On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain

1 Introduction

In this paper, we are concerned with the study of existence and nonexistence of global weak solutions to the wave inequality

Here, □ := ∂_tt − Δ is the wave operator, Ω = {x ∈ ℝ^N : |x| ≥ 1}, N ≥ 2, p > 1, and λ ≥ − N−222 . We will investigate (1.1) under the inhomogeneous boundary condition

where α, β ≥ 0, (α, β) ≠ (0, 0), w ∈ L¹(∂ Ω), and ν denotes the outward unit normal vector on ∂ Ω relative to Ω. Notice that (1.2) includes different types of inhomogeneous boundary conditions. Namely, the Dirichlet type boundary condition (in the case (α, β) = (0, 1))

the Neumann type boundary condition (in the case (α, β) = (1, 0))

and the Robin type boundary condition (in the case α = 1 and β > 0)

Let us consider the semilinear wave equation

where V = V(x) is a potential, and let p_c(N) be the positive root of the quadratic equation

In the special case V ≡ 0, (1.3) has been investigated by several authors. Namely, John [12] proved that, if the initial values are compactly supported and nonnegative, then for N = 3 and 1 < p < p_c(3) = 1 + 2 , nontrivial solutions must blow-up in finite time, while if p > p_c(3), global solutions exist for small initial values. Next, a similar result has been derived by Glassey [6] in the case N = 2. In [19], Shaffer proved that in the case N ∈ {2, 3}, p_c(N) belongs to the blow-up case. Georgiev et al. [5] (see also [15, 21]) proved that, if p > p_c(N) and N ≥ 3, then global solutions exist for small initial values. A blow-up result was shown by Sideris [5] (see also [15, 21]) in the case 1 < p < p_c(N) and N ≥ 4. In [23], Yordanov and Zhang proved that for all N ≥ 4, p_c(N) belongs to the blow-up case.

In [22], Yordanov and Zhang studied (1.3) when N ≥ 3 and V is a nonnegative potential satisfying the following conditions:

“There exist functions ϕ_i ∈ C²(ℝ^N), i = 0, 1, such that

where C0−1 ≤ ϕ₀(x) ≤ C₀ and 0 < ϕ₁(x) ≤ C1(1+|x|)−(N−1)2e|x| with positive constants C_i, i = 0, 1”.

It was shown that, if the initial values are nonnegative and compactly supported, then a blow-up occurs when 1 < p < p_c(N).

In [7], Hamidi and Laptev considered semilinear evolution inequalities of the form

where k ≥ 1 (integer), N ≥ 3 and λ ≥ − N−222 . It was shown that when the initial values are nonnegative, if

or

where s_* < s^* are the roots of the polynomial

then (1.4) admits no nontrivial global weak solution.

The study of blow-up phenomena for semilinear wave equations in exterior domains was considered by many authors (see e.g. [8, 10, 11, 13, 14, 24, 25] and the references therein). In particular, Zhang [24] studied the semilinear wave equation

under the inhomogeneous Neumann boundary condition

where N ≥ 3, w ∈ L¹(∂Ω), w ≥ 0, and w ≢ 0. Namely, it was shown that (1.5)–(1.6) admits as critical exponent the real number p^* = 1 + 2N−2 , i.e. if 1 < p < p^*, then (1.5)–(1.6) admits no global weak solution, while if p > p^*, global solutions exist for some w > 0. Later, the same critical exponent was obtained for (1.5) under the inhomogeneous Dirichlet boundary condition [10]

and the Robin boundary condition [8]

To enlarge the literature review on the main topic of this article, we recall the study of blow-up of solutions carried out by Mohammed et al. [17], for fully nonlinear uniformly elliptic equations. Also, we mention the recent work of Bahrouni et al. [1], where the authors dealt with a class of double phase variational functionals related to the study of transonic flow, and established useful integral inequalities. In a series of remarkable papers, Cîrstea and Rădulescu [2, 3, 4] focused on special classes of semilinear elliptic equations (namely, logistic equations) and linked the nonregular variation of the nonlinearity at infinity with the blow-up rate of the solutions. They also established existence and uniqueness results for related problems, in the cases of homogeneous Dirichlet, Neumann or Robin boundary condition.

To the best of our knowledge, the study of critical behavior for wave inequalities with Hardy potential in an exterior domain was not considered in previous works. In this paper, we investigate the critical behavior for (1.1) under the inhomogeneous boundary condition (1.2). Namely, we will show that there exists a critical exponent p_c(N, λ) ∈ (1, ∞] for which, when 1 < p < p_c(N, λ) and ∫_∂Ω w(x) dσ > 0, (1.1)–(1.2) has no global weak solution; when p > p_c(N, λ), the problem admits global solutions for some w > 0.

Before presenting our results, let us mention in which sense the solutions to (1.1)–(1.2) are considered. Let

We introduce the test function space

where Cc2 (𝓞) denotes the space of C² functions compactly supported in 𝓞. Notice that Ω is closed and ∂𝓞 ⊂ 𝓞.

Now, we are ready to state our main results. We discuss separately the cases λ = − N−222 and λ > − N−222 . For λ ≥ − N−222 , let

The rest of the paper is organized as follows. In Section 2, we establish some lemmas and provide some estimates that will be used in the proofs of our main results. Section 3 is devoted to the proofs of Theorems 1.1 and 1.2. Namely, we first prove the nonexistence results (parts (i) and (ii) of Theorem 1.1, and part (i) of Theorem 1.2), next we prove the existence results (part (iii) of Theorem 1.1 and part (ii) of Theorem 1.2).

2 Preliminaries

For λ ≥ − N−222 , let Δ_λ be the differential operator defined by

For α, β ≥ 0 and (α, β) ≠ (0, 0), we introduce the function H_α,β defined in Ω by

where

and

One can check easily that H_α,β is a nonnegative solution to the exterior problem

We need also to introduce two cut-off functions. Let η, ξ ∈ C^∞(ℝ) be such that

and

For 0 < T < ∞, let

and

where ℓ ≥ 2 and θ > 0 are constants to be chosen later.

Throughout this paper, C denotes a positive constant (independent of T) whose value may change from line to line.