Blowup for semilinear wave equation with space-dependent damping and combined nonlinearities

This paper is concerned with the Cauchy problem for semilinear wave equation with space-dependent scattering damping and combined nonlinearities. The blowup results of solution are established by introducing proper test functions. Moreover, upper bound lifespan estimates of a solution to the Cauchy problem with small initial values are derived. To the best of our knowledge, the results in Theorems 1.1–1.2 are new.

Problem (1.4) is known as the Glassey conjecture (see [6]), where the Glassey critical exponent is p G (n) = n+1 n-1 . It is shown in [4] that the Cauchy problem of heat equation (1.5) possesses the Fujita critical exponent p F (n) = 1 + 2 n . Scholars investigated the blowup dynamics of a semilinear wave equation with damping term (1.6) where h(u t ) = μ (1+t) β u t , μ (1+|x|) β u t (μ > 0, β ∈ R) and f (u, u t ) = |u| p , |u t | p , |u t | p + |u| q (p > 1, q > 1). When the critical exponent of the damped wave equation (1.6) is related to the Srauss exponent p S (n) or the Glassey exponent p G (n), we say that the equation behaves like that of the wave equation. This means that the damping term in the equation makes no effect. When the critical exponent is related to the Fujita exponent p F (n), we say that the damping term makes an effect. According to the range of β, we use the following table to show the effect of damping terms (we can see it in [18,21]).
Blowup and global existence results in connection with the semilinear wave equation with time-dependent damping μ (1+t) β u t are established in [1,13,16,20,31,37,38]. Energy estimates of solution to semilinear wave equation with space-dependent damping are derived in [14,15,36]. Nishihara et al. [32] investigated the blowup and global existence for a semilinear wave equation with space-and time-dependent damping. In the present paper, we mainly concentrate on the problem with space-dependent scattering damping case μ (1+|x|) β u t (β > 1). Namely, the behavior of a solution is similar to that of the wave equation in this case. Lai and Tu [17] considered upper bound lifespan estimates of a solution to the wave equation with space-dependent damping μ (1+|x|) β u t (β > 2, n ≥ 2) and f (u, u t ) = |u| p , |u t | p for both subcritical and critical exponents. Especially, for the power nonlinearity |u| p ( n n-1 < p ≤ p S (n)) and derivative-type nonlinearity |u t | p (1 < p ≤ p G (n)), they obtained the same critical exponents and upper bound lifespan estimates of solutions as in the situation without damping by using the test function method. Lai et al. [17] obtained upper bound lifespan estimate of solution when f (u, u t ) = |u| p and β > 1. Meanwhile, the lifespan estimate for the case 1 < p < n n-1 was also improved. We are in the position to present some known results related to the semilinear wave equation (1.6) with combined nonlinearities f (u, u t ) = |u t | p + |u| q . Han and Zhou [10] obtained an upper bound lifespan estimate of solution to the Cauchy problem without damping term by constructing a proper test function and solving ordinary differential inequalities. Hidano et al. [12] established the sharp lower bound lifespan estimate of a solution to the problem. Dai et al. [2] derived the sharp lifespan estimate of a solution to the nonlinear wave equation when p ≥ q S (n) and q = q S (n) (n = 2, 3), where q S (n) is the Strauss critical exponent of the semilinear wave equation with power nonlinearity |u| q . Lai and Takamura [19] illustrated blowup results and upper bound lifespan estimates of a solution to the problem with time-dependent damping term μ (1+t) β u t (β > 1) by using a multiplier and iteration argument. Blowup of a solution to the problem with scale-invariant damping μ 1+t u t was investigated by applying test function approach (see [7,8]). Liu and Wang [26] consider problem (1.1) for the more general nonlinearity f (u, u t ) = c 1 |u t | p + c 2 |u| q on asymptotically Euclidean manifolds. Upper bound lifespan estimates of solution with different values of c 1 and c 2 are obtained. In addition, the existence of a solution is established.
Inspired by the works [10,17,19,21], we consider blowup and upper bound lifespan estimates of a solution to problem (1.1). To our best knowledge, the blowup for the spacedependent damped wave equation with combined nonlinearities has not been discussed yet. The purpose of this paper is to fill this gap. We establish upper bound lifespan estimates of a solution. It is worth mentioning that in this paper, we employ the test function method different from the technique in [10,19]. We bear in mind that lifespan estimates of solutions to the problems with space-dependent damping μ (1+|x|) β u t (β > 2) and f (u, u t ) = |u| p , |u t | p are investigated in [21]. Thanks to the work [17], we obtain upper bound lifespan estimates of a solution to problem (1.1) with μ (1+|x|) β u t (β > 1) and combined nonlinearities |u t | p + |u| q (see the new results in Theorems 1.1-1.2 in this paper).
The main results in this paper are described as follows.
Theorem 1.1 Let n ≥ 2, μ > 0, and β > 1, and let f and g satisfy (1.2). Suppose that problem (1.1) has an energy solution u such that Then we have the following lifespan estimates of solution: where C is a positive constant. Theorem 1.2 Let n ≥ 2, μ > 0, and β > 1, and let f and g satisfy (1.2). Suppose that problem (1.1) has an energy solution u such that Then the lifespan estimates of solution satisfy (1.8) We use Fig. 1 to make a simple description for n = 2. For p, q ∈ B ∪ C ∪ E, we have the first lifespan estimate in (1.7). For p, q ∈ A ∪ B ∪ C ∪ D, we obtain the second lifespan estimate in (1.7), whereas for p, q ∈ B, the second lifespan estimate in (1.7) is better than the first one. For p, q ∈ C, the first lifespan estimate in (1.7) is better than the second one. .
Throughout this paper, C denotes a positive constant independent of ε, which may vary from line to line.

Preliminaries
In this section, we present several basic definitions and lemmas.

Definition 2.1 A function u is called an energy solution of problem
Here T(ε) represents the upper bound lifespan estimate of a solution to problem (1.1), which satisfies T(ε) = sup{T > 0, there exists an energy solution to problem (1.1)}.

Proof of Theorem 1.1
In this section, we illustrate the proof of Theorem 1.1.

Case p ≥ q
First, we choose ϕ(t, x) = η 2q T as the test function, where q satisfies 1 q + 1 q = 1. From (2.1) we obtain where we have used the fact that ∂ t η T (0) = 0 and η T (T) = 0.
Using the Hölder and Young inequalities, we have that for q > n n-1 ,

2)
where where we have applied Lemma 2.4.

Proof of Theorem 1.2
Taking A direct calculation shows that It follows that T ψ dx dt = I 11 + I 12 + I 13 .