The Move From Fujita Type Exponent to a Shift of It for a Class Of semilinear Evolution Equations With Time-dependent Damping

Resumo. In this paper, we derive suitable optimal L p − L q decay estimates, 1 ≤ p ≤ 2 ≤ q ≤ ∞ , for the solutions to the σ -evolution equation, σ > 1, with scale-invariant time-dependent damping and power nonlinearity | u | p , u


Introduction
In this paper we study the global (in time) existence of small data solutions to the Cauchy problem for the semilinear damped σ-evolution equations with scale-invariant time-dependent damping where µ > 0, σ > 1 and f (u) = |u| p for some p > 1.The nonlinearity may have several shapes, for instance, the derived results in this paper also hold if f (u) = |u| p−1 u or if f is locally Lipschitz-continuous satisfying [10] for some p > 1.The important information is that the nonlinearity is a perturbation which may create blow-up in finite time, well known in the literature as source nonlinearity.For this reason one is often able to prove such a global (in time) existence result in the supercritical range p > p c .Let us introduce some previous results to the Cauchy problem for the semilinear damping-free σ−evolution equations u tt + (−∆) σ u = |u| p , u(0, x) = 0, u t (0, x) = u 1 (x). (1.2) We begin with results for σ = 1.If 1 < p < p K (n) = n+1 [n−1]+ Kato [22] proved the nonexistence of global generalized solutions to (1.2), for small initial data with compact support.On the other hand, John [21] showed that p = 1 + √ 2 is the critical exponent for the global existence of classical solutions with small initial data in space dimension n = 3.A bit later, Strauss [38] conjectured that the critical exponent is given by p = p S (n), n ≥ 2, where p S (k) is the positive root of (k − 1)(p − 1) = 2 1 + 1 p . (1.3) Glassey ([15], [16]) solved this conjecture for classical solutions in space dimension n = 2.However, in space dimensions n > 3, Sideris [35] proved the nonexistence of global solutions in C([0, ∞), L

2(n+1)
n−1 (R n )) with arbitrarily small initial data for 1 < p < p S (n), even under the assumption of spherical symmetry.Later in the supercritical case p > p S (n), Lindblad and Sogge [24](see references therein for further reported results) proved a global existence result up to n ≤ 8 and for all n in the case of radial initial data (see also [23] for the case of odd space dimension).In [17], the authors removed the assumption of spherical symmetry.Then, for σ > 1 and for space dimensions 1 ≤ n ≤ 2σ, in [12] it was obtained the critical exponent to (1.2), p(γ 1 − 1) = 1 + 2 [γ1−1]+ with γ 1 = n σ , which is of Kato type.In [39], the authors proved global existence of small data solutions for the semilinear damped wave equation u tt − ∆u + u t = |u| p , u(0, x) = u 0 (x), u t (0, x) = u 1 (x), in the supercritical range p > 1 + 2/n, by assuming small initial data with compact support from the energy space.A previous existence result in space dimensions n = 1 and n = 2 was proved in [25].The compact support assumption on the initial data can be weakened.By only assuming initial data in Sobolev spaces, the existence result was proved in space dimensions n = 1 and n = 2 in [20], by using energy methods, and in space dimensions n ≤ 5 in [27], by using L r − L q estimates, 1 ≤ r ≤ q ≤ ∞.Nonexistence of the global small data solution is proved in [39] for 1 < p < 1 + 2/n and in [44] for p = 1 + 2/n.The critical case for more general nonlinearities has been recently discussed in [11].The exponent p(n) = 1 + 2/n is well known as Fujita exponent and it is the critical index for the semilinear parabolic problem [14]: The diffusion phenomenon between linear heat and linear classical damped wave models (see [18], [26], [27] and [29]) explains the parabolic nature of classical damped wave models with power nonlinearities from the point of view of decay estimates of solutions.
In [43] Wirth considered a more general model with a class of time dependent damping b(t)u t for which the critical exponent is still Fujita exponent p(n) = 1 + 2/n for the associate semilinear Cauchy problem with power nonlinearity |u| p (see [8] and [10]).
We state now well known results for the semilinear wave equation with scale-invariant time-dependent damping u tt − ∆u + µ 1+t u t = |u| p , u(0, x) = u 0 (x), u t (0, x) = u 1 (x). ( This model is critical, in the sense that it is relevant the size of the parameter µ to describe the asymptotic behavior of solutions.If µ ≥ 5 3 for n = 1 or µ ≥ 3 for n = 2, by assuming initial data in the energy spaces with additional regularity L 1 (R n ), a global (in time) existence result for (1.1) was proved in [4] for p > p(n).This result was extended by own D'Abbicco for higher space dimensions n ≥ 3 by assuming initial data in spaces with weighted norms for µ ≥ n + 2. For large values of µ, the exponent p(n) is critical for this model, that is, for 1 < p ≤ p(n) and suitable, arbitrarily small initial data, there exists no global weak solution [8].In [9] the authors studied the special case µ = 2 and showed that the critical exponent for (1.4) is given by p c = max{p S (n + 2), p(n)}.In the same paper the authors also conjectured that p c = max{p S (n + µ), p(n)} for µ ∈ (2, n + 2).The threshold value µ ⋆ is the solution to p S (n + µ ⋆ ) = p(n) and it is given by In [19], for suitable initial data, the authors obtained blow-up in finite time and gave the upper bound for the lifespan of solutions to (1.4 Recently, in [6] it is proved that for n = 1 the critical exponent to (1.4) is really p c = max{p S (1 + µ), 3}, where p S (k) is the solution to (1.3) and, in [5] it is proved the global existence of small data solutions for p > p(n) and µ ≥ n in space dimension 2 ≤ n ≤ 5.As far as we know, it is still an open problem to prove global existence of small data solutions for p > p(n) in the cases µ ⋆ < µ < n for n ≥ 3 and for p > p S (n + µ) for 0 < µ < µ ⋆ for n ≥ 2.
A related model to (1.4) is the semilinear wave equation with scale-invariant mass and dissipation For results about existence and non-existence of global (in time) small initial data solutions, we address the reader to [28,31,32,33] and the references therein.
The main goals in this paper are to derive L p − L q estimates and energy estimates for solutions to the parameterized linear Cauchy problem associated to (1.1) and to obtain the critical exponent for the global (in time) existence of small initial data solutions to (1.1).We conclude that µ = 1 is the threshold for the asymptotic behavior of solution to (1.1) at low space dimension n with respect to σ, it means that, for m = 1 or m = 2, the critical exponent is the Fujita type index p(γ The plan of the paper is the following: • in Section 2, we collect and discuss our main results; • in Section 3, we derive the L p − L q estimates for solutions to the associate linear Cauchy problem; • in Section 4, we apply the decay estimates previously derived to prove Theorems, from Theorem 2.1 to Theorem 2.4, for the nonlinear problems (1.1); • in Appendix, we include some notations and properties of special functions used to prove our results throughout the paper.

Main results
For some values of σ > 1 and the space dimension n, the critical exponent to (1.1) is the well-known Fujita exponent for µ > 1 and a shift of it for µ ∈ (0, 1).In the main results we are going to use the following notation where m = 1, 2 is related to the assumption on the initial datum Our first result is for small µ > 0. Combined with Proposition 2.1, it shows that for some values of σ > 1 and the space dimension n, the critical exponent is a shift of the Fujita exponent, unlike other cases it may appears a shift of Strauss exponent, for instance see [6,9,19] in the case σ = 1.In the next two theorems we are going to use the following notations and Theorem 2.1.Let σ > 1, 1 ≤ n < σ, and 1 − γ 1 < µ < min{1, µ 1 } with γ 1 and µ 1 given by (2.1) and (2.2).If with p and p 1 given by (2.1) and (2.3), then there exists ϵ > 0 such that for any initial datum (1.1).Moreover, for any δ > 0 and q ≥ 2 the solution satisfies the following estimates ) σ , in order that the range for p in (2.4) is not empty we have to assume that µ < µ 1 , where µ 1 given by (2.2) is the positive root of σµ . We stress out that (2.4) is empty if n ≥ σ.Hence, in general this condition can not be removed in Theorem 2.1.In particular µ 1 ≥ 1 if 3n ≤ 2σ, and the case µ = 1 can be included in Theorem 2.1, but in this case the δ loss of decay also appears in the first line of (2.5) and in (2.6).
In the following result we show that choosing the initial datum u 1 ∈ L 2 but u 1 / ∈ L 1 , we are able to prove global (in time) existence for a larger range of space dimension n and for small values of µ, but bigger bounds (lower and upper) for p come into play.(1.1).Moreover, the solution satisfies for any δ > 0 and q ≥ 2 the following estimates for any δ > 0 and , in order that the range for p in (2.7) is not empty we have to assume that µ < µ 2 , where µ 2 given by (2.2) is the positive root of 2σµ . We stress out that (2.7) is empty if n ≥ 2σ, so that this condition can not be removed in Theorem 2.2, and, in particular µ 2 ≥ 1 if 3n ≤ 4σ.
Remark 2.5.We remark that (2.8) may be written for any δ > 0 and q ≥ 2 as In Theorem 2.3 we take r ∈ [1,2] given by (2.9) Our next result is for sufficiently large µ.Combined with Proposition 2.2, it shows that for some values of σ > 1 and the space dimension n, the critical exponent is a Fujita type exponent.
given by (2.9) and µ > max then there exists ϵ > 0 such that for any initial datum , with p < q < q r .Moreover, for 2 ≤ q ≤ q the solution satisfies the following estimates Remark 2.6.The condition µ > 4n n+2σ implies that the range for p in (2.10), with r = 1, is not empty.To verify that for 2 < µ < 2n σ , we use again that n < 2σ.
Example 2.2.In the case σ = 2, Theorem 2.3 holds for µ > then there exists ϵ > 0 such that for any initial datum (1.1).Moreover, the solution satisfies for any δ > 0 and q ≥ 2 the following estimates (2.13) Remark 2.8.We remark that (2.13) may be written for any δ > 0 and q ≥ 2 as Example 2.3.In the case of the plate equation σ = 2, the global existence of small data solutions holds for all p > 1 + 8 n and µ > 1 for n = 1, 2. For the sake of simplicity, in the next two results about non-existence of solutions we restrict our analysis for integer σ.However, the test function method was recently applied in [7] for a class of σ−evolution operators with non-integer σ.First let us state the result in the case 0 < µ ≤ 1.The proof of the next result can be obtained with a slightly change in the proof of Theorem 1.5 in [41]: then there exists no global (in time) weak solution u ∈ L p loc ([0, ∞) × R n ) to (1.1).Remark 2.9.The proof of Proposition 2.1 also holds for µ > 1, however it is not optimal (see Proposition 2.2).
If µ > 1 then p(γ 1 + µ − 1) < p(γ 1 ) and so that Proposition 2.1 is not the counterpart of Theorem 2.3.Applying Theorem 2.2 in [8], one may have the following improvement of Proposition 2.1 and the counterpart of Theorem 2.3 is obtained. ) ). Remark 2.10.From Theorem 2.3 and Proposition 2.2 we conclude that for µ > 1 and for some values of σ and n, the Fujita type index p(γ 1 ) = 1 + 2σ n is the critical exponent to (1.1), whereas for 0 < µ ≤ 1, Theorem 2.1 and Proposition 2.1 implies that a shift of the Fujita type index is the critical exponent to (1.1) for some values of σ and n.
The next remark was suggested by Prof. M. D'Abbicco and says that Proposition 2.1 could also be obtained by applying Theorem 2.2 in [8].
Remark 2.12.Let us consider with ν > 0. Applying Theorem 2.2 in [8], one may derive a nonexistence result for The left-hand side is clearly true, due to 1 − µ < 2, and the right-hand side gives the condition for the desired critical exponent: 3. L p − L q estimates for solutions Let us consider the Cauchy problem for the linear σ-evolution equation with scale-invariant time-dependent damping Taking the partial Fourier transform with respect to the x variable in (3.1) we obtain According to [30] and [42], we have the following representation for the solution to (3.2) in terms of the Hankel functions H ± ρ : Proposition 3.1.Assume that u solves the Cauchy problem (3.1) for initial datum u 1 ∈ L 2 (R n ).Then the Fourier transform û(t, s, ξ) can be represented as where the multiplier ψ satisfies ) In order to derive estimates for u and its derivatives, we divide the extended phase space into zones to analyse the behavior of the Hankel functions H ± ρ (see Lemma 4.1 in Appendix): where In the following we decompose the multiplier ) as m = (1−χ)m+χm and χm = m χ i and estimate each of the summands (1−χ)m and m i := mχ i , i = 1, 2, 3: 1) .By using Haussdorff-Young inequality and Hölder inequality, setting In particular, if n σ Considerations in Z 2 : In Z 2 we may estimate If µ ̸ = 1 and 0 ≤ ζ + k ≤ 1 then, by using Haussdorff-Young inequality and Hölder inequality, setting We then conclude: If µ = 1 and 0 ≤ ζ + k ≤ 1 we may estimate and obtain Considerations in Z 3 : In this zone, since H ± ρ = J ρ ±iY ρ we use the following representation for the multiplier: if ρ, ρ − k ̸ ∈ Z, with k = 0, 1 and J ρ , Y ρ denote the Bessel functions of the first and second kind, respectively.We apply Lemma 4.1 (see Appendix) in the following estimates to both cases, which are slightly different.In the case ρ, ρ − k ̸ ∈ Z we obtain . By using Haussdorff-Young inequality and Hölder inequality, setting with a ≥ 0. In the case, ρ, ρ − k ∈ Z, we obtain In fact, we use the relation By using Haussdorff-Young inequality and Hölder inequality, setting Hence, if µ ̸ = 1, then we have the same estimate for both cases.
In the following, we state the estimates for solutions to the linear Cauchy problem (3.1) that will be used in Section 4. (i): Moreover, the solution to (3.1) satisfies the below estimates for all 2 ≤ q < 2n [n−2σ]+ : Remark 3.2.If n < rσ then the conclusions of Theorem 3.1 hold for all q ≥ 2.
Demonstração.The proof of (i): Suppose that µ > max{ 2n [n−rσ]+ and, it holds for all q ≥ 2 in the particular case r = 2 and n < 2σ.Applying the derived estimates at zone Z 1 with p = r and r ∈ [1, 2], we may estimate In Z 2 and Z 3 from (3.5) and (3.10) we have the following estimate: and if µ = 2n σ 1 − 1 q > 1 a logarithm term appears in the previous estimate.Thus (i) is concluded.The proof of (ii) and (v): The desired estimates in Z high are obtained by putting p = r or p = 2 in (3.12), respectively.In Z 1 we have Moreover, in Z 1 one may also get the estimate whereas a logarithm term may appears if n σ 1 Hence, we have in The proof of (iii) and (vi): Suppose that 0 whereas a logarithm term may appears if n σ 1 − 1 q = 1.We obtain the following estimates in Z 2 ∪ Z 3 : , a logarithm term appears in (3.13).The proof of (iv): Suppose that µ = 1.We have in Z 3 , and in We complete this section presenting some energy estimates for solutions in the Theorem 3.2 and also L 2 − L q estimates for solutions in the Corollary 3.1.
The solution to (3.1) satisfies the below estimates in according with the following values of µ: Moreover, the ∥∂ t u(t, •)∥ L 2 satisfies the same decay estimates of ∥(−∆) Demonstração.At high frequencies Z high we have and We now use previous estimates on Section 3 with 2 )− γ σ (1 + s)∥u 1 ∥ L p and by using (3.14), in Z 1 we may estimate The proof of (ii): Suppose that 1 in Z 3 by using (3.14).Suppose that max{2 − 2n by using (3.15).
The solution to (3.1) satisfies the below estimates for all 2 ≤ q ≤ 2n [n−2σ]+ in according with the following values of µ: Demonstração.Using the fractional Sobolev embedding for all 2 ≤ q ≤ 2n [n−2σ]+ , the desired estimates may be obtained from Theorem 3.2.□ Remark 3.3.Using the fractional Sobolev embedding and Theorem 3.
for all 2 ≤ q ≤ 2n [n−2σ]+ Hence, Theorem 3.1 (i) for r = 2 is a consequence of Theorem 3.2 (i) with p = 1.The same analysis may be done for small values of µ > 0.

Proof of the Global existence results
According to Duhamel's principle, solutions of (1.1) are interpreted as solutions to the nonlinear integral equation where K 1 (t, s, x) = F −1 (ψ)(t, s, x) and is the solution to the linear Cauchy problem (3.1) with s = 0.The proof of our global existence results is based on the following scheme.We define an appropriate data function space and an evolution space for solutions Z(T ), which is introduced later, equipped with a norm related to the estimates of solutions to the linear problem (3.1) with s = 0 such that We define the operator F such that, for any u ∈ Z, then we prove the estimates ) By standard arguments, since u lin satisfies (4.3) and p > 1, from (4.4) it follows that u lin + F maps balls of Z into balls of Z, and for small data in A, from (4.5) F is a contraction.So, the estimates (4.4)-(4.5)lead to the existence of a unique solution to (4.1), that is, u = u lin + F u, satisfying (4.3).We simultaneously gain a locally in time for large data and globally in time for small data existence result [13].

Appendix
In this section we include notations and properties of special functions used throughout the paper.
Notation 2. We write f ≲ g if there exists a constant C > 0 such that f ≤ Cg, and f ≈ g if g ≲ f ≲ g.
Notation 3. We denote by f = Ff or f (t, •) = Ff (t, •) the partial Fourier transform, with respect to the space variable x, of a tempered distribution S ′ (R n ) or of a function, in the appropriate distributional or functional sense and its inverse transform by F −1 .
Notation 4. By L p = L p (R n ), p ∈ [1, ∞], we denote the space of measurable functions f such that |f | p has finite integral over R n , if p ∈ [1, ∞), or has finite essential supremum over R n if p = ∞.We denote by W m,p , m ∈ N, the space of L p functions with weak derivatives up to the m-th order in L p .We denote by H s (R n ) and Ḣs (R n ), s ≥ 0, the spaces of tempered distributions S ′ (R n ) with (1 + |ξ| 2 ) s 2 û ∈ L 2 and |ξ| s û ∈ L 2 , respectively.