Initial pointwise bounds and blow-up for parabolic Choquard-Pekar inequalities

We study the behavior as $t\to 0^+$ of nonnegative functions \begin{equation}\label{0.1} u\in C^{2,1} (\mathbb{R}^n\times (0,1)) \cap L^\lambda (\mathbb{R}^n\times (0,1)),\quad n\ge 1, \end{equation} satisfying the parabolic Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq u_t-\Delta u\leq(\Phi^{\alpha/n}*u^\lambda )u^\sigma \quad \text{ in }B_1 (0)\times (0,1) \end{equation} where $\alpha\in(0,n+2)$, $\lambda>0$, and $\sigma\geq0$ are constants, $\Phi$ is the heat kernel, and $*$ is the convolution operation in $\mathbb{R}^n\times (0,1)$. We provide optimal conditions on $\alpha,\lambda$, and $\sigma$ such that nonnegative solutions $u$ satisfy pointwise bounds in compact subsets of $B_1(0)$ as $t\to 0^+$. We obtain similar results for nonnegative solutions when $\Phi^{\alpha/n}$ is replaced with the fundamental solution $\Phi_\alpha$ of the fractional heat operator $(\frac{\partial}{\partial t}-\Delta)^{\alpha/2}$.

The regularity condition u ∈ L λ (R n × (0, T )) in (1.1) and the upper bound of n + 2 for α are natural because one does not want the nonlocal convolution operation on the right side of (1.2) to be infinite at every point in R n × (0, T ).
We also obtain results on the behavior as t → 0 + of nonnegative solutions of (1.1),(1.2) when Φ α/n in (1.2) is replaced with the fundamental solution Φ α of the fractional heat operator ( ∂ ∂t − ∆) α/2 . (See Remark 1.2.) A motivation for the study of (1.1),(1.2) comes from the nonlocal elliptic equation was introduced in [16] as a model in quantum theory of a polaron at rest (see also [2]). Later, the equation (1.4) appears as a model of an electron trapped in its own hole, in an approximation to Hartree-Fock theory of one-component plasma [6]. More recently, the same equation (1.4) was used in a model of self-gravitating matter (see, e.g., [5,12]) and it is known in this context as the Schrödinger-Newton equation.
Using nonvariational methods, the authors in [14] obtained sharp conditions for the nonexistence of nonnegative solutions to −∆u ≥ (Γ α/(n−2) * u λ )u σ in an exterior domain of R n , n ≥ 3.
For some very recent results on positive solutions Choquard-Pekar equations and inequalities which have an isolated singularity at the origin see [1] and [4].
Other examples of nonlocal equations which have been studied extensively in recent years are equations containing the fractional Laplacian and some of these equations are equivalent to equations containing convolutions with powers of the fundamental solution Γ of −∆u. For example, see [21] and [9].
On the other hand, we know of no results for nonlocal equations or inequalities when the nonlocal feature of the problem is due to convolutions with powers of the fundamental solution (1.3) of the heat equation. Our results for (1.1), (1.2) are, in this regard, new.
In this paper we consider the following question. and what is the optimal such ϕ when it exists?
We call the function ϕ in (1.5) a pointwise bound for u on compact subsets of Ω as t → 0 + . Remark 1.1. Suppose 0 < λ < (n + 2)/n. Then, since u = Φ, where Φ is the heat kernel given by (1.3), is a solution of (1.1),(1.2) and Φ(0, t) = (4πt) −n/2 , we see that any pointwise bound for nonnegative solutions u of (1.1),(1.2) on compact subsets of Ω as t → 0 + must be at least as large as t −n/2 and whenever t −n/2 is such a bound it is necessarily optimal.
Let ϕ : (0, 1) → (0, ∞) be a continuous function satisfying Then there exists a positive solution u of (1.1),(1.2) with T = 1 and Ω = R n such that u(0, t) = O(ϕ(t)) as t → 0 + . Theorems 1.1-1.3 completely answer Question 1.1 when the point (λ, σ) lies below the graph of g α or above the graph of G α . In particular, if u is a nonnegative solution of (1.1),(1.2) where (λ, σ) lies in the first quadrant of the λσ-plane and either σ < g α (λ) or σ > G α (λ) then according to Theorems 1.1-1.3 either (i) ϕ(t) = t −n/2 is an optimal a priori pointwise bound for u on compact subsets of Ω as t → 0 + ; or (ii) ϕ(t) = t −(n+2)/(2λ) is an optimal a priori pointwise bound for u on compact subsets of Ω as t → 0 + ; or (iii) no pointwise a priori bound exists for u on compact subsets of Ω as t → 0 + , that is solutions can be arbitrarily large as t → 0 + . The regions in which these three possibilities occur are shown in Figures 1 and 2. Also included in Figures 1 and 2 is an open triangular region marked with a question mark. For (λ, σ) in this region we have no results for Question 1.1.
Concerning the case that (λ, σ) lies on the graph of g α we have the following result.
(i) If 0 < λ < n+2−α n and σ = g α (λ) then ϕ(t) = t −n/2 is a poinwise bound for nonnegative solutions u of (1.1),(1.2) on compact subsets of Ω as t → 0 + . (ii) If α ∈ (2, n + 2), λ > n+2 α−2 , and σ = g α (λ) then there does not exist an a priori pointwise bound for nonnegative solutions u of (1.1),(1.2) on compact subsets of Ω as t → 0 + . When a pointwise a priori bound as t → 0 + for nonnegative solutions u of (1.1),(1.2) on compact subsets of Ω does not exist, as in Theorems 1.3 and 1.4(ii), we prove this by constructing for any given continuous function ϕ : (0, 1) → (0, ∞) a nonnegative solution u of (1.1),(1.2) consisting of a sequence of smoothly connected peaks centered at (x j , t j ) where t j → 0 + such that When such a pointwise a priori bound does exist, as in Theorems 1.1 and 1.4(i), we reduce the proof of this fact to ruling out the possibility of such peaked solutions.
If α ∈ (0, n + 2) and λ > 0 then one of the following three conditions holds: n ≤ λ < ∞. The proofs of Theorems 1.1-1.4 in case (i) (resp. (ii), (iii)) are given in Section 3 (resp. 4, 5). In Section 2 we provide some lemmas needed for these proofs. Our approach relies on an integral representation formula for nonnegative supertemperatures (see Appendix A), some integral estimates for heat potentials (see Appendix B), and Moser's iteration (see Lemmas 4.1 and 5.2).
In this paper, we denote by P r (x, t) the open circular cylinder in R n × R of radius √ r, height r, and top center point (x, t). Thus However, by checking the proofs of our results, we find that Theorems 1.1, 1.3, and 1.4 remain correct if Φ(x, t) α/n in (1.2) is replaced with any function of the form where C 1 (n, α) and C 2 (n, α) are any given positive constants. In particular, since the fundamental solution Φ α of the fractional heat operator ( ∂ ∂t − ∆) α/2 , α ∈ (0, n + 2), is given by where Φ is the heat kernel (1.3) (see [18,Chapter 9, Section 2]), we find for 0 < α < n + 2 that is of the form (1.10). Thus Theorems 1.1, 1.3, and 1.4
Proof. Make the change of variables z = √ γ(x − y).
The following lemma will be needed to estimate the last integral in (2.15).
Lemma 2.4. Suppose u ∈ L p (Ω × (0, T )) (2.18) for some open subset Ω of R n , n ≥ 1, and some constants p ∈ [1, ∞) and T > 0. Assume also that for some finite positve Borel measure µ on R n . Then for each compact subset K of Ω we have Proof. The proof consists of two steps.
Step 1. In this step we prove Lemma 2.4 in the special case that for some x 0 ∈ R n and some r > 0. Clearly we can assume to complete step 1, it suffices to prove v and w satisfy (2.19) when Ω and K are given by (2.20).
Since for |x − y| ≥ r and t > 0 Thus w satisfies (2.19) when Ω and K are given by (2.20).
For |y| ≤ 2r and τ > 0 it follows from Lemma 2.2 that We obtain therefore from Jensen's inequality and Fubini's theorem that We now use (2.21) to show v satisfies (2.19). For 0 < τ < t and x ∈ R n it follows from standard L p -L q estimates with q = ∞ (see [17,Prop. 48 → 0 as t → 0 + by (2.18) and (2.21). Thus v satisfies (2.19) when Ω and K are given by (2.20).
Step 2. We now use Step 1 to complete the proof. For each
Proof. When β = 0, Lemma 2.7 follows directly from Lemma 2.2. Hence we can assume β ∈ (0, n + 2). Under the change of variables we see that the left side of (2.23) equals Proof. We consider three cases.
Lemma 2.9. Suppose α > 0 and T are constants. Then for s < t ≤ T and |x| ≤ √ where the last two inequalities need some explanation. Since |x| ≤ √ T − t < √ T − s, the center of the ball of integration in (2.24) is closer to the origin than the center of the ball of integration in (2.25). Thus, since the integrand is a decreasing function of |z|, we obtain (2.25). Since √ T − s ≥ √ t − s, the ball of integration in (2.25) contains the ball of integration in (2.26) and hence (2.26) holds.
Then f j and u j are C ∞ and Hu j = f j in R n × R.
Since |x| ≥ r 0 and 0 < s < t < 1 we have and thus by the definition of z 0 we obtain from (3.31) that
By (3.25) we find that f j (y, s) dy ds < ∞ provided we take a subsequence if necessary. Hence, since the C ∞ functions f j have disjoint supports, we see that the function u : is C ∞ and by (3.18) we have From (3.26) we have u ∈ L λ (R n × (0, 1)) provided we take a subsequence of u j if necessary. Thus (3.6) holds.
It therefore follows from (2.14) and Remark 2.
Since by (2.14) and Lemma 2.8 we have Φ(y −ȳ, s −s)Hv(ȳ,s) dȳ ds for (y, s) ∈ P Rt j /4 (x j , t j ) it follows from Lemma 2.6 that for (x, t) ∈ P Rt j /4 (x j , t j ) we have Also by Jensen's inequality, (4.5) and Lemma 2.7 we have for x ∈ R n , t > 0, and λ ≥ 1 that We claim that (4.19) also holds for 0 < λ < 1. To see this, let x ∈ R n and t > 0 be fixed and define f (y, s) = Φ(x − y, t − s) α/n and g(y, s) = Then by Lemma 2.7 with β = 0 and β = n we have respectively, where C depends on neither x nor t. Thus by Jensen's inequality we find for (x, t) ∈ R n × (0, ∞) and 0 < λ < 1 that That is (4.19) also holds for 0 < λ < 1.
It therefore follows from (2.15), (4.17), (4.18), and Lemma 2.7 that for (x, t) ∈ P Rt j /8 (x j , t j ) we have (y,s)∈P 8 where C > 0 depends on R but not on j.
Also, similar to the way (4.9) was derived, we obtain We see therefore from (2.13) that for (x, t) ∈ P Rt j /8 (x j , t j ) and R ∈ (0, 1] we have Hence under the change of variables (4.13), we obtain from (4.12) and (4.6) that Hv(x, t) for (ξ, τ ) ∈ P R (0, 0) and R ∈ (0, 1] where C > 0 depends on R but not on j. To complete the proof of Theorem 4.1 we will need the following lemma. for some constants p ∈ [1, n+2 2 ] and R ∈ (0, 1]. Then there exists a positive constant C 0 = C 0 (n, λ, σ, α) such that the sequence for some q ∈ (p, ∞) satisfying Proof. For R ∈ (0, 1] we formally define operators N R and I R by where ε is as in (4.6). Then p 2 ∈ (p, ∞) and thus by Theorem B.2 we have and where · p := · L p (P 4R (0,0)) . Since we see by (4.6) that p 2 λ > 1. (4.27) Now there are two cases to consider.
Thus for σ ≤ 1 we have and for σ > 1 it follows from (4.32) and (4.6) that Thus defining q ∈ (p,q) by That is (4.23) holds.
We return now to the proof of Theorem 4.1. By (4.14) the sequence {f j } is bounded in L 1 (P 4 (0, 0)). (4.35) Starting with this fact and iterating Lemma 4.1 a finite number of times (m times is enough if m > 1/C 0 ) we see that there exists R 0 ∈ (0, 1) such that the sequence Thus (4.20) implies the sequence Since by Lemma 2.8, we see that (4.35) and (4.36) contradict (4.15). This contradiction completes the proof of Theorem 4.1.

5.
The case λ ≥ n+2 n In this section we prove Theorems 1.1-1.4 when λ ≥ n+2 n . For these values of λ, Remark 2.1 and the following theorem imply Theorem 1.1.
Also by (5.14) we have lim inf To complete the proof of Theorem 5.1 we will require the following lemma.
By taking a subsequence we can assume the sets Ω j are pairwise disjoint. Let χ j : R n × R → [0, 1] be a C ∞ function such that χ j ≡ 1 in ω j and χ j ≡ 0 in R n × R\Ω j . Define f j , u j : R n × R → [0, ∞) by Then f j and u j are C ∞ and Hu j = f j in R n × R. (5.59) By Theorem B.2 with p = n + 2 and q = ∞ we see that provided we decrease ε j if necessary because |Ω j \ω j | → 0 as ε j → 0. Also, it follows from (5.43) 2 , (5.50) 1 , (5.37), (5.56) 1 , (5.54), and (5.53) that there exists a positive constant M , independent of j, such that for (x, t) ∈ Ω j we have provided we take a subsequence if necessary, where Ψ is defined by (5.33).
In order to obtain a lower bound for u j in ω j , note first that for s < t ≤ a j + ε j and |x| ≤ H j (t) we have by Lemma 2.9 that Next using (5.62) and (5.63), we find for (x, t) ∈ Ω j that It therefore follows from (5.58), (5.57), and (5.60) that for (x, t) ∈ Ω j we have  Repeating the derivation of (5.70) with β replaced with 1, we find that R n ×R f j (y, s) dy ds → 0 as j → ∞.
f j (y, s)dηds < ∞ provided we take a subsequence if necessary. Hence, since the C ∞ functions f j have disjoint supports, it follows from Theorem 5.2 that the function u : R n × (0, ∞) → (0, ∞) defined by is C ∞ and from (5.59) and Theorem 5.2 we have By (5.71) and Theorem 5.2, u ∈ L λ (R n × (0, 1)) provided we take a subsequence of u j if necessary. Thus (5.44) holds.

Appendix A. Representation formula
In this appendix we provide the following representation formula for nonnegative supertemperatures.

Appendix B. Heat potential estimates
In this appendix we provide estimates for the heat potentials (J α f )(x, t) = where Φ is given by (1.3), Ω = R n × (a, b), and α ∈ (0, n + 2). The proofs of these estimates are given in [3, Appendix B].
Then J α f L q (R n ×R) ≤ C f L p (R n ×R) where C = C(n, p, α) is a positive constant.
Theorem B.2 is weaker than Theorem B.1 in that the second inequality in (B.1) cannot be replaced with equality. However it is stronger in that the cases p = 1 and q = ∞ are allowed.