Nonexistence of global solutions of wave equations with weak time-dependent damping and combined nonlinearity

In our previous two works, we studied the blow-up and lifespan estimates for damped wave equations with a power nonlinearity of the solution or its derivative, with scattering damping independently. In this work, we are devoted to establishing a similar result for a combined nonlinearity. Comparing to the result of wave equation without damping, one can say that the scattering damping has no influence.


Introduction
Recently, the small data Cauchy problem of damped semilinear wave equations with time dependent variable coefficients attracts more and more attention. The works of Wirth [18,19,20] showed that the behavior of the solution of the following linear problem heavily relies on the decay rate β and the size of the positive constant µ.
Then people get interested in the corresponding nonlinear problem, i.e. the following small data Cauchy problem where µ > 0, n ∈ N and β ∈ R, and ε measures the smallness of the data. Before going on, it is necessary to mention two corresponding nonlinear problems without damping and Remark 1.2 It is easy to prove that p F (n) < p S (n) for n ≥ 2.
Now we come back to Cauchy problem (1.1). It is interesting to study the relation of the critical exponents among (1.1), (1.2) and (1.3). For β ∈ [−1, 1), due to the works [3,13,17,8,4,7], we know that it admits the same critical exponent as that of problem (1.2). For β > 1, since the authors showed blow-up result for 1 < p < p S (n) in [11], we may believe that it has the same critical exponent as that of (1.3).
If we consider the case β = 1 for Cauchy problem (1.1), the size of the positive constant µ should also be taken into account. Generally speaking, if µ is relatively large, the term {µ/(1 + t)}u t in the equation will have the main influence on the behavior of the solution, which means that this case has the same critical exponent as that of problem (1.2). See the works [1,2]. But, if µ is relatively small, we may conjecture that the influence of u tt will dominate over {µ/(1 + t)}u t , which means that the critical exponent is related to p S (n). See the work [10] by the authors and Wakasa for 0 < µ < (n 2 +n+2)/{2(n+2)}, which was extended to 0 < µ < (n 2 +n+2)/(n+2) by Ikeda and Sobajima [9] and Tu and Lin [15,16]. Unfortunately, till now we are not clear of the boardline of µ, which determines that the critical power of Cauchy problem (1.1) with β = 1 will be Fujita or Strauss. We refer the reader to a very recent work by Palmieri and Reissig [14].
In a recent work [12] by the authors, we study the blow-up for the small data Cauchy problem If β > 1, then we showed that the problem has no global solution for 1 < p ≤ p G (n), where which denotes the critical exponent for Glassey conjecture. In this work, we are devoted to studying the small data Cauchy problem with combined nonlinear terms, that is: where β > 1. Inspired by the work [5], in which Han and Zhou studied the Cauchy problem (1.5) without damping and obtained the blow-up result for we want to show that whether we have the same blow-up result for Cauchy problem (1.5). The difficulty comes from the damping term, which prevents us from getting the lower bound of some functional by using the test function method, and we overcome this by using a multiplier which was first introduced in the authors [11]. Also, due to the damping term, we can't get the blow-up result and lifespan estimate by using Kato's Lemma, and we do it by using an iteration argument similar to that in [11]. Remark 1. 3 Hidano, Wang and Yokoyama [6] established global existence result for Cauchy problem (1.5) without damping for n = 2, 3 and In the following we are going to find out that whether the global existence result holds for Cauchy problem (1.5).

Main Result
First we introduce the definition of the solution as follows.
Definition 2.1 As in [11], we say that u is an energy solution of (1 ) and any t ∈ [0, T ). Employing the integration by parts in (2.1) and letting t → T , we get the weak solution of (1.5) Our main theorem is the following.
Theorem 2.1 Let µ > 0, β > 1 and n ≥ 1. Assume that both f ∈ H 1 (R n ) and g ∈ L 2 (R n ) are non-negative, compactly supported, and g does not vanish identically. Suppose that an energy solution u of (1.5) on [0, T ) satisfies Remark 2.2 The restriction q < 2n/(n − 2) for n ≥ 2 is necessary to guarantee the integrability of the nonlinear term |u| q .

Remark 2.3
As in [5], we should point out that there exist pairs of (p, q) satisfying p > p G (n), q > q S (n), but still blow-up will occur. For example, since we may choose such an appropriate pair (p 0 , q 0 ) by setting small constants , δ 1 and δ 2 , such that We also have an improvement on the estimate of the lifespan for relatively large p and small q as follows.

Lower bound of the first functional
One of the key ingredients to the blow-up result is to get the lower bound of which was first introduced in Yordanov and Zhang [21]. Another key point is a multiplier, which is crucial for our proof and was first introduced in [11]. We note that m(t) is bounded as 0 < m(0) ≤ m(t) ≤ 1.
Then we have the following lemma.
Proof. The proof of Lemma 3.1 is almost the same as that of Lemma 3.1 in [12], which is established by neglecting the spatial integral of the nonlinear term R n |u(x, t)| p dx due to its positivity. Replacing this quantity by we get the desired proof immediately.

Lower bound of the second functional
With Lemma 3.1 in hand, we may prove a key inequality for Lemma 4.1 Let u(x, t) and ψ(x, t) be as in section 3. Under the same assumption of Theorem 2.1, it holds that Proof. Actually Lemma 4.1 is a partial result of the proof of Theorem 2.1 in [12]. For convenience we rewrite the detail. By direction calculation we have

(4.2)
Replacing the test function φ in the definition (2.1) with ψ and taking derivative to both sides with respect to t, we have that Since for ψ(x, t) we have ψ t = −ψ, ψ tt = ∆ψ = ψ, then by integration by parts in the first term in the second line of the last equality yields that for t ≥ 0. Then (4.5) and the positivity of F 1 by Lemma 3.1 yield On the other hand, noting that Multiplying the above equality by m(t), we get Adding (4.6) and (4.7) together, we obtain that (4.8) Setting then we have It is easy to get from (4.8) that Hence, by the definition (4.9), it holds that which implies that which is exactly the desired inequality in Lemma 4.1.

Iteration argument
As mentioned in the introduction, we can't establish the blow-up result and lifespan estimate by using Kato's lemma, instead of which we will use an iteration argument, following the idea in [11]. Set

Choosing the test function
which implies that by taking derivative with respect to t on the both sides Multiplying with m(t) on the both sides yields which means that Lemma 5.1 (Inequality (2.5) of Yordanov and Zhang [21]) There exists a constant C 1 = C 1 (n, p, R) > 0 such that By Hölder's inequality, (5.3) and (4.1), we may estimate the nonlinear term Plugging which into (5.2) we have where C 3 := m(0)C 2 n(n + 1) .
By Hölder's inequality again, it follows from (5.2) that with some positive constant C 4 independent of ε. In this way, we find two key ingredients for our iteration argument. Assuming that Plugging (5.6) into (5.5) we have (qb j + 2) 2 , a j+1 = qa j + n(q − 1), b j+1 = qb j + 2. (5.8) By combining (5.7) and (5.8) we come to Hence we have Repeating this procedure, we have which yields that By d'Alembert's criterion we know that S q (j) converges for q > 1 as j → ∞.
Hence we get the lifespan estimate in Theorem 2.1.

Remark 5.1
In the last line of (5.10), we should require that which leads to the restriction (2.4) for q in the case n ≥ 2.

Proof of Theorem 2.2
Due to (5.4), we roughly get an estimate of the form, for large t with some positive constant C independent of ε. So if p > 2n n − 1 , then we have n + 1 − (n − 1)p/2 < 1, which means that (5.4) is weaker than the linear growth. And hence it is natural to get a better result if we have linear growth in the first step in the iteration argument. Actually, due to the assumption of the initial data, we get from (5.1) that which implies that F 0 (t) ≥ C 8 εt, t ≥ 0, (6.1) where C 8 := m(0) R n g(x)dx.