Nonexistence of solutions for quasilinear hyperbolic inequalities

In this paper, we study the Cauchy problems for quasilinear hyperbolic inequalities with nonlocal singular source term and prove the nonexistence of global weak solutions in the homogeneous and nonhomogeneous cases by the test function method.


Introduction
In this paper, we consider the Cauchy problems for quasilinear hyperbolic homogeneous and nonhomogeneous inequalities with nonlocal singular source term of the forms, respectively, u tt ≥ u m + |x| α |u| q β where S ∞ = R N × (0, ∞), m, q, r, s > 0, α > sσ r , the initial functions u 0 , u 1 ∈ L 1 loc (R N ) are nonnegative, the weight function β is positive and singular at the origin, that is, there exist constants c > 0 and σ ∈ R + such that The existence and nonexistence of global and nonglobal solutions for nonlinear evolution equations (or inequalities) have been assiduously investigated during the past decades. The nonexistence of global solutions is a nonlinear Liouville-type theorem, and thus we can prove some properties of solutions in a bounded domain, which is a kind of essential reflection of blowup or singularity theory as well; see [1]. In this paper, we would like to investigate the nonexistence of global solutions of the Cauchy problems for nonlocal hyperbolic differential inequalities.
For the model this problem was first studied by John [2]. The author proved that if 1 < p < 1 + √ 2 and N = 3, then all local solutions to (1.5), (1.3) with u 0 , u 1 ∈ C ∞ 0 (R 3 ) blow up at a finite time. Later, Glassy [3] showed the nonexistence of global solution for the critical value p = 1 + √ 2 under the additional assumption that the initial values u 0 and u 1 both have positive average. Presumably, then there exists a critical value of p, say p 0 (N), such that that the existence of all small global solutions holds if p > p 0 (N), whereas the nonexistence of most global solutions holds if 1 < p < p 0 (N); the critical value p 0 (N) satisfies Recently, Strauss and Tsutaya [4] proved that in case N = 3, u 0 ≡ 0, and u 1 (x) ≥ C(1 + |x| 2 ) -1+k 2 , the problem has no global solution for any p > 1 and any k < 2 p-1 . The critical case k = 2 p-1 was treated in [5], where the global existence is proved in three space dimensions for p > 1+ √ 2 and for small data u 0 ∈ C 3 (R 3 ) and u 1 ∈ C 2 (R 3 ) satisfying D α u 0 (x), D β u 1 (x) = O(|x| -1-k ) as |x| → ∞, |α| ≤ 3, |β| ≤ 2. Pohozaev and Veron [6] considered the quasilinear differential inequalities where h(x) ≡ 1, ϕ is a locally bounded real-valued function that satisfies |ϕ(r)| ≤ C|r| q (∀r ∈ R) for some C > 0 and q > 0, and L m (ζ ) = |α|=m D α (a α (x, t)ζ ) is a homogeneous differential operator of order m in which the a α are merely bounded measurable functions. They proved that there is no weak global solution to (1.6) No assumption on the sign of the average of u 0 (which may not be integrable) or on its support are required. Later, Guedda [7] obtained the nonexistence of local and global solutions by the test function method.
For the study of the problem with nonhomogeneous term, Laptev [8] considered the semilinear hyperbolic-type inequalities with nonhomogeneous term where K is a conic domain. They proved that there is no nontrivial global solution by the test function method. Motivated by the above works, we can find that the study on nonexistence of global solutions for the quasilinear hyperbolic type inequalities (1.1) and (1.2) with nonlocal singular source term has not been started yet. The purpose of this paper is finding the influence of nonlocal singular source term and nonhomogeneous term on the nonexistence of nontrivial global weak solution by the test function method developed in [8,9]. This method has two merits. Firstly, it is simple and accurate. In fact, the nonexistence of global solution is transformed into algebraic inequality analysis. Secondly, the comparison results and energy functional are not required, so it can be widely applied to nonlinear differential inequalities. There is no regularity assumption about initial data, and the range of parameters is completely different in the homogeneous and nonhomogeneous cases in the paper.
The rest of our paper is organized as follows. In Sect. 2, we introduce some definitions and main results of solutions for the homogenous and nonhomogenous problems. The proofs of the main results are given in Sect. 3.

Preliminaries and main results
Similarly, a nonnegative function u(x, t) is called a weak solution of nonhomogeneous problem (1.2), (1.3) in S ∞ if the following inequality holds: Our main results are as follows. Remark 1 In fact, if we choose m = 1 in Theorem 1, then we get that the homogeneous problem (1.1), (1.3) has no global solution if either N = 1 or 1 < N ≤ 1 + 1 q , which improves the results of Pohozaev and Veron [6]. u(x, t) be a solution of the homogeneous problem (1.1), (1.3). If the initial data u 1 are compactly supported, then Indeed, retaining only the first summand on the left-hand side of the inequality in Theorem 2, we obtain that σ 4 < 0, then the right-hand side obviously tends to zero as ρ → ∞, whereas the left-hand side is increasing (more precisely, it is nondecreasing if the initial data u 1 are compactly supported). Therefore, for any initial data, there is ρ 0 such that this inequality does not hold for ρ > 2

Proof of Theorem 1 Let u(x, t) be a nontrivial solution of homogeneous problem (1.1), (1.3), and let ζ be a nonnegative smooth test function. Then
Now we estimate the right-hand side of inequality (3.1). First, applying the Hölder inequality to the first term in the right-hand side of (3.1), we get

By the Hölder and Young inequalities again, we obtain
where We now focus on the second term on the right-hand side of (3.1). Applying similar arguments like for the first term of (3.1), we obtain where We assume that ζ is also chosen such that where ϕ ∈ C 2 0 (R + ) satisfies 0 ≤ ϕ ≤ 1 and ρ is a positive parameter, whereas k > 1 and μ > 0 will be determined later on.
Since ζ t = kt k-1 ρ -2 ϕ (ρ -2 t k )ϕ(ρ -2 |x| μ ), estimate (3.4) holds. To estimate the right-hand side of (3.5), we consider the change of variables ρ -2 t k = τ k and ρ -2 |x| μ = |y| μ . Then where C 1 and C 2 are constants that do not depend ρ. From (3.5)-(3.7) we have We choose k such that that is, Such a choice gives a common value γ of the exponents of ρ in (3.8), namely, The sign of (3.9) does not depend on μ > 0 and k > 1, and This is ensured by taking μ large enough.
If γ < 0, then the right-hand side of (3.8) goes to 0 as ρ goes to infinity, which clearly implies that u cannot exist.
We return to inequality (3.1), which in fact reads However, This infers that Remark 2 The integrability assumption on u 1 can be relaxed by replacing the sign condition by the following weaker one: Proof of Theorem 3 Following the proof of Theorem 1, we obtain the following estimate analogous to the estimate in Theorem 2: if ρ is such that Assuming that we obtain a contradiction by letting ρ → ∞. Next, from (3.13) we have r(q + s)(N -2) < m(rαsσ ) + mN(r + s) ⇔ N < 2r(q + s) + m(rαsσ ) r(q + s)m(r + s) . (3.14) Notice that if N ≤ 2, then (3.14) is obviously fulfilled. The proof of Theorem 3 is completed.