Global non-existence for some nonlinear wave equations with damping and source terms in an inhomogeneous medium
- Yanjin Wang^{1}Email author and
- Yufeng Wang^{2}
Received: 22 August 2016
Accepted: 28 February 2017
Published: 22 March 2017
Abstract
For the low initial energy case, which is the non-positive initial energy, based on a concavity argument we prove the blow-up result. As for the high initial energy case, we give sufficient conditions of the initial data such that the corresponding solution blows up in finite time. In other words, our results imply a complete blow-up theorem in the sense of the initial energy, \(-\infty< E(0)<+\infty\).
Keywords
MSC
1 Introduction
- (H)
\(\rho(x)>0\) for every \(x\in\mathbb{R}^{n}\), \(\rho\in \mathcal{C}^{0,\gamma}(\mathbb{R}^{n})\) with \(\gamma\in (0,1)\), and \(\rho\in L^{n/2}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})\).
The wave equations (1) appear in applications in various areas of mathematical physics (see, for example, [1–3]), as well as in geophysics and ocean acoustics, where, for example, the coefficient \(\rho(x)\) represents the speed of sound at the point \(x\in\mathbb {R}^{N}\) (see [4]), in other words, \(\rho(x)\ne\mathrm{constant}\) implies that the medium where the sound travels is inhomogeneous.
For the case \(\rho(x)=1\) and \(m=0\), we also note that the equations were also considered with \(f(u)=|u|^{p}\) by some authors. Here we just refer to [16] and the references therein.
Now we return to equation (1) with some general \(\rho(x)\). For the linear case, \(f(u)=0\), Eidus [17] first studied the existence of solutions for the linear wave equation (1). Then Karachalios and Stavrakakis [18] studied the existence of the solution of the damped wave equation (1) with some nonlinear power. And they [19] also established the results as regards the global existence and blow-up of solutions for equation (1) with (H) and (4) in the free mass case by the potential well method, which was firstly developed by Sattinger [20]. Their blow-up result was under the condition that the initial energy was negative. Recently, for equation (1) with (H) and (4) Zhou [21] investigated the global existence and blow-up result including the mass free case and mass case. Zhou established the blow-up result when the initial energy was less than a positive constant, which is independent from the initial data. But in all the work mentioned above, equation (1) with high initial energy was not considered under the assumption (H).
The main purpose of this paper is to establish the blow-up result for equation (1) with (H) and (2) when the initial energy is high. Based on a concavity argument, which was used to establish blow-up of solutions to nonlinear damped wave equations (see e.g. [21, 22] and the references therein), we firstly establish the blow-up result when the initial energy is non-positive. Because of the nonlinearity, \(\rho(x)\) is not constant. We cannot directly apply the proof method of [13, 14] for equations (1) in this paper. Note that, when the initial energy is zero, the blow-up result was also established in [21], where the assumption is needed that \(\rho(x)\in L^{1}(\mathbb{R} ^{n})\) with (H) and \(\int \rho(x)u_{0}(x)u_{1}(x)\, dx\geq0\). Thus in some sense (see Remark 1) we improve the blow-up result in [21]. As for the case with the arbitrarily positive initial energy, we establish the blow-up result for (1) under some conditions of \((u_{0}, u_{1})\) on the whole space \(\mathbb{R}^{n}\). To the best of our knowledge this is the first blow-up result with high initial energy for equation (1) with (H).
Moreover, we note that if \(\rho(x)\) satisfies (H) then the mass m does not affect the blow-up result, but if \(\rho=\mathrm{constant}\ne 0\) it will affect the blow-up result, that is, if \(m=0\) and \(\rho(x)=1\) then the blow-up result is obtained only on a bounded subset of \(\mathbb{R}^{n}\). Indeed, by (H) we see that \(\rho(x)\) will be a rapidly enough decreasing at infinite time, thus it makes it possible to consider equation (1) on the whole space \(\mathbb{R}^{n}\) in the mass free case.
The paper is composed of three sections. In the next section we introduce some notations and well-known results, and we also state our main results. In the last section we prove our results.
2 Principal results
In order to state our main results, we briefly mention some facts, notations, and well-known results. We denote by \(\|\cdot\|_{q}\) the \(L^{q}(\mathbb{R}^{n})\) norm for \(1\leq q\leq\infty\), and we define the functional spaces: \(H^{1}(\mathbb{R}^{n})=\{u\in L^{2}(\mathbb{R}^{n}); \|u\| _{H^{1}(\mathbb{R}^{n})}=\|(1-\varDelta )^{1/2}u\|<\infty\}\), and \(H_{0}^{1}(\mathbb{R}^{n})=\{u\in H^{1}(\mathbb{R}^{n}); \operatorname {supp}(u) \mbox{ is compact in }\mathbb{R}^{n}\}\). For simplicity we will denote \(\int_{\mathbb{R}^{n}}\) by ∫. The notation \(t\rightarrow T^{-}\) means \(t< T\) and \(t\rightarrow T\).
Lemma 1
Theorem 1
This result can be proved by the Banach fixed point theorem. The proof follows from the weighted-norm Lebesgue space of the corresponding theorem for the wave equations of Kirchhoff type [25].
If \(T_{{\mathrm{max}}}<\infty\), then the local solution is said to blow up in finite time \(T_{{\mathrm{max}}}\). Otherwise, \(T_{{\mathrm {max}}}=\infty\), the corresponding local solution is global.
Next we state our first blow-up result for equation (1) with (H) and (2) in the case of non-positive initial energy.
Theorem 2
Remark 1
In the case \(E(0)<0\), Zhou [21] also established the blow-up result for equation (1) with (H) and (4) under the further assumptions \(\int\rho(x)u_{0}(x)u_{1}(x)\,dx\geq0\) and \(\rho\in L^{1}(\mathbb{R}^{n})\). But in the above theorem, we remove \(\rho\in L^{1}(\mathbb{R}^{n})\). Thus, in this sense we improve the result [21].
Now we introduce our main blow-up result for equation (1) with arbitrarily positive initial energy, as far as we know, which is the first blow-up result for equation (1) with (H) on the whole space \(\mathbb{R}^{n}\).
Theorem 3
Remark 2
We note that, for the case \(E(0)<0\), by (15) and (14), \(I(u(t,\cdot))<0\) for every \(t\in[0, T_{{\mathrm{max}}})\).
Reading Theorems 2, 3 and Remark 2, naturally one considers the local solution when the initial data satisfies \(E(0)>0\) and \(I(u_{0})>0\). Indeed, for this case, being similar to the argument with \(m=0\) and \(f(u)=|u|^{p-1}u\) in [19], by a potential well method we can also obtain the global existence of solutions of equation (1) with (H) and (2) when the positive initial energy is small enough. Here we omit it. Furthermore, it is still open whether there exists a global solution for wave equations when the initial energy is arbitrarily high.
3 Proof of the main theorems
In this section, we prove Theorems 2 and 3 based on a concavity argument. Firstly, we introduce a lemma concerned with the concavity argument [5].
Lemma 2
We next show a lemma, which plays a role in our proofs of Theorems 2 and 3.
Lemma 3
Proof
Here we let \(H(t)=\frac{d}{dt}\int \rho(x)|u(t,x)|^{2}\,dx\), then as in [19] we see that the function \(H(t)\) is a Lipschitzian function over \([0, T_{{\mathrm {max}}})\). Thus, by (28) it follows that \(\|u(t,\cdot)\|^{2}_{L_{\rho}^{2}}\) is strictly increasing on \([0, T_{{\mathrm{max}}})\). □
Proof of Theorem 2
Thus by (17) and Lemma 3 we see that \(\|u(t,\cdot)\|_{L_{\rho}^{2}}^{2}\) is strictly increasing on \([0, T_{{\mathrm{max}}})\).
In this case we still use the auxiliary function \(G(t)\) as (29).
Thus, according to the proof of Case I, by (17) we see that \(G(t)>0\), \(G^{\prime}(t)>0\), \(G^{\prime\prime}(t)>0\) on \((0, T_{{\mathrm{max}}})\), that is to say, \(G(t)\) and \(G^{\prime}(t)\) are strictly increasing over \([0, T_{{\mathrm{max}}})\).
Then by the same argument as Case I, we can claim that the corresponding local solution of equation (1) blows up in finite time.
Thus the proof of Theorem 2 is completed. □
In the following part we will address Theorem 3. The next lemma is the crux to prove Theorem 3.
Lemma 4
Proof
Thus the proof for Case I, \(m\ne0\), is complete.
Thus the proof of Lemma 4 has been completed. □
Proof of Theorem 3
And from (21) we see that \(G^{\prime}(t)>0\) for every \(t\in(0, T_{{\mathrm{max}}})\). Thus, it turns out that \(G(t)\) and \(G^{\prime}(t)\) is strictly increasing on \([0, T_{{\mathrm{max}}})\).
We next let A, B, C denote the same terms as (38), (39), and (40), respectively.
By Lemma 2 we see that there exists a finite time \(T_{{\mathrm{max}}}<\infty\) such that the corresponding solution blows up in finite time \(T_{{\mathrm{max}}}\). □
Declarations
Acknowledgements
This work was supported by National Natural Science Foundation of China (11371069, 11671413, 11001020). The authors would like to express their thanks to the referees for their constructive suggestion.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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