Effect of decay rates of initial data on the sign of solutions to Cauchy problems of polyharmonic heat equations

In this paper, we consider the sign of solutions to Cauchy problems of linear and semilinear polyharmonic heat equations. Cauchy problems for higher order parabolic equations have no positivity preserving property in general, however, it is expected that solutions to these Cauchy problems are eventually globally positive if initial data decay slowly enough. We first show the existence of the threshold of the decay rate of initial datum which separates whether the corresponding solution to the Cauchy problem of the linear polyharmonic heat equation is eventually globally positive or not. Applying this result, we construct eventually globally positive solutions to the Cauchy problem of the semilinear polyharmonic heat equation under the super-Fujita condition.


Introduction
This paper is concerned with the sign of solutions to Cauchy problems of linear and semilinear polyharmonic heat equations.
Contrary to Cauchy problems of second order parabolic equations, it is known that those of higher order parabolic equations do not enjoy a positivity preserving property (see e.g. [4]). Here, the positivity preserving property means that non-negative and non-trivial initial data always yield solutions which are positive in the whole space and any times. In particular, this property does not hold for the Cauchy problem of the B Nobuhito Miyake n-miyake@g.ecc.u-tokyo.ac.jp 1 Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8914, Japan polyharmonic heat equations (see e.g. [2,9,10,14]): where N ≥ 1, m ∈ N with m ≥ 2 and u 0 ∈ (L 1 + L ∞ )(R N ).
The loss of the positivity preserving property for problem (P1) is also derived from the oscillation property of the fundamental solution G m of the operator ∂ t + (− ) m in R N × (0, ∞) (see e.g. [2,4,8,19]). Indeed, it was shown in [14,Theorem 1] and [9,Theorem 1.3] that, by the use of the existence of the negative part of G m , for any non-negative and non-trivial function u 0 ∈ C ∞ c (R N ) there exists T > 0 satisfying the following: where This means that we cannot expect global (in space and time) positivity of the solution to problem (P1) if the initial datum has compact support in R N . It is also known that, after enough time, solutions to problem (P1) with suitable initial data is locally (in space) positive. Indeed, it was shown in [14, Theorem 1] and [9, Theorem 1.1] that for any non-negative and non-trivial function u 0 ∈ C ∞ c (R N ) the corresponding solution to problem (P1) is eventually locally positive. Here, we say that a function u = u(x, t) is eventually locally positive if for any compact set V ⊂ R N there exists T > 0 satisfying u(x, t) > 0 for all (x, t) ∈ V × [T , ∞). ( Moreover, [10] introduced a new class of initial data C β := 1 g(x) + |x| β g ∈ C(R N ), g ≥ 0 on R N , g(0) > 0, lim |x|→∞ |x| β g(x) = 0 and proved the following: for any β ∈ (0, N ), u 0 ∈ C β and compact set V ⊂ R N it holds that lim t→∞ t β/4 S 2 (t)u 0 (x) = A β uniformly on V for some A β > 0. This result means that S 2 (t)u 0 (x) is eventually locally positive if u 0 ∈ C β . They also mentioned that A β > 0 does not depend on V , however, it was not known whether the above limit is uniform on R N and hence S 2 (t)u 0 (x) is eventually globally positive or not. Here, we say that a function u = u(x, t) is eventually globally positive if there exists T > 0 satisfying (2) There are several results for such a "global in space" positivity of higher order parabolic equations. As for the eventual global positivity, [1, Theorems 9, 10 and 11] considered problem (P1) with an inhomogeneous term, and gave sufficient conditions for initial data and inhomogeneous terms such that the corresponding solution is eventually globally positive. In this case, however, it is necessary to assume that the inhomogeneous term has a "strictly positive impact" (see also the comment in [16, p.354]). In the homogeneous case, it is clear that S m (t)u 0 (x) ≡ 1 if u 0 ≡ 1. Hence it presumably holds that S m (t)u 0 (x) is eventually globally positive if the decay rate of u 0 is sufficiently slow. Indeed, [10,Proposition A.6] showed that [S m (t)| · | −β ](x) is global (in space and time) positive if N = 1, m = 2 and β is sufficiently small. Recently, [16,Theorem 1.2] and [22,Theorem B] generalized this result and gave some properties of [S m (t)| · | −β ](x). In order to describe the precise statement, it is convenient to define the following: (ii) If β ∈ P, then They also obtained a lower bound of β 1 , however, it was left open what happens in the case β ∈ [β 1 , β 2 ] or whether possibly β 1 = β 2 . The first purpose of this paper is to improve Proposition 1 as follows: Then there exists β * ∈ (0, N ) satisfying the following: Theorem 2 implies that β * is the threshold which determines whether S m (t)| · | −β (x) is globally (in space and time) positive or not. As an effective application of Theorem 2, we prove that β * is also the threshold for the decay rate of u 0 such that S m (t)u 0 (x) is eventually globally positive or not. More precisely, we show the following: Then S m (t)u 0 (x) is eventually locally positive. Moreover, the following hold: (ii) If β ∈ N , then there exists T > 0 satisfying (1).
We remark that, at least the case m = 2, (5) does not hold in general if we take T = 0 (see [10,Theorem 1.2]). In order to prove Theorem 3, we first prove the statement of Theorem 3 except for assertion (i) under a "weak" decay condition (18) instead of (4), see Proposition 7. Combining Proposition 7 with Lemma 9, we prove Theorem 3. Proposition 7 is also useful to consider an asymptotic behavior of S m (t)u 0 (x) when u 0 is given by convolutions with | · | −β and measures, see Proposition 10 and Corollary 12.
The second purpose of this paper is to prove an analogous result to Theorem 3 for the following Cauchy problems of semilinear polyharmonic heat equations: where N ≥ 1, m ∈ N with m ≥ 2, p > 1 + 2m/N , ε > 0 is sufficiently small constant and u 0 ∈ C b (R N ), which is the set of bounded continuous functions on R N . We define the solution to problem (P2) as follows: and ε > 0. We say that u : and u satisfies . We also say that u is an eventually locally (resp. globally) positive solution to problem (P2) if u is a global-in-time solution to problem (P2) and eventually locally (resp. globally) positive.
We remark that super-Fujita condition p > 1 + 2m/N is necessary to exist an eventually globally positive solution to problem (P2) (see [5,Theorem 1]). See the ground breaking work [11] for second order analogues.
As for the eventual local positivity, the following was proved in [10,Theorem 1.4]: if m = 2, u 0 ∈ C β , β ∈ (4/( p−1), N ) and ε > 0 is sufficiently small, then there exists an eventually locally positive solution to problem (P2). Recently, [16,Theorem 1.3] and [22,Theorem C] showed that there exists a globally (in space and time) positive solution to problem (P2) with u 0 (x) = |x| −2m/( p−1) if 2m/( p − 1) ∈ P. To the best of our knowledge, however, there is no result on the sufficient condition for initial data such that the corresponding solution to problem (P2) is eventually globally positive.
As an analogous result to Theorem 3, we obtain: (4). Then there exists ε 1 > 0 such that for any ε ∈ (0, ε 1 ) there exists a unique (in the sense of Definition 1) eventually locally positive solution u to problem (P2). Moreover, u satisfies the following: Before showing Theorem 4, we construct a unique global-in-time solution to problem (P2), see Theorem 13 (which is a generalization of [10, Theorem 1.5]). There are several studies for the existence and the asymptotic behavior of global-in-time solutions (which may change its sign) to Cauchy problems of higher order parabolic equations (see e.g. [3,6,12,17,18,21]). In contrast to these results, in Theorem 13, we prove the existence of a global-in-time solution to problem (P2) which satisfies a pointwise estimate (see (26)).
The rest of this paper is organized as follows. In Sect. 2, we recall several properties of Morrey spaces and of fundamental solutions G m . We also set up some notation related to S m (t)| · | −β (x). In Sect. 3, we prove Theorem 2. Section 4 is divided into three subsections. In Sect. 4.1, we give a proof of Proposition 7 which is needed to show Theorem 3. As an application of Proposition 7, we prove Theorems 3 in Sect. 4.2. In Sect. 4.3, we give an another application of Proposition 7. Section 5 is devoted to the proof of Theorems 4 and 13 . In Sect. 6, we give proofs of technical propositions which are needed to prove Lemma 9 and Theorem 13.

Preliminaries
In this section, we first recall several properties of Morrey spaces and of fundamental solutions G m . We also introduce some notation related to S m (t)| · | −β (x). In what follows, by the letter C we denote generic positive constants, and they may have different values also within the same line.

Morrey spaces
For q ∈ [1, ∞], we define the Morrey space M q as the set of Lebesgue measurable functions f on R N such that we see that M 1 and M ∞ coincide with L 1 (R N ) and L ∞ (R N ), respectively. We also define M + as the set of (non-negative) finite Borel measures on R N . Then, for β ∈ (0, N ) it holds that where

Estimates of fundamental solutions G m
Let J ν be the ν-th Bessel function of the first kind. We first recall that G m can be rewritten as where Moreover, it is known that G m satisfies the following pointwise estimate (see e.g. [ where E m (r ) := exp[−c 2 r 2m/(2m−1) ] and c 1 , c 2 > 0 are constants depending only on N and m. Then we obtain the following L p -L q type estimates: Proof (11) immediately follows from the Young inequality and (10). (12) follows from the argument similar to that in the proof of [15, Proposition 3.2-(ii)]. Indeed, for any q ∈ [1, ∞) and f ∈ M q we observe from (10) and the layer cake representation that This implies that (12) holds for all q ∈ [1, ∞). In the case q = ∞, (12) is equivalent to (11) with q = ∞. Therefore, the proof of Proposition 5 is complete.
We finally give two estimates related to E m .

Proposition 6
Let N ≥ 1 and m ∈ N.
Proposition 6 is derived from an argument similar to that in [10, Propositions 6.1 and 6.2] (see also [13,Propositions 2 and 3]). For the completeness, we give a simpler proof in Section 6.

Existence of the thresholdˇ *
In this section, we prove Theorem 2. In the proof, we use the following: for any N ≥ 1, where A > 0 is a positive constant depending only on N , γ 1 and γ 2 (see e.g. [20, p.134]). For the sake of completeness, we give a proof of (16) in Appendix.

Proof of Theorem 2
We first prove the following claim: Fix β ∈ (0, β 0 ) arbitrarily. Applying Fubini's theorem, we see that for all r ≥ 0 and x ∈ R N with r = |x|. Here, we used (16) with γ 1 = β 0 and Therefore, the claim (17) follows. Set In order to complete the proof, it is left to show that β * does not belong to P and N . As mentioned in the proofs of [

Eventual global positivity of solutions to problem (P1)
In this section, we consider the asymptotic behavior of solutions to problem (P1) under three different conditions for initial data.

Case I: u 0 decays like |x| −ˇi n a weak sense
We first consider the case where u 0 satisfies for some c * > 0, where B R = {x ∈ R N | |x| < R} and χ A is the characteristic function of the subset A ⊂ R N . We remark that (18) is a weaker assumption than (4). Moreover, since (18) implies that u 0 must be close to c * |x| −β for sufficiently large x ∈ R N . The aim of this subsection is to prove the following proposition: Assume that there exist β ∈ (0, N ) and c * > 0 which satisfy (18). Then S m (t)u 0 (x) is eventually locally positive. Moreover, if β ∈ N , then there exists T > 0 satisfying (1).
In order to prove Proposition 7, we provide the following lemma: Lemma 8 Assume the same condition as in Proposition 7. Then Proof Fix R > 0 arbitrarily. It follows from Proposition 5 and (14) that Since R > 0 is arbitrary, the desired conclusion follows from the assumption (18).
We now turn to the proof of Proposition 7.

Proof of Proposition7
We first show the eventual local positivity of S m (t)u 0 (x). Fix a compact set V ⊂ R N arbitrarily. Then by H β (0) > 0 there exists T 1 > 0 such that On the other hand, by Lemma 8 there exists T 2 > 0 such that Setting T := max{T 1 , T 2 } > 0, we observe from (19) and (20) that . This means that S m (t)u 0 (x) is eventually locally positive. Assume that β ∈ N . Then there exists r 0 > 0 such that H β (r 0 ) < 0. Then by Lemma 8 there exists T > 0 such that In particular, for any (x, t) ∈ R N × [T , ∞) with |x| = r 0 t 1/2m we see that (1). Thus, the proof of Proposition 7 is complete.

Case II: u 0 decays like |x| −ˇi n a strong sense
In this subsection, applying Proposition 7, we prove Theorem 3. We first show the following lemma.

Lemma 9
Assume the same condition as in Theorem 3. Then Proof Fix ε > 0 arbitrarily. Then by (4) there exists R > 1 such that By (10) and (14) we have On the other hand, it follows from R > 1 and Proposition 6-(i) that Combining the above estimate with (21) and (22), we obtain Since ε > 0 is arbitrary, the desired conclusion follows.
We now turn to the proofs of Theorem 3.

Proof of Theorem 3
Since u 0 satisfies the assumption in Proposition 7, we see that S m (t)u 0 (x) is eventually locally positive and assertion (ii) is valid. Therefore, it suffices to show that S m (t)u 0 (x) is eventually globally positive if β ∈ P.
Since β ∈ P, it follows from (15) that K β > 0. Hence by Lemmas 8 and 9 there exists T > 0 satisfying . Then we observe from (15) that . This means that S m (t)u 0 (x) is eventually globally positive and satisfies (5). Hence the proof of Theorem 3 is complete.

Remark 1
We consider the case where u 0 satisfies (4) Since G m satisfies (9) and has a negative part, we deduce from the argument similar to that in the proof of Proposition 7 that there exists T > 0 satisfying (1) if M > 0.

Proof of Proposition 11
Fix R > λ > 0 arbitrarily. For x ∈ R N \B R and y ∈ B λ we see that Since by the above inequality for all x ∈ R N \B R , we deduce from (7) and (8) that and hence lim sup Since λ > 0 is arbitrary, we obtain (23). Thus, the proof of Proposition 11 is complete.
As a corollary of Proposition 10, we have: Corollary 12 Let N ≥ 1 and m ∈ N with m ≥ 2. Then the following hold: This corollary immediately follows from the following fact: for any N ≥ 1 and β ∈ (0, N ), it holds that where B > 0 is a constant depending only on N and β. For the sake of completeness, we give a proof of (24) in Appendix.

Semilinear problem (P2)
In this section, we consider the semilinear problem (P2). We first prove the existence of a unique global-in-time solution to problem (P2) with a pointwise estimate: Then there exists ε 2 > 0 which satisfies the following: for any ε ∈ (0, ε 2 ) there exists a unique (in the sense of Definition 1) global-in-time solution u to problem (P2) satisfying the following estimate: where W β is given in (13) and M > 0 is a constant given in (28).
Proof We first prove the uniqueness of global-in-time solutions to problem (P2). Let ) be solutions to problem (P2). Then it follows from (10) that we observe from the above inequality that u 1 = u 2 on R N ×(0, T * ]. Since u j ( j = 1, 2) satisfies for (x, t) ∈ R N × (0, ∞), we deduce from the argument similar to (27) that u 1 = u 2 on R N × (T * , 2T * ]. Iterating this argument, we see that u 1 = u 2 on R N × (0, ∞). This shows the uniqueness of global-in-time solutions to problem (P2). In the following, we prove the existence of a solution to problem (P2) which satisfies the desired properties stated in Theorem 4 via the contraction mapping theorem. By Proposition 6-(i), (10) and (25) we see that Fix ε > 0 arbitrarily and set Note that (X , d X ) is a complete metric space. We first consider the case β ∈ [2m/( p −1), N ). We observe from Proposition 6-(ii) and (10) that By a similar argument we see that Therefore, there exists ε 2 > 0 such that, for any ε ∈ (0, ε 2 ), is a contraction on X and hence there exists u ∈ X satisfying u = [u]. This means that u is a globalin-time solution to problem (P2) and u satisfies (26). Hence the proof of Theorem 13 with β ∈ [2m/( p − 1), N ) is complete.
In the case β > N , we observe from Proposition 6-(ii) and (10) that The rest of the proof is similar to that in the case β ∈ [2m/( p − 1), N ). Therefore, the proof of Theorem 13 is complete.
We now turn to prove Theorem 4. (4), there exists a unique global-in-time solution u to problem (P2) satisfying (26) if ε ∈ (0, ε 2 ), where ε 2 > 0 is given in Theorem 13. Moreover, by the same argument as in the proof of Theorem 13 we have

Proof of Theorem 4 By Theorem 13 and
We first prove that, taking ε > 0 small enough if necessary, u is eventually locally positive. Fix a compact set V ⊂ R N arbitrarily. Then, by the proof of Proposition 7 there exists T > 0 such that This together with (29) implies that is independent of T and V , there exists ε 1 > 0 such that for any ε ∈ (0, ε 1 ) the corresponding solution u is eventually locally positive. We next consider the case β ∈ P and prove assertion (i). By Theorem 3 there exists T > 0 such that This together with (29) implies that and ε ∈ (0, ε 1 ). Therefore, taking ε 1 > 0 small enough if necessary, we see that for any ε ∈ (0, ε 1 ) the corresponding solution u satisfies (6).
We finally consider the case β ∈ N and prove assertion (ii). By the proof of Proposition 7 there exist T > 0 and r 0 > 0 such that This together with (29) implies that with |x| = r 0 t 1/2m and ε ∈ (0, ε 1 ). Therefore, taking ε 1 > 0 small enough if necessary, we see that for any ε ∈ (0, ε 1 ) the corresponding solution u satisfies This implies that assertion (ii) is valid. Hence the proof of Theorem 4 is complete.

Proof of Proposition 6
The aim of this section is to prove Proposition 6.

Proof of Proposition 6-(i)
Then we have where, we used the estimate E m (r ) it follows from (31) that we deduce from (33) that On the other hand, we see that and Therefore, it follows from (34), (35) and (36) that Proposition 6-(i) is valid.
Acknowledgements The author is grateful to Professor Shinya Okabe (Tohoku University) for giving him much helpful comments. The author was supported in part by the Grant-in-Aid for Research Activity Start-up (No. JP21K20321) and JSPS Fellows (No. JP22J00221) from Japan Society for the Promotion of Science.
Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflict of interest
There is no conflict of interest regarding the content of this article.
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