On the critical regularity of nonlinearities for semilinear classical wave equations

In this paper, we consider the Cauchy problem for semilinear classical wave equations \begin{equation*} u_{tt}-\Delta u=|u|^{p_S(n)}\mu(|u|) \end{equation*} with the Strauss exponent $p_S(n)$ and a modulus of continuity $\mu=\mu(\tau)$, which provides an additional regularity of nonlinearities in $u=0$ comparing with the power nonlinearity $|u|^{p_S(n)}$. We obtain a sharp condition on $\mu$ as a threshold between global (in time) existence of small data radial solutions by deriving polynomial-logarithmic type weighted $L^{\infty}_tL^{\infty}_r$ estimates, and blow-up of solutions in finite time even for small data by applying iteration methods with slicing procedure. These results imply the critical regularity of source nonlinearities for semilinear classical wave equations.


Introduction
In the last forty years, the Cauchy problem for semilinear classical wave equations with power nonlinearity, namely, with p > 1, has been deeply studied by the mathematical community.For example, the questions on global (in time) existence of solutions, blow-up of solutions in finite time and sharp lifespan estimates of solutions were of interest.In particular, the critical exponent for the semilinear Cauchy problem (1.1) is given by the so-called Strauss exponent p S (n), which was proposed by Walter A. Strauss in [23].Nowadays, the correctness of the Strauss exponent is well-known.The Strauss exponent p S (n) is the positive root of the quadratic equation (n − 1)p 2 − (n + 1)p − 2 = 0 (1.2) for n 2, that is, when n 2, and we put p S (1) := +∞.On one hand, for blow-up results when 1 < p p S (n), we refer interested readers to the classical papers [12,13,8,22,21,11,26,27] and the new proofs proposed in [25,10].
On the other hand, concerning global (in time) existence results when p > p S (n), we refer to [12,9,19,7,24] and references therein.Summarizing these known results, in the scale of power nonlinearities {|u| p } p>1 , the critical exponent p = p S (n) for semilinear classical wave equations (1.1) has been found, to be the threshold condition between global (in time) existence of solutions and blow-up of local (in time) solutions with small initial data.Nevertheless, to determine the critical nonlinearity or the critical regularity of nonlinearities, it seems too rough to restrict the consideration of semilinear wave equations (1.1) to the scale of power nonlinearities {|u| p } p>1 .The question of the critical regularity of nonlinearities for semilinear classical wave equations is completely open as far as the authors know.For this reason, our contribution of this paper is to give an answer to this question for a class of modulus of continuity.Furthermore, we will suggest a candidate for the general critical nonlinearity via our derived results.
In this manuscript, we consider the following Cauchy problem for semilinear classical wave equations with modulus of continuity in the nonlinearity: x ∈ R n , t > 0, u(0, x) = u 0 (x), u t (0, x) = u 1 (x), x ∈ R n , (1.3) for n 2 (due to p S (1) = +∞), where p S (n) stands for the Strauss exponent, and µ = µ(τ ) is a modulus of continuity.To be specific, a function µ : [0, +∞) → [0, +∞) is called a modulus of continuity, if µ is a continuous, concave and increasing function satisfying µ(0) = 0.The additional term of modulus of continuity provides an additional regularity of the nonlinear term in u = 0 in the Cauchy problem (1.3) comparing with the power nonlinearity |u| p S (n) .Note that the critical nonlinearity has been studied recently in semilinear classical damped wave equations [6] and the corresponding weakly coupled systems [5].Nevertheless, due to the lack of crucial damping mechanisms, the study of the semilinear Cauchy problem (1.3) is not a generalization of those of [6,5], e.g. the usual test function methods and the Matsumura type L p − L q estimates do not work for our model (1.3).
The main purpose of this paper is to derive the critical regularity of nonlinearities for the semilinear Cauchy problem (1.3), namely, the threshold condition for modulus of continuity µ.First of all, by applying iteration methods with slicing procedure (motivated by [1,25]) for a new weighted functional, which contains a local (in time) solution and a modulus of continuity, under some conditions of initial data, we will prove a blow-up result in Section 3 when with a suitably large constant c l ≫ 1. Next, we will study the three dimensional Cauchy problem (1.3) with modulus of continuity satisfying = 0 in the radial case.By developing polynomial-logarithmic type weighted L ∞ t L ∞ r estimates via refined analysis in the (t, r)-plane, we will demonstrate global (in time) existence of small data radial solution in Section 4. The typical example is that for a modulus of continuity µ = µ(τ ) with µ(0) = 0 which satisfies Our results of this paper ensure that the critical regularity of nonlinearities |u| p S (3) µ(|u|) in semilinear classical wave equations with the modulus of continuity satisfying (1.4) is described by the threshold γ = 1 p S (3) .Namely, global (in time) existence of solutions holds when γ > 1 p S (3) and blow-up of solutions holds when 0 < γ 1 p S (3) .Other examples will be shown in Section 2. To end this paper, we will give a conjecture for general conditions of the critical nonlinearity for the semilinear Cauchy problem (1.3) as final remarks in Section 5.
Notation: Firstly, c and C denote some positive constants, which may be changed from line to line.We write f g if there exists a positive constant C such that f Cg.The relation f ≃ g holds if and only if g f g.Moreover, B R (0) denotes the ball around the origin with radius R. We denote y := 3 + |y| for any y ∈ R throughout this manuscript.

Main results
Before stating our blow-up result, we firstly introduce the notion of energy solutions to the Cauchy problem (1.3) that we are going to use later.
Theorem 2.1.Let us consider a modulus of continuity µ = µ(τ ) with µ(0) = 0 satisfying with a suitably large constant c l ≫ 1.We assume that the function g : is convex on R. Let u 0 ∈ H 1 and u 1 ∈ L 2 be non-negative, non-trivial and compactly supported functions with supports contained in B R (0) for some R > 0. Let ) be an energy solution to the semilinear Cauchy problem (1.3) on [0, T ) for n 2 according to Definition 2.1.Then, the energy solution u blows up in finite time.
Note that the modulus of continuity in the last cases can be continued to τ ∈ [0, +∞) in such a way that µ = µ(τ ) is a continuous, concave and increasing function, for example, a smooth and concave continuation function with µ(0) = 0 such that with a suitably large constant c l ≫ 1 and 0 < γ n) .A counterexample for the condition (2.2) in Theorem 2.1 is µ(τ ) = τ ν with ν > 0. This is not surprising due to the global (in time) existence results [12,9,19,7,24] for the semilinear classical wave equations (1.1) with power nonlinearity |u| p S (n)+ν .Remark 2.1.Concerning the semilinear wave equation (1.1) with the critical exponent p = p S (n), by taking the additional term of modulus of continuity µ(|u|) fulfilling (2.2) in the nonlinearity, Theorem 2.1 shows that the energy solutions still blow up in finite time.
To indicate the sharpness of the condition (2.2), we next study the three dimensional semilinear Cauchy problem (1.3) with a modulus of continuity satisfying (2.4).Before showing our result, taking r = |x|, let us introduce a definition of radial solutions to our aim model in three dimensions, namely, In the above, E 0 = E 0 (t, r) and E 1 = E 1 (t, r) are the fundamental solutions to the corresponding linear Cauchy problem to (2.3) with vanishing right-hand side.
We turn to the global (in time) existence of radial solutions in the subsequent theorem.
Example 2.2.The hypotheses (2.4) and (2.5) hold for the following functions µ = µ(τ ) on a small interval [0, τ 0 ] with 0 < τ 0 ≪ 1: ) γ with γ < 0 and k 3. Remark 2.3.By assuming additionally decay properties for initial data with respect to the radial behavior, we also can derive some pointwise decay estimates for the global (in time) radial solutions.More details will be given in Corollary 4.1 and in our proof in Section 4.

Remark 2.4. The key tool to prove Theorem 2.2 is to derive polynomial-logarithmic type weighted L ∞
t L ∞ r estimates.Concerning higher dimensional cases, one may recall more general representations of radial solutions to the linear wave equation associated with polynomial type weighted L ∞ t L ∞ r estimates (see [14,16] for odd dimensions and [17] for even dimensions).Furthermore, by setting suitable logarithmic factors to be the additional part of weighted functions, one may derive some weighted L ∞ t L ∞ r estimates to get a global (in time) existence result for higher dimensions n, nevertheless, this purpose is beyond the scope of this manuscript.Remark 2.5.Let us summarize the given results in Theorems 2.1 and 2.2.We recall the typical modulus of continuity proposed in Examples 2.1 and 2.2.In the consideration of semilinear wave equations (1.3) for n = 3 with the modulus of continuity satisfying (1.4), we may conclude that the critical regularity of nonlinearities is described by the threshold γ = 1 p S (3) .This is one of the main contributions of this paper and it answers the open question proposed in the introduction.
Remark 2.6.Motivated by the global (in time) existence condition (2.4) as well as the blow-up condition (2.2), one may introduce the following possible quantity: to describe the critical regularity of nonlinearities for semilinear wave equations (1.

and the global (in time) existence result holds when
Explanations more in detail will be provided in Section 5.

Blow-up of energy solutions
This section is organized as follows.In Subsection 3.1, we will introduce a test function, and derive sharp estimates for it in L 1 (B R+t (0)).Then, thanks to some estimates for auxiliary functions, the iteration frame and lower bound estimates for a time-dependent functional will be established in Subsections 3.2 and 3.3, respectively.Finally, in Subsection 3.4, we will demonstrate the lower bound of this functional blows up in finite time by using iteration methods with slicing procedure.

Preliminaries and auxiliary functions
Let us set a non-negative parameter Next, we recall the following pair of auxiliary functions from [25]: where λ 0 is a fixed positive parameter and the test function Φ = Φ(x) defined by was introduced by [26].The test function Φ is positive, smooth, and satisfies ∆Φ = Φ with By introducing the function with separate variables it is the solution to the free wave equation Ψ tt − ∆Ψ = 0 and has the next property.

Lemma 3.1. The test function fulfills the sharp estimates
for any t 0 and n 2.
Proof.By using integration by parts, we arrive at Shrinking the domain of integration to [t, R + t], one notices Therefore, the previous sharp estimates imply because of (3.4).The proof is completed.
Additionally, some useful estimates of ξ q and η q are stated in the following lemma, whose proof can be found in [25,Lemma 3.1].Note that our setting of q fulfills all assumptions in Lemma 3.2.Moreover, we recall the notation y = 3 + |y|.

Lemma 3.2.
There exists λ 0 > 0 such that the following properties hold for n 2: Here, A 0 and B k , with k = 0, 1, 2, are positive constants depending only on λ 0 , q and R.
To end this subsection, we include the following generalized version of Jensen's inequality [20], whose proof also has been shown in [6,Lemma 8].Lemma 3.3.Let g = g(τ ) be a convex function on R. Let α = α(x) be defined and non-negative almost everywhere on Ω, such that α is positive in a set of positive measure.Then, it holds provided that all the integral terms are meaningful.

Construction of an iteration frame
In order to prove Theorem 2.1, we are going to use an iteration argument to derive lower bound estimates for the weighted space average of a local (in time) solution containing modulus of continuity.For this reason, we first derive a nonlinear integral inequality to get an iteration frame.Proposition 3.1.Let u 0 ∈ H 1 and u 1 ∈ L 2 be non-negative, non-trivial and compactly supported functions with supports contained in B R (0) for some R > 0. Let u be an energy solution to the semilinear Cauchy problem (1.3) on [0, T ) according to Definition 2.1.Then, the following integral identity holds: for any t ∈ (0, T ), where ξ q and η q are defined in (3.2) and (3.3), respectively.
Proof.From finite propagation speed for solutions of wave equations, u(t, •) has compact support contained in B R+t (0) for any t 0. Therefore, we may employ (2.1) for a non-compactly supported test function.We now define the test function As Φ is an eigenfunction of the Laplacian and y(t, s; λ) solves (∂ 2 s − λ 2 )y(t, s; λ) = 0 with the endpoints y(t, t; λ) = 0 and y s (t, t; λ) = −1, the function ψ solves the free wave equation ψ ss − ∆ψ = 0 and satisfies Applying the test function ψ in (2.1) with an integration by parts once more, we may derive Multiplying both sides of the last equality by e −λ(R+t) λ q , integrating the resultant with respect to λ over [0, λ 0 ] and applying Tonelli's theorem, we complete the derivation of (3.5).
Hereafter until the end of this section, we shall assume that u 0 , u 1 satisfy the assumptions from Theorem 2.1.Let u be an energy solution to the semilinear Cauchy problem (1.3) on [0, T ).Inspired by the modulus of continuity in its nonlinearity, let us introduce the time-dependent functional with the parameter q defined in (3.1).Moreover, it follows immediately the non-negativity of the functional for any t 0 by where we employed as a direct consequence R n u(t, x)η q (t, t, x)dx 0 from (3.5), with the help of non-negative data and non-negative nonlinearity.A further step is to derive some estimates involving U(t) both in the left-and right-hand sides, which will establish an iteration frame.According to (3.5) and non-negativity of initial data, we may claim (3.7) Using Hölder's inequality, we arrive at Remark that p ′ S (n) denotes Hölder's conjugate of p S (n).With the aid of the properties (ii) and (iii) in Lemma 3.2 (both q > n−3 2 and q > −1 are always fulfilled), we obtain due to our choice of q in (3.1) and Note that log s log 3 > 0. Plugging the previous estimates in (3.7), it leads to Moreover, thanks to the support condition of u(t, •), let us apply Lemma 3.3 with Ω = B R+t (0), α = η q (t, t, x), v = u(t, x) and the convex function Note that the function g = g(τ ) is strictly monotonic from the monotonically increasing property of µ = µ(|τ |).After combining (3.8) and (3.9) it follows 1 The action of the mapping g on both sides of the last estimate yields Employing the non-negativity of U(t) stated in (3.6) as well as from Lemma 3.2, in conclusion, we obtain the iteration frame for any t 0, with positive constants C 0 and C 1 .

Derivation of a first lower bound estimate
By applying (3.7) and the property (ii) in Lemma 3.2, we may arrive at An application of Hölder's inequality gives where we employed the next inequality (e.g. the proof was shown in [26,18] by using an integration by parts): That is to say due to the fact that Let us apply Lemma 3.3 again with g(τ A further step of integration by parts to (2.1) shows Again, since u is supported in a forward cone, we may apply the definition of energy solutions even though the test function is not compactly supported.Taking as test function φ = φ(t, x) the function Ψ = Ψ(t, x), it holds due to the non-negativity of nonlinearity and Ψ tt = ∆Ψ.By multiplying e 2t on both sides of the last inequality, we can find From our assumption on initial data, one gets where the unexpressed multiplicative constant may depend on u 0 as well as u 1 .With the aid of Lemma 3.1 and (3.14), we are able to estimate from (3.13) and (3.12) that with a positive constant C 2 .According to (3.9), we derive Furthermore, recalling the increasing property of µ and shrinking the interval of integration [0, t] to [1, t] for t 1, one obtains

U(t)
B R+t (0) η q (t, t, x)dx g 1 Taking account of with a positive constant C 3 .
Our assumption (2.2) shows that there is a suitably large constant c l ≫ 1 such that the next estimate holds: Let us choose a large constant t 0 1 such that for any t 3 2 t 0 , the following inequalities hold (later, we will take t 0 to be suitably large): Note that the second inequality in the above (it will be used for reducing the argument) can be guaranteed since According to our assumption (2.2) for t 3  2 t 0 , it follows .
Summarizing, we deduce the following first lower bound estimate: for any t 3 2 t 0 with a large parameter t 0 1, with a positive constant M 0 independent of c l .Here, ǫ 0 > 0 is a sufficiently small constant.We have to underline that such a small ǫ 0 does not bring any influence on the blow-up condition.

Iteration procedure and blow-up phenomenon: Proof of Theorem 2.1
Up to now, we have determined among other things the iteration frame (3.11) for the functional U(t) and the first lower bound estimate (3.17) containing a logarithmic factor and a factor depending on the given modulus of continuity.In this part, we are going to prove a sequence of lower bound estimates for U(t) by applying the so-called slicing procedure, which has been introduced in [1].
Let us choose the sequence {ℓ j } j∈N 0 with ℓ j := 2 − 2 −(j+1) .Our goal is to derive the sequence of lower bound estimates for the functional U(t) as follows: for t ℓ 2j t 0 with a suitably large constant t 0 ≫ 1, where {M j } j∈N 0 , {a j } j∈N 0 , {b j } j∈N 0 and {σ j } j∈N 0 are sequences of non-negative real numbers that we shall determine recursively throughout the iteration procedure.From the first lower bound estimate (3.17), we may choose with j = 0 the parameters

.19)
We are going to prove the validity of (3.18) for any j ∈ N 0 by using an inductive argument.As we have already shown the validity of the basic case (3.17), it remains to prove the inductive step.Let us assume that (3.18) holds for j 1, our purpose is to demonstrate it for j + 1.First of all, via the lower bound estimate (3.18), we know ds for t ℓ 2j+2 t 0 , where we shrank the interval of integration [0, t] to [ℓ 2j t 0 , t] so that s = 3 + |s| 3s for any s ℓ 2j t 0 .By employing integration by parts, we may derive for t ℓ 2j+2 t 0 so that ℓ 2j t 0 ℓ 2j ℓ 2j+2 t.For this reason, the last relation for t ℓ 2j+2 t 0 implies immediately where we introduce and estimate .
Taking a suitably large t 0 such that for t ℓ 2j+2 t 0 , recalling the conclusion (3.15) from our assumption (2.2) and the condition (3.16), then the following inequality holds: for a fixed j, because the polynomial decay factor (R+t) −ǫ 0 plays from the point of decay a dominant role in comparison with all logarithmic factors on the right-hand side of the last inequality.Later, we will verify the last inequality (3.22) uniformly for all j ≫ 1 by choosing suitable parameters a j , b j , σ j and estimating M j .According to the last lower bounds estimates, it provides for t ℓ 2j+2 t 0 with a suitably large t 0 ≫ 1. Summarizing the above estimates (3.21) as well as (3.23), we claim the lower bound estimate for t ℓ 2j+2 t 0 .In other words, we have proved (3.18) for j + 1 provided that By using recursively the relations and the initial exponents (3.19), we deduce to the facts that ℓ 2j 2 and a j p S (n) p S (n)−1 p j S (n), the lower bound of M j can be estimated by > 0, which depends on n but is independent of j.Applying the logarithmic function to both sides of the last inequality and using iteratively the resulting inequality, we may obtain with a suitable constant C 7 > 0 independent of j.Let us take account of the property (3.27) of κ = κ(τ ), namely, n) , where C 7 only depends on M 0 , C 0 , p S (n), n and is independent of j.Then for large time t max{2t 0 , t 1 , t 2 }, we find that in the last estimate Finally, taking the limit as j → +∞ in the above estimate the lower bound for U(t) blows up in finite time.This completes the proof of Theorem 2.1.

Global (in time) existence of radial solutions in three dimensions
Firstly, by introducing a polynomial-logarithmic type weighted Banach space, we will prepare uniform bounded L ∞ estimates of the radial solution to the three dimensional free wave equation in Subsection 4.1.Then, the philosophy of the proof for Theorem 2.2 and its key tool will be stated in Subsection 4.2.We will demonstrate the global (in time) existence result in Subsection 4.3 by applying a refined analysis in the (t, r)-plane to estimate the nonlinear terms.

Preliminary and weighted L ∞
t L ∞ r estimates for the linearized model As preparations for studying nonlinear models, we will state some polynomial-logarithmic type weighted L ∞ t L ∞ r estimates for the linear wave equation in the radial case.Let us first extend initial data u 0 (r) and u 1 (r) by even reflections, namely, u 0 (−r) = u 0 (r) and u 1 (−r) = u 1 (r) for r < 0. Note that our assumptions u 0 ∈ C 2 and u 0 radially symmetric ensure u ′ 0 (0) = 0. Due to our interest of radial solutions and the application of even reflections, we may rewrite the semilinear Cauchy problem (1.3) in three dimensions as Now we turn our focus to the linear model with vanishing right-hand side and the same initial data as those of (4.1), namely, Let us recall u 0 ∈ C 2 0 as well as u 1 ∈ C 1 0 .According to the well-known d'Alembert's formula, the solution to the linear Cauchy problem (4.2) can be represented as follows: where we denoted Its proof is standard by taking the new variable rv(t, r) and the representation of solution for the one dimensional free wave equation.Motivated by the papers [2,15,4], we are able to derive some decay estimates in some weighted L ∞ t L ∞ r for radial solutions of the linear Cauchy problem (4.2).In order to overcome some difficulties from the influence of modulus of continuity when we consider the nonlinear model (4.1), we will include an additional logarithmic type weighted function in the solution space.To be specific, we introduce the Banach space with a polynomial-logarithmic type weighted norm .
Here, the new weighted factor is defined by ω(τ ) := (log τ ) for any τ 3. (4.4) Then, we have the next result for bounded estimates in the Banach space X κ .
Proposition 4.1.Let u 0 ∈ A κ ∩ C 2 and u 1 ∈ B κ+1 ∩ C 1 with κ > 1.Then, the following estimate holds: where the Banach spaces for initial data are defined as follows: A κ := h ∈ C 1 : h is an even function and h Aκ < +∞ , B κ := h ∈ C : h is an even function and h Bκ < +∞ , carrying the corresponding norms In the above, we used the property to be even for the functions r and |h(r)|, |h ′ (r)| so that we just need to consider r 0 in these norms.
Proof.By using the definitions of u 0 Aκ and u 1 B κ+1 , respectively, we may estimate Let us employ the triangle inequality to observe According to the solution formula (4.3), one may derive because of κ > 1, where we used the relation (4.5).Let us separate our next consideration into two situations with respect to the interplay between t and |r|.
• When t 2|r|, since the integrand takes its maximum for ρ = t − |r| and t + |r| ≈ t − |r| , thanks to the representation (4.3), we may estimate where we employed the mean value theorem with ζ ∈ (t − r, t + r), (4.5) and • When t 2|r|, we have some further discussions.
-If |r| 1, since t − |r| ≈ t + |r| ≈ 3 and the compact (t, r)-zone, then we get whose approach is the same as the one for t 2|r|. - In other words, we at which completes our proof.

Philosophy of our approach
By Duhamel's principle, the solution to the inhomogeneous linear Cauchy problem with F = F (t, r) as a source term and vanishing data is given by Concerning the semilinear Cauchy problem (4.1), inspired by the last representation, we introduce with the nonlinear term In view of Duhamel's principle as well as the above setting, we expect that if we find u ∈ X κ with κ > 1 such that u(t, r) = v(t, r) + Lu(t, r) Motivated by the above explanation, we believe the blow-up result still holds in the intermediate case C Str ∈ (0, c l ).However, due to some technical difficulties, the rigorous justification is still challenging.
• Global (in time) existence of solutions when C Str = 0: Due to the proposed condition (2.4), i.e.C Str = 0, and the decay assumption (2.5) in Theorem 2.2, our conjecture is partially verified from the global (in time) existence perspective.
Lastly, we underline that the validity of our conjecture (5.1) has been verified in the present manuscript for the semilinear three dimensional Cauchy problem (1.3) with a modulus of continuity fulfilling µ(0) = 0 and µ(τ ) = c l (log 1 τ ) −γ with c l ≫ 1 when τ ∈ (0, τ 0 ], because the global (in time) existence result for γ > 1 p S (3) and the blow-up result for 0 < γ 1 p S (3) have been rigorously demonstrated in Theorems 2.2 and 2.1, respectively.