Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.


Introduction
This paper is concerned with the positivity of solutions to Cauchy problems for fourth order parabolic equations.
We say that a parabolic Cauchy problem has a positivity preserving property if nonnegative and non-trivial initial data always yield solutions which are positive in the whole space and for any positive time. It is well known that second order parabolic Cauchy problems enjoy a positivity preserving property.
On the other hand, it follows from [1, Theorems 7.1 and 9.2] that the elliptic operator being of second order is not only sufficient but also necessary for the corresponding Cauchy problem to enjoy a positivity preserving property. This means that this property does not hold for the Cauchy problem for the biharmonic heat equation (see also [2][3][4]): where ϕ is a suitable measurable function and N ≥ 1. "Suitable" means locally integrable and less than exponential growth at infinity. One should keep in mind that small times are particularly sensitive for change of sign. For large times, at least in bounded domains, the behaviour is more and more dominated by the elliptic principal part (and a strictly positive first eigenfunction would yield eventually positive solutions to the initial boundary value problem). The loss of the positivity preserving property for (1.1)-(1.2) is reflected by the sign change of the fundamental solution G( · , t) of the operator ∂ t + (−∆) 2 in R N × (0, ∞) for all t > 0. See Section 2.1 below. Moreover, it was even shown in [3, Theorem 1] that for any non-negative and non-trivial function ϕ ∈ C ∞ c (R N ) there exists T > 0 satisfying the following: inf The issue of eventual local positivity was studied further in [4] for initial data with specific polynomial decay at inifinty: For β > 0, initial data ϕ(x) := 1 |x| β + g(x) (1. 5) with g ∈ A β := {h ∈ C(R N ) | h(x) > 0, h(x) = o(|x| β ) as |x| → ∞} were considered. It was proved in [4, Theorem 1.1] that eventual local positivity holds locally uniformly and at an explicit asymptotic decay rate. At the same time this eventual positivity cannot be expected to be global, see [4,Theorem 1.2]: For each β ∈ (0, N ) and t > 1 there exists a radially symmetric function g ∈ A β such that (1.3) holds for ϕ as in (1.5).
In order to understand the underlying reason for this change of sign even for large times and how initial data could look like to avoid this, a first step was made also in [4]: So, it is natural to ask the following general question like Barbatis and Gazzola in [5,Problem 13]: Problem A. For N ≥ 1, can one find suitable classes of initial data ϕ such that the corresponding solutions (1.4) to (1.1)-(1.2) are globally positive?
To the best of our knowledge, the existence of globally (in space) positive solutions to (1.1)-(1.2) has received only little attention. Beside Proposition 1.1, we mention Berchio's paper [6]. In [6,Theorem 11] she considered the initial datum ϕ(x) := |x| −β for β ∈ (0, N ) and introduced a right hand side with a strictly positive impact. (The reader should notice that the actual formulation of [6,Theorem 11] is not correct. A vanishing right hand side e.g. is not admissible.) In this situation she obtained eventual global positivity.
In Theorem 1.2 below we shall prove that this positivity is even global in time (i.e. even for arbitrarily small t > 0) and holds for the homogeneous biharmonic heat equation, provided that β > 0 is small enough. This, however, will follow from our first result which gives an affirmative answer to Problem A.
Let S be the Schwartz space and S be the space of tempered distributions. We define for (x, t) ∈ R N × (0, ∞), where ·, · is the duality pairing between S and S. For ϕ ∈ S we denote by F[ϕ] the Fourier transform of ϕ. If ϕ ∈ S is even smooth then this is given by: (1.6) Theorem 1.1. Let N ≥ 3 and ϕ ∈ S . Assume that all of the following conditions hold : is real valued, radially symmetric and positive.
Moreover, Theorem 1.2 is applied to show (to the best of our knowledge for the first time) the existence of global-in-time positive solutions to the Cauchy problem for the following fourth order semilinear parabolic equation: 12) where N ≥ 1, ϕ > 0 is a "suitable" measurable function, ε > 0 is a parameter, and This "super-Fujita" condition is necessary in order to have global positive solutions because Egorov and coauthors showed in [7, Theorem 1.1] finite time blow up of any positive solution in the "sub-Fujita" case 1 < p ≤ 1 + 4/N . See the ground breaking work [8] of Fujita for second order analogues. We first make clear that we understand the notion of solution to problem (1.11)-(1.12) in the strong sense: Definition 1.1. Let ϕ be locally integrable and bounded at infinity and ε > 0. We say that u ∈ C((0, ∞); BC(R N )) is a global-in-time solution to problem (1.11)-(1.12) if u satisfies Here, BC(R N ) denotes the space of bounded continuous functions. Global existence of presumably sign changing solutions for similar problems was studied first by Caristi and Mitidieri in [9]. As for eventual local positivity the following was proved in [4, Theorem 1.4]: For ϕ given by (1.5) with β ∈ (4/(p − 1), N ) and g ∈ A β and ε > 0 small enough, there exists a global-in-time solution u to problem (1.11)-(1.12), which is eventually locally positive. However, to the best of our knowledge, there is no result for the existence of globally positive solutions to problem (1.11)-(1.12). Therefore, similarly to Problem A, it is also natural to ask the following question: Problem B. Are there initial data ϕ such that there exists a global-in-time positive solution to problem (1.11)-(1.12)?
As an application of Theorem 1.2 (ii), we have: (1.14) Then for sufficiently small ε > 0, there exists a global-in-time solution u to problem (1.11)-(1.12) such that where M * > 0 depends only on N and p.
Let ϕ be as in Theorem 1.3. Then ϕ belongs to the weak Lebesgue space L rc,∞ (R N ), where The existence of a global-in-time solution to problem (1.11)-(1.12) with sufficiently small ε > 0 and ϕ ∈ L rc,∞ (R N ) is obtained in [10, Theorem 3.4 and Remark 3.7] (see also [11, Theorem 1.1]). However, in order to prove Theorem 1.3, we need to study the decay of global-in-time solution to (1.11)-(1.12) (which are not necessarily positive). (1.17) Remark 1.1. It is a natural question to ask whether our results can be generalised to Cauchy problems where the biharmonic operator is replaced by the polyharmonic operator (−∆) m with m > 1. For related questions, results in this direction have already been obtained. Indeed, Ferreira and Villamizar-Roa show in [10] well-posedness for problem (1.11)-(1.12) with (−∆) m instead of (−∆) 2 and assuming p > 1 + (2m)/N . They allow even for any fractional m > 0. Concerning problem (1.1)-(1.2) with (−∆) m instead of (−∆) 2 and compactly supported nonnegative initial datum, Ferreira and Ferreira prove in [12] eventual local positivity for any fractional polyharmonic operator (i.e. m > 1) thereby solving [5, Problem 10] mentioned by Barbatis and Gazzola. In view of the techniques developed in these papers we are confident that the present paper can be extended to the general polyharmonic framework.
The rest of this paper is organised as follows. In Section 2 we recall several properties of the fundamental solution G, of the Fourier transform of radially symmetric functions, and of Bessel functions. In Section 3 we prove Theorems 1.1 and 1.2. Section 4 is devoted to the proofs of Theorems 1.3 and 1.4.

Preliminaries
In this section, we recall some properties of the fundamental solution G, of the Fourier transform of radially symmetric functions, and of Bessel functions which will be useful in order to prove our results.

Fundamental solution G
We collect properties of the fundamental solution G without proof (for details, see e.g. [3,4,13]). Let J µ be the µ-th Bessel function of the first kind. Then G is given by It is known that f N changes sign infinitely many times, see [4,Theorem 2.3].
In what follows the constants c i > 0 (i = 1, 2, 3) depend only on N .
• For t > 0, the function G(·, t) belongs to Schwartz space S. More precisely, f N satisfies Here, F denotes the Fourier transform defined in (1.6).

Fourier transform of radially symmetric function
To show positivity of S(t)ϕ, we use the representation of the Fourier transform of radially symmetric functions. According to [14,Theorem 9.10.5] the Fourier transform of f (x) = g(|x|) ∈ L 1 (R N ) is given by
3 Existence of positive solutions to problem (1.1)-(1.2) In this section, we prove the sufficient condition on ϕ to ensure [S(t)ϕ](x) > 0 for (x, t) ∈ R N × (0, ∞). In what follows, the letter C denotes generic positive constants and they may have different values even within the same line.

General initial data
This section is devoted to the proof of Theorem 1.1.
Proof of Theorem 1.1. Since by (2.3) +ix·ξ for x ∈ R N and ξ ∈ R N , we deduce from (2.4) and the assumption in Theorem 1.1 that

Special initial data ϕ(x) = |x| −β
In this section, we prove Theorem 1.2. To this end we consider another representation of S(t)ϕ. Let β ∈ (0, N ). Since we see that for (x, t) ∈ R N × (0, ∞). Thus, in order to prove Theorem 1.2, it suffices to show that F N,β > 0.
The positivity statement will then be a direct consequence of Proposition 2.1 and Remark 2.2 provided that N ≥ 3. In order to cover also the small dimensions N = 1, 2, we need some preparations. We remark that by a change of variables F N,β satisfies for η > 0. We remark that F N,β can be also defined for β ≥ N . In the following, we consider F N,β with N ≥ 1 and β > 0.
for η > 0, N ≥ 1 and β > 0. In the following two lemmas we study the asymptotic behaviour of F N,β at 0 and at ∞. Proof. We prove this lemma by means of an inductive argument. Let N ≥ 1 and β > 0. We first claim that for k ∈ N ∪ {0} there exists {a k l } k l=0 ⊂ R such that for η > 0  It is clear that (3.7) holds for k = 0.
We now turn to the proof of Theorem 1.2.
In the case N = 2, we deduce from (2.9) that for η > 0 and β > 0. Since we have already proved in (3.15) that F 4,β+2 (η) is positive for We prove that we can extend the positivity result to β > β 0 . Assume that there exist If {η m } ∞ m=1 is bounded then η m converges, after passing to a subsequence, to some η 0 ∈ [0, ∞). Otherwise, a subsequence of {η m } ∞ m=1 goes to infinity. In what follows it is important that a careful inspection of the proofs of Lemmas 3.1 and 3.2 shows that the arguments are uniform with respect to β in a neighbourhood of β 0 . By an argument similar to that of the proof of Lemma 3.2, if η 0 = 0, This contradicts the positivity of F N,β 0 or A N,β 0 , respectively. In the case η 0 = 0, it follows with the same arguments as in Lemma 3.2 that again a contradiction. Therefore, we can find β 1 > β 0 which satisfies (1.7). One may observe that this argument even proves that the set {β ∈ (0, N ) : (1.7) is satisfied } is open in (0, N ). Finally, we show the existence of β 2 which satisfies (1.8). Since where f N is as in (2.1), F N,N has a nontrivial negative part. Since F N,β (η) is continuous with respect to β, F N,β has also a nontrivial negative part if β < N is sufficiently closed to N . Therefore, we obtain β 2 ≥ β 1 which satisfies (1.8).
By Lemma 3.2,(3.4) and (3.19) we also obtain (1.10) in (iii). Here, K * in (1.10) is a constant depending only on N and β. We remark that the upper bound holds irrespective of whether [S(t)ϕ](x) is positive or not. This proves (iii). The proof of Theorem 1.2 is complete.
As a direct consequence of Theorem 1.2-(i) we have: Proof. By the Hardy-Littlewood-Sobolev inequality (see e.g., [17,Theorem 4.3]) we see that (3.22) holds. From Fubini's theorem we deduce that In this section, we consider the semilinear equation (1.11) and prove Theorem 1.3. Set where c 2 is given by (2.2). We remark that the function H appears when we estimate the second term of the right hand side of (1.13) by (2.2) and |u(x, t)| ≤ C |x| β + t β/4 for R N × (0, ∞).
Regarding A 1 and A 2 , we have