and similarly, which implies (3.35). Thus, we conclude Proposition 3.6.

Unconditional uniqueness and uniqueness criterion
In this section, we prove Theorem 1.
Then we have the following nonlinear estimates, which are used to prove unconditional uniqueness in the double subcritical case and in the single critical case I.
In addition, we prepare the nonlinear estimates of the following type. These estimates are used to prove uniqueness criterion in the single critical case I, and unconditional uniqueness and uniqueness criterion in the scale-critical case.
For the assertion (iii), the proof can be done in the same way as the above (ii), but the endpoint case σ d + α * q = 1 appears in the proof. For this, we use Proposition 3.6 with the endpoint case (3.32), which requires the stronger restriction on r , and finally the condition r ≤ α * − 1 appears. Thus, (iii) is also proved.

4.2.
Proofs of Theorems 1.2, 1.3, 1.7 and Proposition 1.5. To begin with, we prepare the following lemma. Lemma 4.3. Let d ∈ N, 1 ≤ q, q ≤ ∞, 0 < r ≤ ∞ and s ∈ R, and let β be given by (4.6). Assume that Then, given a compact set K of L q,r s (R d ), there exists a function µ : for any t ∈ (0, 1) and any f ∈ K (replace L q,r The proof of this lemma can be done as in [7,Lemma 8,page 283] and [27].
We are now in a position to prove the theorems.
Proof of Theorem 1.2. We give the proof only for the case (2), since the proof of the case (1) is similar. Let T > 0 and u 1 , u 2 ∈ L ∞ (0, T ; L q,α s (R d )) be mild solutions to (1.1) with initial data u 1 (0) = u 2 (0). By Lemma 4.1 (ii), we have for any t ∈ (0, T ), where δ > 0 is given in (4.1). If we choose t 0 ∈ (0, T ] such that then we can derive that u 1 = u 2 on [0, t 0 ]. We can repeat this argument until we reach t = T , and hence, we arrive at u 1 = u 2 on [0, T ]. Thus, we conclude Theorem 1.2. The proof of Proposition 1.5 is similar to that of Theorem 1.2, and we have only to use Lemma 4.2 (i) instead of Lemma 4.1 (ii).
Proof of Theorem 1.3. We give the proof only for the case (2), since the proof of the case (1) is similar. Let T > 0 and u 1 , for any t ∈ (0, T ), where q ∈ (q, ∞). Since u 0 ∈ L q,α * −1 s (R d ), we see that for i = 1, 2. Since u 1 , u 2 , e τ ∆ u 0 ∈ C([0, T ]; L q,α * −1 s (R d )), the right-hand side converges to zero as t → 0, and hence, On the other hand, we deduce from Lemma 4.3 that Theorem 1.7 is similarly proved, and so we omit the proof.

Non-uniqueness
In this section, we prove Theorem 1.6, i.e., non-uniqueness for (1.1) in the double critical case s d + 1 q = 1 qc = 1 Qc (i.e. α = α * ). For this purpose, we shall show the existence of two kind of mild solutions (regular and singular) to (1.1) for arbitrary initial data. For convenience, we define Then we note that

Existence of the regular solution.
In this subsection, we prove the local in time existence of a mild solution u to (1.1) in C([0, T ]; L q,r s (R d )) with the auxiliary condition for q > q , where β is given in (4.6). The goal of this subsection is to prove the following: Then, for any u 0 ∈ L q,r s (R d ), there exist a time T = T (u 0 ) > 0 and a unique mild solution u ∈ C([0, T ]; L q,r s (R d )) to (1.1) with u(0) = u 0 satisfying (5.1) (replace L q,r s (R d ) by L q,∞ s (R d ) if r = ∞). Once the following nonlinear estimates are established, the proof can be done by the standard fixed point argument (see, e.g., [10,Subsection 3.1]). Hence, we only give a proof of the following. that q satisfies (5.2). Then there exists a constant C > 0 such that for any t ∈ (0, T ) and any functions u 1 , u 2 satisfying Then there exists a constant C > 0 such that for any t ∈ (0, T ) and any functions u 1 , u 2 satisfying (5.4).

Remark 5.3. Note that (5.2) implies (5.5).
Proof. We first prove the assertion (i). We set σ := αs − γ and take Here, there exists a q as above if (5.2) holds. In a similar way to (4.4), we estimate , where (5.7) and (5.8) are required in the first and second steps, respectively.
Similarly, we can prove the assertion (ii). In fact, taking , where (5.9) and (5.10) are required in the first and second steps, respectively. Here, there exists a q as above if (5.5) holds. Thus, the proof is complete.

5.2.
Existence of singular solution. The mild solution u obtained in Subsection 5.1 is a bounded solution (see [4, Remark 1.1 and Proposition 3.2] and also [40, the remark after Definition 2.1]). In this subsection, we find a singular mild solution v to (1.1) for any initial data u 0 ∈ L q,r s (R d ). Here, the singular mild solution means that v ∈ L q,∞ s (R d ) for any q satisfying (5.2) (in particular, this solution has a singularity at x = 0). The goal of this subsection is to prove the following: Qc . Then, for any u 0 ∈ L q,r s (R d ), there exist T = T (u 0 ) > 0 and a mild solution v ∈ C([0, T ]; L q,r s (R d )) to (1.1) with v(0) = u 0 such that v ∈ L q,∞ s (R d ) for any q satisfying (5.2) and v(t) − e t∆ u 0 ∈ L q,α * −1 s (R d ) and v(t) − e t∆ u 0 ∈ L q,r s (R d ) for any r > α * − 1 (5.11) . The proof is based on the argument in [27,38]. In order to construct the singular solution v , we use a positive, radially symmetric and singular stationary solution of (5.12) where d ≥ 3, γ > −2 and B := {x ∈ R d ; |x| < 1}. We have the results on the existence of the singular stationary solution and the sharp bound of its behavior at x = 0. The proofs of (i) and (ii) can be found in [1, Example 1] and [12, Theorem 1.1 (ii)], respectively. For completeness, we give the proof of (ii) in Appendix B. Therefore, we denote by U 0 the singular stationary solution with We extend U 0 to a function V 0 on R d as follows.
Proposition 5.7. Let d, γ, α, q, s be as in Theorem 5.4. Then there exists a function with compact support such that in a neighborhood of x = 0, and for any q > q. Here, w = w(t) is a (regular) solution to the perturbed problem More precisely, we have the following: Lemma 5.8. Let d, γ, α, q, r, s be as in Theorem 5.4. Then, for w 0 ∈ L q,r s (R d ), there exist T > 0 and a unique solution w ∈ C([0, T ]; L q,r s (R d )) to (5.19) with w(0) = w 0 such that it satisfies (5.1) for any q satisfying (5.2).
The proof of this lemma is based on the fixed point argument as in [27]. Hence, it is sufficient to show some estimates for the term N (w). To prove the nonlinear estimates, we use the following decomposition of V 0 . By density and (5.16), for any Then we have the following estimates for N (w).
Proof. We write By the decomposition (5.20) together with the inequality (5.23) First, we prove the estimate (5.21). In the same way as in the proof of Lemma 4.2, the norms of the terms I(t) and II(t) can be estimated as (5.24) As to the term III(t), we use Proposition 3.1 with (q 1 , r 1 , s 1 ) = ( q, ∞, s + γ − ) and (q 2 , r 2 , s 2 ) = ( q, ∞, s) to obtain where we required that Here, thanks to (5.2) and γ − ∈ [0, 2), the above conditions are satisfied.
As to the term IV (t), thanks to (5.2), we can take σ := α * s − γ and Then we use Proposition 3.6 with (q 1 , r 1 , s 1 ) = (p, ∞, σ) and (q 2 , r 2 , s 2 ) = ( q, ∞, s) to obtain Finally, in a similar way to the proof of [36,Proposition 8.2], we can prove the properties (5.11) of v . In fact, we decompose the Duhamel term v(t) − e t∆ u 0 into the following three terms: The first term w(t) − e t∆ w 0 can be rewritten as We see from Lemma 5.9 and the property (5.15) of R that both terms in the righthand side belong to L q, r s (R d ) for any r > 0 and any t ∈ (0, T ], and hence, w(t) − e t∆ w 0 also belongs to L q, r s (R d ). As to the second term e t∆ V 0 , we estimate where we used Propositions 3.1 and V 0 ∈ L 1 (R d ) with compact support in Proposition 5.7. Hence, e t∆ V 0 ∈ L q, r s (R d ) is also shown for any r > 0 and t ∈ (0, T ]. In contrast, the third term V 0 satisfies V 0 ∈ L q,α * −1 s (R d ) and V 0 ∈ L q,r s (R d ) for any r > α * − 1 by Proposition 5.7. Therefore, (5.11) is proved for any t ∈ (0, T ]. Thus, Theorem 5.4 is proved. Proof of Theorem 1.6. The proof is a combination of Proposition 5.1 and Theorem 5.4. In fact, by these results, there exist a regular mild solution u and singular mild solution v to (1.1) with the same initial data u 0 . When r = ∞, the above arguments are also valid if L q,r s (R d ) is replaced by L q,∞ s (R d ).

Scale-supercritical case
In this section we discuss the scale-supercritical case. We use the self-similar solution of (1.1) to show the existence of a non-trivial mild solution of (1.1) with initial data 0. More precisely, we have the following: Assume that there exists a solution W of Let be the positive self-similar solution of (1.1). Then Ψ ∈ C([0, ∞); L q,r s (R d )) satisfies the equation for any t ∈ (0, ∞). In particular, Ψ is a non-trivial mild solution to (1.1) with initial data 0 in C([0, ∞); L q,r s (R d )). Proof. By the assumptions (i)-(iii) on W, it follows that Then W satisfies the equation ( for 0 < ε < t in the sense of distributions. It is clear that Finally, we prove that the integral t 0 e (t−τ )∆ (| · | γ |Ψ(τ )| α Ψ(τ )) dτ (6.4) converges absolutely in L q,r s (R d ). By Proposition 3.1, we have where we require that If α, q , s, q and s satisfy d 2 then (6.4) converges absolutely in L q,r s (R d ). To check these conditions, let us choose q and s such that It is obvious that under the assumptions in Proposition 6.1, it is possible to take q, s satisfying (6.7) and (6.8). We now show that (6.5) and (6.6) hold if (6.3), (6.7) and (6.8) are satisfied. Indeed, (6.5) is already in (6.7) and the first inequality in (6.8). For (6.6), we have < 1. Thus, we conclude Proposition 6.1.
The existence of positive self-similar solutions Ψ of (1.1) with (i)-(iii) in Proposition 6.1 is proved for any α satisfying α F < α < α HS (6.9) with γ = 0 by [18, Propositions 3.1, 3.4 and 3.5] and with γ satisfying From Proposition 6.1 and this result, it immediately follows that the equation (1.1) has three different solutions 0 and ±Ψ with initial data 0 in C([0, ∞); L q,r s (R d )) under the assumptions (6.9) and (6.10) for d, γ, α, q, r, s as in Proposition 6.1. Thus, Proposition 1.9 is proved. The situation of the case α > α HS is different from the case (6.9). In this case, the nonexistence of positive self-similar solution Ψ satisfying (i)-(iii) in Proposition 6.1 is proved by the following result on uniqueness in the Sobolev space H 1 (R d ): Lemma 6.3. Let T > 0 and u = u(t, x) be a mild solution to (1.1) satisfying The proof of Lemma 6.3 is almost the same as that of [18,Theorem 2], and so we omit the proof. If α > α HS and there exists a positive self-similar solution Ψ satisfying (i)-(iii) in Proposition 6.1, then Ψ satisfies all assumptions in Lemma 6.3, and hence, Ψ ≡ 0. This contradicts Ψ > 0. Thus, we see the nonexistence of such a Ψ. 7. Additional results and remarks 7.1. Double critical case. We give a remark on the number of solutions in the double critical case. Theorem 1.6 shows that the problem (1.1) has two different solutions, where one is regular and the other is singular (see Section 5). In fact, however, (1.1) has an uncountable infinite number of different mild solutions in C([0, T ]; L q,r s (R d )) for any initial data u 0 ∈ L q,r s (R d ). This can be confirmed by constructing the family {u t 0 } t 0 ∈(0,T ) of solutions to (1.1) such that u t 0 is a singular solution for 0 < t ≤ t 0 and a regular solution for t 0 < t < T . 7.2. Case γ = − min{2, d}. The problem on well-posedness for (1.1) in the critical singular case γ = − min{2, d} has not been studied. Establishing the weighted linear estimates (3.1) with the double endpoint s 1 d + 1 q 1 = 1 and s 2 d + 1 q 2 = 0, we can present the following results on uniqueness for the case d = 1 and γ = −1.
In the case d = 1 and γ = −1, the existence of a solution has not been proved, but Theorem 7.1 implies that only one solution exists at most. It remains open whether unconditional uniqueness holds in the critical singular case d ≥ 2 and γ = −2.
Once the weighted Meyer inequality (3.34) with the endpoint case s 2 d + 1 q 2 = 0 is proved, this problem is solved, but we do not know if the endpoint inequality holds. 7.3. Case of the exterior problem. It is also interesting to analyze in more detail the influence of the potential |x| γ at the origin or at infinity. For this, we discuss unconditional uniqueness for the initial-boundary value problem of the Hardy-Hénon parabolic equation on the exterior domain Ω : where T > 0, d ∈ N, γ ∈ R, α > 1, q ∈ [1, ∞], r ∈ (0, ∞] and s ∈ R. Here, ∂Ω denotes the boundary of Ω. In conclusion, the critical exponents (1.4) and (1.5) with γ = 0 (i.e. q c (0) = d(α−1) 2 and Q c (0) = α) appear in the results on unconditional uniqueness for (7.1), since the effect near the origin x = 0 has been eliminated. The results of this subsection can be extended to more general situations such as the initial-boundary value problem on general domains Ω not containing the origin with the Robin boundary condition (cf. [22,Section 6]).
In the following, we shall prove the result on unconditional uniqueness.
Then the following assertions hold: or q = Q c (0) > q c (0) and r = α.
or q = q c (0) = Q c (0) and r = α − 1. Then unconditional uniqueness holds for (7.1) in C([0, T ]; L q,r s (Ω)). Remark 7.3. Since L q,r s 1 (Ω) ⊂ L q,r s 2 (Ω) if s 2 ≤ s 1 , the exponent s should be taken as close to max{− d q , γ α−1 } as possible in the above proposition from the point of view of unconditional uniqueness.
We denote by −∆ D the Laplace operator with the homogeneous Dirichlet boundary condition on Ω and by {e t∆ D } t>0 the semigroup generated by −∆ D . The integral kernel G D (t, x, y) of e t∆ D satisfies the Gaussian upper bound for any t > 0 and almost everywhere x, y ∈ Ω. Then, we have the following linear estimates.

(B.4)
Proof. It is obvious that u is positive and of C 2 , and a straightforward calculation gives that u satisfies the nonlinear ordinary differential equation (B.3). It is shown by Theorem B.1 that u d+γ d−2 ∈ L 1 ((0, ∞)). We shall prove that u is strictly decreasing on (0, ∞) by contradiction. Suppose that u is not strictly decreasing on (0, ∞). Then there exist t 0 , t 1 such that 0 < t 0 < t 1 and u t (t 0 ) = u t (t 1 ) = 0 and u t ≥ 0 on (t 0 , t 1 ).

(B.5)
Since u is positive, we find from (B.5) that which implies that u(t 0 ) > u(t 1 ). This is a contradiction to u t ≥ 0 on (t 0 , t 1 ). Therefore, u is strictly decreasing on (0, ∞). In addition, it is also shown by the inverse function theorem that u is a C 1 -diffeomorphism from (0, ∞) to (0, u(0)). Lastly, since u ∈ C 2 ((0, ∞)), the fundamental theorem of calculus gives for t ≥ t > 0, and as t → ∞, for t > 0. Since u t < 0, the convergence u t (t) → 0 as t → ∞ must hold. Noting u(t), u t (t) → 0 as t → ∞, and integrating (B Then the assertion (ii) in Theorem 5.5 holds.
Finally, we conclude the proof of (ii) of Theorem 5.5 by showing the following.