Critical exponent for semi-linear structurally damped wave equation of derivative type

Main purpose of this paper is to study the following semi-linear structurally damped wave equation with nonlinearity of derivative type: $$u_{tt}- \Delta u+ \mu(-\Delta)^{\sigma/2} u_t= |u_t|^p,\quad u(0,x)= u_0(x),\quad u_t(0,x)=u_1(x),$$ with $\mu>0$, $n\geq1$, $\sigma \in (0,2]$ and $p>1$. In particular, we are going to prove the non-existence of global weak solutions by using a new test function and suitable sign assumptions on the initial data in both the subcritical case and the critical case.


Introduction
This paper is concerned with the Cauchy problem for semi-linear structurally damped wave equation with the power nonlinearity of derivative type (powers of the first order timederivatives of solutions as nonlinear terms) as follows: # u tt´∆ u`µp´∆q σ{2 u t " |u t | p , x P R n , t ą 0, up0, xq " u 0 pxq, u t p0, xq " u 1 pxq, where µ ą 0, σ P p0, 2s, n ě 1 and p ą 1. Here p´∆q σ{2 is the fractional Laplacian defined as in Definition 2.1 below when σ P p0, 2q, and when σ " 2 it is the classical Laplacian. Our main goal is to investigate the so-called critical exponent for (1). By critical exponent p c " p c pn, σq we mean that global (in time) solutions cannot exist (it sometimes called blow-up is some cases), under suitable sign assumption on the initial data, in the critical and subcritical cases p ď p c , whereas small data global (in time) solutions exist in the supercritical case p ą p c .
Regarding the structurally damped wave equation (1) with the power nonlinearity |u| p , the critical exponent has been investigated by D'Abbicco and Reissig [7], where they proposed to distinguish between "parabolic like models" in the case σ P p0, 1s, the so-called effective damping, and "hyperbolic like models"in the remaining case σ P p1, 2s, the so-called noneffective damping according to expected decay estimates (see more [3]). In the former case, they proved the existence of global (in time) solutions when p ą p 0 pn, σq :" 1`2 pn´σqf or the small initial data and low space dimensions 2 ď n ď 4 by using the energy estimates. Here we denote prq`:" maxtr, 0u as its positive part for any r P R. Afterwards, D'Abbicco and Ebert [2] extended their global existence results to higher space dimensions by using L r´Lq estimates for solutions to the corresponding linear equation. On the other hand, the authors indicated in [7] the non-existence of global (in time) solutions, just when σ " 1, if the condition p ď p 0 pn, 1q " 1`2 n´1 holds by using the standard test function method via the non-negativity of the fundamental solution (see also [6]). In these cited papers, one should recognizes that the assumptions u 0 " 0 and u 1 ě 0 come to guarantee the non-negativity of the fundamental solution, which cannot be expected for any σ P p0, 2s. Quite recently, the global non-existence result for any σ P p0, 2s has been completed by Dao and Reissig [10] when p ď p 0 pn, σq and for all n ě 1 by using a modified test function which deals with sign-changing data condition, namely u 0 " 0 and u 1 P L 1 satisfying Again, we can see that assuming the first data u 0 " 0 is necessary to require. It seems that the previous used approaches do not work so well if u 0 is not identically zero. For the non-effective case σ P p1, 2s, the global existence results were also shown by [7] only for p ą 1`p1`σq{pn´1q with n ě 2, while the blow-up of solutions has been obtained by [10] when p ď 1`2{pn´1q. Unfortunately, there appears a gap between the two exponents 1`p1`σq{pn´1q and 1`2{pn´1q. This is naturally due to the hyperbolic-like structure of the problem which seems not suitable with the standard test function method used to prove blow-up results.
Let us come back our interest to consider the structurally damped wave equation with the power nonlinearity of derivative type (1). At present, there do not seem to be so many related manuscripts. D'Abbicco and Ebert [4] proved the global (in time) existence of small data solutions for any p ą p 1 pn, σq :" 1`σ{n in the case of σ P p0, 1q and lower space dimensions, as well as for any p ą p 1 pn, 1q " 1`1{n in the case of σ " 1 and all n ě 1. For the purpose of looking for the global (in time) existence of small data Sobolev solutions to (1), with σ P p0, 1q, from suitable function spaces basing on L q spaces, with q P p1, 8q, we address the interested readers to the new papers of Dao and Reissig [8]). When σ P p1, 2s, the only global existence results known up to our knowledge can be found in [9] for any p ąp, wherep are a suitable exponent, under small initial data in Sobolev space. From these observations, it still keeps an open problem so far to indicate a non-existence result for (1) in all cases σ P p0, 2s.
For this reason, our main motivation of this paper is to fill this lack. Especially, we would like to face up to dealing with the fractional Laplacian p´∆q σ{2 , the well-known nonlocal operators, where σ is supposed to be a fractional number in p0, 2q. As we can see, this case was not included in [4] since the standard test function method seems difficult to be directly applied to these fractional Laplacian. To overcome this difficulty, the application of a new modified test function developed by Dao and Fino in the recent work [5], and mentioned in [1], comes into play. Moreover, as analyzed above, we want to point out that it is challenging to follow the recent papers ( [7,10]) in terms of the treatment of u 0 ‰ 0. Hence, the other point worthy of noticing in the present paper is that our method can be applicable effectively to relax the limitation of the assumption for u 0 " 0, which plays an important role in the proofs of blow-up results in severval previous literatures (see, for example, [4,6,7,10]).

Notations
‚ We denote the constantσ :" mintσ, 1u, where σ P p0, 2s. ‚ For later convenience, C and C i with i P Z stand for suitable positive constants. ‚ For given nonnegative f and g, Our main result reads as follows.
Theorem 1.1 (Blow-up). Let σ P p0, 2s. We assume that pu 0 , u 1 q P`L 1 pR n q X L 2 pR n q˘2 satisfying the following condition: then, there is no global (in time) weak solution to (1).  [4], it is clear that p c :" 1`σ{n is the critical exponent of (1) when σ P p0, 1s. It is still an open problem whether 1`1{n is the critical exponent of (1) when σ P p1, 2s.

Preliminaries
In this section, we collect some preliminary knowledge needed in our proofs.

Definition 2.1 ([11]
). Let s P p0, 1q. Let X be a suitable set of functions defined on R n . Then, the fractional Laplacian p´∆q s in R n is a non-local operator given by p´∆q s : v P X Ñ p´∆q s vpxq :" C n,s p.v.
Then, φ P C 2 pR n q and the following estimate holds: Lemma 2.2. [10, Lemma 2.4] Let s P p0, 1q. Let ψ be a smooth function satisfying B 2 x ψ P L 8 pR n q. For any R ą 0, let ψ R be a function defined by ψ R pxq :" ψpx{Rq for all x P R n .
Then, p´∆q s pψ R q satisfies the following scaling properties: p´∆q s pψ R qpxq " R´2 s pp´∆q s ψqpx{Rq for all x P R n .

Proof of the main result
Before starting our proof, we define weak solutions for (1).
Definition 3.1. Let T ą 0, p ą 1, and pu 0 , u 1 q P L 2 pR n qˆL 2 pR n q. A function u is said to be a global weak solution to (1) if u P L 1 loc`p 0, 8q, L 2 pR n q˘satisfying u t P L p loc`p 0, 8q, L 2p pR n q˘X L 1 loc`p 0, 8q, L 2 pR n q˘, and the following formulation holds for any test function ϕ P C`r0, 8q; H 2 pR n q˘X C 1`r 0, 8q; L 2 pR n q˘such that its support in time is compact.
Proof of Theorem 1.1. First, we introduce the function φ " φpxq as defined in (4) with s " σ{2 and the function η " ηptq having the following properties: For the existence of such function, see e.g. [12,Chapter 1]. Let R be a large parameter in r0, 8q. We define the following test function: where η R ptq :" η`R´σt˘and φ R pxq :" φ`R´1K´1x˘for some K ě 1 which will be fixed later. Moreover, we introduce the function Ψ R ptq " Here we also notice that the relation Ψ 1 R ptq "´η R ptq holds. We define the functionals and I R,t :" ż Rσ Rσ{2 ż R n |u t pt, xq| p ϕ R pt, xq dxdt and I R,x :" Let us assume that u " upt, xq is a global weak solution to (1), then Using integrating by parts, we conclude that Applying Hölder's inequality with 1 p`1 p 1 " 1 we may estimate J 1 as follows: By the change of variablest :" R´σt andx :" R´1K´1x, a straight-forward calculation gives Here we used η 1 R ptq " R´ση 1 ptq and the assumption (6). Now let us turn to estimate J 2 and J 3 . Applying Hölder's inequality again as we estimated J 1 leads to In order to control J 2 , we derive the following estimate: where we have used the fact that η R is a non-increasing function satisfying η R ď 1. Then, carrying out the change of variablest :" R´σt,x :" R´1K´1x and Lemma 2.3 with s " 1 we arrive at |J 2 | À I 1 p R,x R´2`σ`n`σ p 1 K´2`n p 1 .
Next carrying out again the change of variablest :" R´σt andx :" R´1K´1x and employing Lemma 2.2, then Lemma 2.3, with s " σ{2, we can proceed J 3 as follows: Combining the estimates from (7) to (10) we may arrive at Moreover, it is clear that Ψ R p0q ď Rσ. By the change of variables, using Lemma 2.1 we can easily check that |∆φ R pxq| ď R´2φ R pxq. Therefore, this implies that Because of the assumption (2), there exists a sufficiently large constant R 1 ą 0 such that it holds ż for all R ą R 1 . Since u 0 P L 1 , it implies immediately that Rσ´2 ż R n |u 0 pxq|φ R pxq dx Ñ 0 as R Ñ 8.
Summarizing, the proof Theorem 1.1 is completed.