Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 67, pp. 1–11. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.67 EXISTENCE AND NON-EXISTENCE OF SOLUTIONS FOR HARDY PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA ALDRYN APARCANA, BRANDON CARHUAS-TORRE, RICARDO CASTILLO, MIGUEL LOAYZA Abstract. We establish the existence, non-existence and uniqueness of the local solutions of the Hardy parabolic equation ut −∆u = h(t)| · |−γg(u) on Ω× (0, T ) with Dirichlet boundary conditions. We assume that Ω with 0 ∈ Ω is a smooth domain bounded or unbounded, h ∈ C(0,∞), g ∈ C([0,∞)) is a non-decreasing function, 0 < γ < min{2, N}, and the initial data have a singularity at the origin. 1. Introduction Let Ω ⊂ RN be a domain (bounded or unbounded) with a smooth boundary ∂Ω whenever it exists. We assume that 0 ∈ Ω and consider the parabolic problem ut −∆u = h(t)|x|−γg(u) in Ω× (0, T ), u = 0 on ∂Ω, u(0) = u0 in Ω, (1.1) where h ∈ C(0,∞), 0 < γ < min{2, N}, g ∈ C([0,∞)) is a non-decreasing function, and u0 ∈ Lr(Ω) with u0 ≥ 0, 1 ≤ r <∞. Throughout the work, we consider only non-negative solutions. The first equation of (1.1) with h ≡ 1 and g(t) = tp, t ≥ 0, p > 1 is known as the Hardy parabolic equation and it has been considered by many authors; see, for instance, [6, 19, 20, 22] and the references therein. Its elliptic version, that is −∆u = | · |−γup was proposed by Hénon [10] as a model for studying spherical-state stellar systems. Problem (1.1), with γ = 0, h ≡ 1 and initial data in Lebesgue spaces, has been extensively studied, see [2, 9, 15, 16, 24, 25] for g(t) = tp, p > 1, and [14] for g ∈ C([0,∞)) a non-decreasing function. Problem (1.1), with γ > 0, h ≡ 1 and g(t) = tp, p > 1 was treated firstly in [22, Theorem 2.3] for non-negative initial data in the continuous bounded functions space CB(RN ) with γ < 2. For non-negative initial data in the Lebesgue space Lr(Ω) and 0 < γ < min{2, N} there is a non-negative solution if and only if p ≤ p⋆γ if r > 1 or p < p⋆γ if r = 1. where p⋆γ = 1 + (2− γ)r N , (1.2) see [6, 19]. Moreover, for u0 ∈ L1 loc(RN ), p > p⋆γ and 0 ≤ u0(x) ≤ c⋆|x|− 2−γ p−1 for c⋆ sufficiently small, then problem (1.1) has a global solution in the class C((0,∞), Lm(RN )), m > N(p − 1)/(2 − γ), see [19, Theorem 1.3]. Subsequently, in [12] was obtained necessary 2020 Mathematics Subject Classification. 35A01, 35A02, 35B33, 35D30, 35K58. Key words and phrases. Local existence; Hardy parabolic equation; Lebesgue spaces; critical values; uniqueness. ©2025. This work is licensed under a CC BY 4.0 license. Submitted January 21, 2025. Published July 4, 2025. 1 2 A. APARCANA, B. CARHUAS-TORRE, R. CASTILLO, M. LOAYZA EJDE-2025/67 conditions for the local existence considering a non-negative Radon measures in RN as initial data. Their results imply that there exists a constant c⋆ > 0 sufficiently large such that if u0(x) ≥ c⋆|x| − 2−γ p−1 χB(0,l) if p > p⋆γ , c⋆|x|−N log(e+ |x|−1)− N 2−γ −1χB(0,l) if p = p⋆γ , f or l > 0, then problem (1.1) does not admit a solution. Here, χB(0,l) denotes the characteristic function of the open ball B(0, l) centered at the origin and radius l > 0. For similar results to γ = 0, see [11], and for γ > 0 and initial data singular at some point z ∈ RN , see [13]. For results with initial data u0 belonging to the weighted Lebesgue space, see [7]. The above results imply that |x|−(2−γ)/(p−1) is the optimal singularity for problem (1.1) with g(t) = tp, h ≡ 1, p > p⋆γ , and r = 1. Motivated for this fact, our main concern in this work is to analyze the existence/non-existence of solutions for problem (1.1) when the initial datum u0 ∈ Lr(Ω) is compared with the singular function ψκ,β = κ| · |−βχB(0,l), (1.3) for some κ, β > 0 and l > 0 such that B(0, l) ⊂ Ω. It is worth mentioning that the initial data of the form (1.3) were used first in [17] to show the non-existence of non-negative solution for a system related to (1.1) with γ = 0 and g(u) = up. With this in mind, assuming h(t) = ta and g(t) = tp, new solutions are obtained and we show that |x|−[(2−γ+2a)r]/(p−1) is the optimal singularity for the problem (1.1) with r ≥ 1, see Section 5. In our first result, we study the non-existence of solutions for the problem (1.1). These solutions are understood in the sense of mild solutions (see Definition 2.2). To do this, we assume that G(τ) = ∫ ∞ τ dt g(t) <∞, (1.4) for all τ > 0. We also consider the set Iβ(κ) = {φ ∈ Lr(Ω) : φr ≥ ψκ,β in Ω, for some κ, β > 0}. Theorem 1.1. Assume that 0 < γ < min{2, N}, h ∈ C(0,∞) and g ∈ C([0,∞)) is a non- decreasing function such that g(0) = 0. Suppose also that g satisfies condition (1.4) and g1−ϵ, for ϵ > 0 sufficiently small, is a convex function. There exists a constant c0, depending on N , such that if u0 ∈ Iβ(κ) with 0 < β < N and lim t→0+ [ G(c0κ1/rt−β/(2r)) ]−1 ∫ t 0 h(σ)(t− σ)−γ/2dσ = +∞, (1.5) where G is given by (1.4), then the problem (1.1) does not admit a non-negative solution. Remark 1.2. Here are some comments about Theorem 1.1. (i) The convexity condition of g1−ϵ, with 0 < ϵ < 1 sufficiently small, is necessary in our approach. Clearly g(t) = tp, t ≥ 0, p > 1, verifies this condition since g1−ϵ is convex for ϵ > 0 sufficiently small. In general, this assumption is satisfied when g is twice differentiable with gg′′ − ϵg′2 ≥ 0 because (g1−ϵ)′′ = (1− ϵ)g−1−ϵ[gg′′ − ϵg′2]. (ii) A non-existence result for problem (1.1) can be obtained without the convexity assumption of g1−ϵ, ϵ > 0, adapting the arguments of [6, proof of Theorem 1.4] and [1, Proof of Theorem 1.5, Case 2], under the condition lim t→0+ t−N/(2r′) ∫ t 0 h(σ)g(κ1/rc′Nσ −β/(2r))σ(N−γ)/2dσ = +∞, (1.6) for u0 ∈ Iβ(κ), 0 < β < N , and some constant c′N , depending only on N . Although more general, the result is weaker when we apply it to the prototypical case g(t) = tp, p > 1 and h(t) = ta; see Remark 5.1. EJDE-2025/67 HARDY PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA 3 (iii) Another situation where it is possible to obtain non-existence results for problem (1.1) without the convexity condition of g1−ϵ, ϵ ≥ 0, was obtained in [6] when lim sup t→∞ t−p⋆ γg(t) = +∞, if r > 1 or∫ ∞ 1 t−p⋆ γG0(t)dt = +∞, if r = 1 (1.7) where G0(t) = sup1≤σ≤t g(σ)/σ and p⋆γ given by (1.2), but there it is considered an initial datum of the form u0 = ∑∞ k=1 akχB(0,rk) ∈ Lr(Ω), where ak > 0 and rk > 0 are chosen appropriate. A similar situation occurs when Ω = RN and condition (1.7) is satisfied. For the existence of solutions we consider the set Iβ(κ) = {φ ∈ Lr(Ω) : 0 ≤ φr ≤ ψκ,β in Ω, for some κ, β > 0}. The sets Iβ(κ) and Iβ(κ) were considered in [3, 4] to analyze the existence and non-existence of solutions, respectively, for a related problem with (1.1) and γ = 0. Theorem 1.3. Assume that 0 < γ < min{2, N}, h ∈ C(0,∞), and g ∈ C([0,∞)) is a non- decreasing function. There exists a constant C0 depending on N, β and r, such that for every u0 ∈ Iβ(κ) with 0 < β < N , problem (1.1) admits a non-negative solution u ∈ L∞((0, T ), Lr(Ω)) if lim t→0+ tγ/2 ∫ t 0 h(σ)G(C0κ 1/rσ−β/(2r))σ−γ/2(t− σ)−γ/2dσ = 0. (1.8) The function G : (0,∞) → [0,∞) is given by G(t) = sup0 0 such that tβ/2r∥u(t)∥L∞ ≤ C for all t ∈ (0, T ). (b) u ∈ C([0, T ], Lr(Ω)). Remark 1.4. In the Theorem 1.3, when g is a convex function and g(0) = 0 we have that G(t) = g(t)/t for t > 0. It is worth mentioning that we have considered the sets Iβ(κ) and Iβ(κ) with the singularity localized in 0 ∈ Ω. We can obtain the same result considering a singularity in any fixed point x0 ∈ Ω taking the sets {φ ∈ Lr(Ω) : φr(x) ≥ κ |x− x0|−βχB(x0,l)(x) a.e. in Ω, for some κ, β > 0}, {φ ∈ Lr(Ω) : 0 ≤ φr(x) ≤ κ |x− x0|−βχB(x0,l)(x) a.e. in Ω, for some κ, β > 0}, see [13], for h = 1, g(u) = up with Ω = RN , and [4] for γ = 0, h = 1 with Ω a bounded domain. We now analyze the uniqueness. It was shown in [19, Theorem 1.1(ii)] that problem (1.1) with g(t) = tp, t ≥ 0, p > 1, and h ≡ 1 has a unique solution in the class C([0, T ], Lr(RN )) if p < p⋆γ and p r < 1− γ N . Moreover, the uniqueness also holds for p ≤ p⋆γ with the additional condition sup t∈[0,T ] { t N 2 ( 1 r− 1 q )∥u(t)∥Lq } <∞ and q > r. It is important to mention that new advances on the uniqueness of problem (1.1) have been obtained in Lorentz spaces in [20], in weighted Lebesgue spaces in [7], in weighted Lorentz spaces in [8], and in uniformly local Lebesgue spaces in [5]. To establish our uniqueness result we assume that g ∈ C([0,∞)) is locally Lipschtiz and define L : [0,∞) → [0,∞) by L(t) = sup 0≤u,v≤t u̸=v g(u)− g(v) u− v , for t > 0, L(0) = 0. Our uniqueness result reads as follows. 4 A. APARCANA, B. CARHUAS-TORRE, R. CASTILLO, M. LOAYZA EJDE-2025/67 Theorem 1.5. Assume 0 < γ < min{2, N} with 1/r + γ/N < 1, 0 < θ ≤ N , h ∈ C(0,∞), and g ∈ C([0,∞)) is a non-decreasing and locally Lipschitz function. Problem (1.1) admits a unique solution in the class {u ∈ L∞((0, T ), Lr(Ω)) : sup t∈(0,T ) tθ/2r∥u(t)∥L∞ ≤ C} (1.9) if the map t 7→ h(t)L(Ct−θ/2r) belongs to Lq(0, T ) (1.10) for some q > 2/(2− γ). Remark 1.6. Here are some comments on Theorem 1.5. (i) The solution given by Theorem 1.3 belongs to the class defined by (1.9) with θ = β and u0 ∈ Iβ(κ). The same occurs for θ = N with u0 ∈ Lr(Ω) and h = 1, see [6, Theorem 1.2]. (ii) Note that condition (1.10) depends on the constant C of the set defined in (1.9). It is clear, by a change of variable, that if L or h are homogeneous, then the uniqueness holds in the class {u ∈ L∞((0, T ), Lr(Ω)) : sup t∈(0,T ) tθ/2r∥u(t)∥L∞ <∞}, if t 7→ h(t)L(t−θ/2r) belongs to Lq(0, T ), q > 2/(2− γ). The remainder of this article is organized as follows. In Section 2, we present the notion of a solution used in the work and establish some useful technical results. In Section 3, we give the proofs of Theorems 1.1 and 1.3. In section 4, we prove the uniqueness, And in Section 5, give some applications. 2. Preliminaries Throughout this work, PΩ(x, y; t) is the Dirichlet heat kernel associated with the operator ∂t − ∆Ω, where −∆Ω is the Dirichlet Laplacian for the open set Ω ⊂ RN . The Dirichlet heat semigroup is defined for all ϕ ∈ M+, the set of non-negative a.e. finite measurable functions on Ω by [SΩ(t)ϕ](x) = ∫ Ω PΩ(x, y; t)ϕ(y)dy <∞. (2.1) It is well known, see [21, Lemma 7], that PΩ1(x, y; t) ≤ PΩ2(x, y; t) ≤ PN (x, y; t) (2.2) for x, y ∈ Ω1 ⊂ Ω2, where Ω1 and Ω2 are open subsets of RN , and PN is the heat kernel defined by PN (x, y; t) := PRN (x, y; t) = (4πt)−N/2e−|x−y|2/4t. (2.3) Sometimes, when the domain Ω considered is clear, we denote SΩ(t)ϕ by S(t)ϕ. The following result is used in the proof of the non-existence of solutions, see [1]. Lemma 2.1. Let l, δ > 0 be such that B(0, l + 2δ) ⊂ Ω and 0 < γ < N . There exists a constant c′N > 0, depending only on N , such that S(t)| · |−γχB(0,l) ≥ c′N t − γ 2 χB(0, √ t) , for all 0 < t ≤ min{δ2, l2}. The notion of solution used in the work is the following. Definition 2.2. Let u0 ∈ Lr(Ω), u0 ≥ 0, 1 ≤ r < ∞, γ > 0, g ∈ C([0,∞)) and h ∈ C(0,∞). A non-negative measurable function u ∈ L∞((0, T ), Lr(Ω)), defined a.e. in Ω × (0, T ) for some T > 0, is called a solution (resp. supersolution) of problem (1.1) if u(t) = F(u, u0)(t) (resp. u(t) ≥ F(u, u0)) a.e. in Ω× (0, T ), where F(u, u0)(t) = S(t)u0 + ∫ t 0 S(t− σ)h(σ)| · |−γg(u(σ))dσ. (2.4) EJDE-2025/67 HARDY PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA 5 The following result is an adapted version of [6, Lemma 2.4]. Lemma 2.3. Assume that g ∈ C([0,∞)) is non-decreasing, h ∈ C(0,∞), γ > 0, and u0 ∈ Lr(Ω), 1 ≤ r <∞, with u0 ≥ 0. If u is a supersolution of problem (1.1) in Ω× (0, T ), then there exists a solution u of problem (1.1) defined on Ω× (0, T ) such that 0 ≤ u ≤ u. Proof. u ≥ F(u, u0), since u is a supersolution of (1.1). Moreover, F(·, u0) is non-decreasing on u, since g is non-decreasing, h ≥ 0 and the monotonicity property of the heat semigroup {S(t)}t≥0. Consider the sequence {un}n≥0, given by u0 = u, and F(un−1, u0) = un for n ≥ 1. Since F is non-decreasing, the sequence {un}n≥0 is non-increasing a.e. in Ω × (0, T ) and u ≥ un ≥ un+1 ≥ 0. Let u(x, t) = limn→∞ un(x, t), whenever it exists. The continuity of g, the monotonicity of semigroup {S(t)}t≥0, and the monotone convergence theorem allow us to conclude that u = limn→∞ un = limn→∞ F(un−1, u0) = F(u, u0). In addition, since 0 ≤ u ≤ u we conclude that u ∈ L∞((0, T ), Lr(Ω)). □ Let Ω ⊂ RN be a smooth domain (possibly unbounded). We recall the well-known smoothing effect of the heat semigroup on Lebesgue spaces, that is, ∥S(t)ϕ∥Lq2 ≤ (4πt)− N 2 ( 1 q1 − 1 q2 )∥ϕ∥Lq1 , for 1 ≤ q1 ≤ q2 ≤ ∞, t > 0 and ϕ ∈ Lq1(Ω), see [2, Lemma 7]. We also use the following estimate, which can be obtained from estimates (2.2), see [6, Lemma 2.5] and [19, Proposition 2.1]. Lemma 2.4. Let γ ∈ (0, N), and let q1, q2 ∈ (1,∞] satisfy 0 ≤ 1 q2 < γ N + 1 q1 < 1. Then there exists a constant C0 > 0, depending on N, γ, q1 and q2, such that ∥S(t)(| · |−γϕ)∥Lq2 ≤ C0t −N 2 ( 1 q1 − 1 q2 )− γ 2 ∥ϕ∥Lq1 , for all t > 0 and ϕ ∈ Lq1(Ω). Lemma 2.5. Assume that g ∈ C([0,∞)) is a convex function with g(0) = 0 and ϕ ∈ M+. Then g(S(t)ϕ) ≤ S(t)g(ϕ). Proof. From inequality (2.2) we have that η = ∫ Ω PΩ(x, y; t)dy ≤ 1. Using expression (2.1), Jensen’s inequality, the convexity of g and g(0) = 0, we obtain g(S(t)ϕ) = g (∫ Ω PΩ(x, y; t)ϕ(y)dy ) = g ( η ∫ Ω PΩ(x, y; t) η ϕ(y)dy + (1− η)0 ) ≤ ηg (∫ Ω PΩ(x, y; t) η ϕ(y)dy ) ≤ ∫ Ω PΩ(x, y; t)g(ϕ(y))dy = S(t)g(ϕ). □ The following result can be found at [6, Lemma 4.1]. Lemma 2.6. Assume that ϕ ∈ M+ and 0 < γ < N . Then there exists a constant C1 = C1(N, γ) > 0 such that SRN (t)| · |−γSRN (s)ϕ ≤ C1 (1 t + 1 s )γ/2 SRN (t+ s)ϕ for all t, s > 0. 6 A. APARCANA, B. CARHUAS-TORRE, R. CASTILLO, M. LOAYZA EJDE-2025/67 3. Existence and non-existence Proof of Theorem 1.1. We adapt the arguments used in [3, Proposition 2.6] and [18, Lemma 15.6]. Since u0 ∈ Iβ(κ), u0 ≥ κ1/r| · |−β/rχB(0,l) = v0. Note that v0 ∈ Lr(Ω) because 0 < β < N . We argue by contradiction and assume that there exists a non-negative solution u ∈ L∞((0, T ), Lr(Ω)) for problem (1.1) with initial data u0. Since u0 ≥ v0, by (2.4) we have u(t) = F(u, u0)(t) ≥ F(u, v0)(t) for a.e. t ∈ (0, T ). Let t ∈ (0, s) with s ∈ (0, T ) and 1/q = 1− ϵ > 0. From (2.4) S(s− t)u(t) ≥ Θ(·, t), (3.1) where Θ(·, t) = S(s)v0 + ∫ t 0 h(σ)S(s− σ)| · |−γg(u(σ))dσ. Note that for x ∈ Ω fixed, the function Θ(x, t) is absolutely continuous on (0, s). Consequently, it is differentiable a.e. in (0, s). Thus, by the reverse Hölder inequality and (3.1) we have Θ′(t) = h(t)S(s− t)| · |−γg(u(t)) ≥ h(t)[S(s− t)| · |−γ/(1−q)]1−q[S(s− t)g1/q(u(t))]q ≥ h(t)[S(s− t)| · |−γ/(1−q)]1−qg(S(s− t)u(t)) ≥ h(t)[S(s− t)| · |−γ/(1−q)]1−qg(Θ(t)) for t ∈ (0, T ). Here, we have used Lemma 2.5 for the convex function g1/q. Then [G(Θ)]′(t) = −Θ′(t) g(Θ) ≤ −h(t)[S(s− t)| · |−γ/(1−q)]1−q. Integrating from 0 to s, we obtain −G(Θ(0)) ≤ G(Θ(s))− G(Θ(0)) ≤ − ∫ s 0 h(σ)[S(s− σ)| · |−γ/(1−q)]1−qdσ. Hence ∫ t 0 h(σ)[S(t− σ)| · |γ/(q−1)]1−qdσ ≤ G(S(t)v0), for t ∈ (0, T ) and all x ∈ Ω (by continuity). Due to (2.2) and (2.3) we obtain∫ t 0 h(σ)[SRN (t− σ)| · |γ/(q−1)(0)]1−qdσ ≤ G([S(t)v0](0)). Since [SRN (t− σ)| · |γ/(q−1)]1−q(0) = η1−q(t− σ)−γ/2, with η = (4π)−N/2 ∫ RN exp(−|z|2/4)|z|γ/(q−1)dz, we conclude that η1−q ∫ t 0 h(σ)(t− σ)−γ/2dσ ≤ G([S(t)v0](0)). (3.2) On the other hand, by Lemma 2.4, S(t)v0 ∈ L∞(Ω) for all t > 0, and by Lemma 2.1 we have S(t)v0 ≥ c′Nκ 1/rt−β/2rχB(0, √ t) , for all 0 < t < min{(l/3)2, T}. Thus, {G([S(t)v0](0))}−1 ∫ t 0 h(σ)(t− σ)−γ/2dσ dσ ≥ [G(c′Nκ1/rt−β/2r)]−1 ∫ t 0 h(σ)(t− σ)−γ/2dσ > ηq−1 as t→ 0+, by condition (1.5). This contradicts estimate (3.2). □ EJDE-2025/67 HARDY PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA 7 Proof of Theorem 1.3. Let u0 ∈ Lr(Ω), u0 ̸= 0, with u0 ∈ Iβ(κ) and 0 < β < N . Consider ũ0 the zero extension to RN of u0. Because u0 ∈ Iβ(κ), there exists κ, l > 0 such that ũ0(x) ≤ κ1/r|x|−β/rχB(0,l)(x) for x ∈ RN . Thus, ũ0 ∈ Lr(RN ) for r ≥ 1. By Lemma 2.4, ∥SRN (σ)ũ0∥L∞ ≤ C0κ 1/rσ−β/2r. (3.3) The function w(t) = 2[SRN (t)ũ0] ∣∣ Ω is a supersolution of (1.1). Indeed, using inequality (3.3), the estimate provided by Lemma 2.6 and condition (1.8), we have∫ t 0 S(t− σ)h(σ)| · |−γg(w(σ))dσ ≤ ∫ t 0 h(σ)G(∥w(σ)∥L∞)S(t− σ)| · |−γ w(σ)dσ ≤ 2 ∫ t 0 h(σ)G(2C0κ 1/rσ−β/2r)S(t− σ)| · |−γSRN (σ)ũ0dσ ≤ 2C1[SRN (t)ũ0]|Ωtγ/2 ∫ t 0 h(σ)G(2C0κ 1/rσ−β/2r)(t− σ)−γ/2σ−γ/2dσ ≤ δw(t) (3.4) for all t ∈ (0, T ) with T > 0 sufficiently small and 0 < δ ≤ 1/2. Hence, from (2.4) and (3.4), we have F(w, u0) = S(t)u0 + ∫ t 0 S(t− σ)h(σ)| · |−γg(w(σ))dσ ≤ 1 2 w(t) + δw(t) ≤ ( 1 2 + δ)w(t) ≤ w(t), for t ∈ (0, T ) with T > 0 sufficiently small. Thus, F(w, u0) ≤ w in (0, T ) and w is a supersolution of (1.1) in (0, T ). Lemma 2.3 assures that problem (1.1) admits a solution defined on (0, T ) and 0 ≤ u(t) ≤ w(t) for t ∈ (0, T ). Moreover, from estimate (3.3), ∥u(t)∥L∞ ≤ 2∥SRN (t)ũ0∥L∞ ≤ Ct−β/2r, for some constant C > 0. This shows item (a). To show item (b) we write the solution of problem (1.1) in the form u(t) = S(t)u0 + ∫ t 0 S(t− σ)h(σ)| · |−γf(u(σ))dσ := u1(t) + u2(t), for t ∈ [0, T ]. Since u1 ∈ C([0, T ], Lr(Ω)), we need only to show the continuity of u2. To this end, we argue as in [23, p. 285]. Using the facts that g and G are non-decreasing and estimate (3.3) we have g(u(σ)) ≤ g(w(σ)) ≤ w(σ)G(w(σ)) ≤ 2G(2C0κ 1/rσ−β/2r)[SRN (σ)ũ0] ∣∣ Ω . Here, we have used that w(t) = 2[SRN (t)ũ0] ∣∣ Ω is a supersolution of problem (1.1) on (0, T ). Taking 0 ≤ τ < t < T and arguing as in the derivation of (3.4) we obtain ∥ ∫ t τ S(t− σ)h(σ)| · |−γg(u(σ))dσ∥Lr ≤ 2∥ ∫ t τ h(σ)G(2C0κ 1/rσ−β/2r)S(t− σ)[| · |−γSRN (σ)ũ0]dσ∥Lr ≤ 2C1∥u0∥Lr tγ/2 ∫ t τ h(σ)G(2C0κ 1/rσ−β/2r)(t− σ)−γ/2σ−γ/2dσ. 8 A. APARCANA, B. CARHUAS-TORRE, R. CASTILLO, M. LOAYZA EJDE-2025/67 It follows that ∥ ∫ t τ S(t− σ)h(σ)| · |−γg(u(σ))dσ∥Lr → 0 as t→ τ . Indeed, this is clear if τ > 0, and when τ = 0 we use the hypothesis (1.8). □ 4. Uniqueness We use the following singular Gronwall Lemma, see [2, p. 288]. Lemma 4.1. Let T > 0, A ≥ 0, 0 ≤ α, ζ ≤ 1 and let f be a non-negative function with f ∈ Lq(0, T ) for some q > 1 such that q′ max{α, ζ} < 1. Consider a non-negative function φ ∈ L∞(0, T ) such that φ(t) ≤ At−α + ∫ t 0 (t− σ)−ζf(σ)φ(σ)dσ for t ∈ (0, T ). Then there exists a constant C > 0, depending only on T, α, ζ and ∥f∥Lq , such that φ(t) ≤ ACt−α a.e. t ∈ (0, T ). Proof of Theorem 1.5. Suppose that problem (1.1) has two solutions u and v in the class (1.9) defined on some interval (0, T ) with the same initial data u0, that is, u(t) = S(t)u0 + ∫ t 0 S(t− σ)h(σ)| · |−γg(u(σ))dσ, (4.1) v(t) = S(t)u0 + ∫ t 0 S(t− σ)h(σ)| · |−γg(v(σ))dσ. (4.2) Subtracting (4.2) from (4.1), we have u(t)− v(t) = ∫ t 0 S(t− σ)h(σ)| · |−γ [g(u(σ))− g(v(σ))]dσ ≤ ∫ t 0 h(σ)∥g(u(σ))− g(v(σ)) u(σ)− v(σ) ∥L∞S(t− σ) { | · |−γ [u(σ)− v(σ)] } dσ ≤ ∫ t 0 h(σ)L(Cσ−θ/2r)S(t− σ) { | · |−γ [u(σ)− v(σ)] } dσ. Since γ < N , γ/N + 1/r < 1 and 1/q + γ/2 < 1, using Lemma 2.4 we obtain ∥u(t)− v(t)∥Lr ≤ C0 ∫ t 0 h(σ)L(Cσ−θ/2r)∥S(t− σ)| · |−γ [u(σ)− v(σ)]∥Lrdσ ≤ C0 ∫ t 0 h(σ)L(Cσ−θ/2r)(t− σ)−γ/2∥u(σ)− v(σ)∥Lrdσ. Hence, the uniqueness follows from Gronwall’s singular Lemma (Lemma 4.1) and condition (1.10). □ 5. Applications We apply Theorems 1.1 and 1.3 to some classical examples of nonlinear heat equations. 5.1. Case g(t) = tp with p > 1. Non-existence. In this case G(τ) = τ1−p/(p−1). Thus, condition (1.5) is equivalent to lim t→0+ t−β(p−1)/2r ∫ t 0 h(σ)(t− σ)−γ/2dσ = +∞. (5.1) In particular, when h(t) = ta condition (5.1) is satisfied if 1 + a > γ/2 and (2− γ + 2a)r p− 1 < β < N. Hence, p > 1 + [(2− γ + 2a)r]/N . EJDE-2025/67 HARDY PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA 9 Existence. Since G(t) = tp−1 condition (1.8) is equivalent to lim t→0+ tγ/2 ∫ t 0 h(σ)σ−β(p−1)/(2r)σ−γ/2(t− σ)−γ/2dσ = 0. (5.2) In particular, when h(t) = ta, condition (5.2) is satisfied if 1 + a > γ/2, and β < (2− γ + 2a)r p− 1 < N whenever p > 1 + [(2− γ + 2a)r]/N . In summary, when h(t) = ta and 1 + a > γ/2 we obtain a critical exponent β⋆ = (2− γ + 2a)r p− 1 , (5.3) such that for β ∈ (0, N) and p > 1 + [(2− γ + 2a)r]/N we have: • If β < β⋆ and u0 ∈ Iβ(κ), then problem (1.1) admits a non-negative solution. • If β > β⋆ and u0 ∈ Iβ(κ), then problem (1.1) does not admit a non-negative solution. This shows that |x|−β⋆ is the optimal singularity for problem (1.1). The critical case β = β⋆ was treated in [13, Theorem 1.2] in the particular case where h = 1 and r = 1. Uniqueness. When h(t) = ta, t > 0 with 1 + a > γ/2, we have that L(s) = psp−1 and condition (1.10) is verified for β < β⋆, with β⋆ given by (5.3). Indeed, in this case, it is possible to choose q > 1 such that β(p− 1) 2r − a < 1 q < 1− γ 2 . The arguments used in this case can be used to treat the function g(t) = (1 + t)q[ln(1 + t)]p with p, q > 1. Remark 5.1. If in the nonexistence part, we use condition (1.6) in place of (1.5), with g(t) = tp, p > 1, and h(t) = ta, a > −1 we have t− N 2r′ ∫ t 0 h(σ)g(κ1/rc′Nσ − β 2r )σ N−γ 2 dσ ≥ (κ1/rc′N )pt− N 2r′ − β 2r ∫ t 0 h(σ)σ N−γ 2 dσ = Ct− N 2r′ − β 2r+1+a+N−γ 2 → +∞ as t→ 0 for 2r p ( 2− γ + 2a+ N r ) < β < N, whenever p > 1 + [(2− γ + 2a)r]/N . However, β⋆ < 2r p ( 2− γ + 2a+ N r ) . 5.2. Case g(t) = eαt with α > 0 and h(t) = ta. Although g1−ϵ is not a convex function, we can apply Theorem 1.1 to conclude that problem (1.1) does not admit a non-negative solution for u0 ∈ Iβ(κ) if β > (2− γ + 2a)r α , 1 + a > γ/2. To show this, we argue by contradiction and assume that problem (1.1) has a non negative solution v on a some interval (0, T ). Since exp(αv) ≥ ( α α+ 1 )α+1 vα+1, for v ≥ 0, we conclude that v is a supersolution of problem (1.1) with g(t) = (α/(α+1))α+1tα+1. By Lemma 2.3, problem (1.1) with g(t) = (α/(α+ 1))α+1tα+1 admits a solution which contradicts the result obtained in Subsection 5.1. 10 A. APARCANA, B. CARHUAS-TORRE, R. CASTILLO, M. LOAYZA EJDE-2025/67 6. Concluding remarks We establish new results on the existence and non-existence of a solution for the Hardy parabolic equation ut −∆u = h(t)| · |−γg(u) in Ω × (0, T ), where Ω is a smooth (bounded or unbounded) domain (0 ∈ Ω), g ∈ C([0,∞)) non-decreasing, h ∈ C(0,∞) and 0 < γ < min{2, N} (see Theorem 1.1 and 1.3 for specific conditions on g). In our approach, we consider initial data with a singularity at the origin, that is, in the sets Iβ(κ) and Iβ(κ) with 0 < β < N and κ > 0. As a consequence of the results, considering g(t) = tp with p > 1, h(t) = ta for all t > 0, we determine a new critical value β⋆, given by (5.3), for the existence of solutions, see Section 5. Finally, we establish a conditional uniqueness analyzing the behavior of the Lipschitz constant of function g (see Theorem 1.5). class where the solutions obtained are defined. Acknowledgments. This work was partially supported by UNICA (Universidad Nacional San Luis Gonzaga de ICA, Perú) Res. No.083-VRI-UNICA-2022 (A. Aparcana). The authors want to thank professors Javier Magallanes Yui and Roberto Yactayo for their collaboration in the preparation of the project. B. Carhuas-Torre was supported by CNPq/Brazil - 142441/2020- 1. R. Castillo was supported by ANID-FONDECYT - 11220152. M. Loayza was supported by CAPES-PRINT, 88881.311964/2018-01, MATHAMSUD 88881.520205/2020-01, and CNPq Proc. 313382/2023-9. References [1] A. Aparcana, R. Castillo, O. Guzmán-Rea, M. Loayza; On the local existence for a weakly parabolic system in Lebesgue spaces. J. Differential Equations, 268 (6) (2020), 3129-3151. [2] H. Brezis, T. Cazenave; A nonlinear heat equation with singular initial data. J. Anal. 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EJDE-2025/67 HARDY PARABOLIC EQUATIONS WITH SINGULAR INITIAL DATA 11 [21] M. Van den Berg; Gaussian bounds for the Dirichlet heat kernel. J. Funct. Anal., 88 (1990), 267-278. [22] X. Wang; On the Cauchy problem for reaction-diffusion equations, Trans. Am. Math. Soc., 337 (1993) 549-590. [23] F. B. Weissler; Semilinear evolution equations in Banach spaces, J. Funct. Anal., 32 (1979) 277-296. [24] F. B. Weissler; Local existence and nonexistence for semilinear parabolic equations in Lp. Indiana Univ. Math. J., 29 (1980), 79-102. [25] F. B. Weissler; Existence and nonexistence of global solutions for a semilinear heat equation. Isr. J. Math., 38 (1981) 29-40. Aldryn Aparcana Facultad de Ciencias, Universidad Nacional San Luis Gonzaga, Ica, Perú Email address: aldryn.aparcana@unica.edu.pe Brandon Carhuas-Torre Departamento de Matemática, Universidade Federal de Pernambuco, Av. Jornalista Anibal Fernandes, Cidade Universitáa, Recife, Pernambuco, Brasil Email address: brandon.carhuas@ufpe.br Ricardo Castillo Facultad de Ciencias, Departamento de Matemática, Universidad del B́ıo-B́ıo, Avenida Collao 1202, Casilla 5-C, Concepción, B́ıo-B́ıo, Chile Email address: rcastillo@ubiobio.cl Miguel Loayza Departamento de Matemática, Universidade Federal de Pernambuco, Av. Jornalista Anibal Fernandes, Cidade Universitáa, Recife, Pernambuco, Brasil Email address: miguel.loayza@ufpe.br 1. Introduction 2. Preliminaries 3. Existence and non-existence 4. Uniqueness 5. Applications 5.1. Case g(t)=tp with p>1. 5.2. Case g(t)=et with >0 and h(t)=ta 6. Concluding remarks Acknowledgments References