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Global nonexistence for some nonlinear wave equations with damping and source terms in an inhomogeneous medium
Boundary Value Problems volumeÂ 2017, ArticleÂ number:Â 39 (2017)
Abstract
In this paper we study the initial value problem for the nonlinear wave equation with damping and source terms
with some \(\rho(x)\) and \(f(u)\) on the whole space \(\mathbb{R}^{n}\) (\(n\geq 3\)).
For the low initial energy case, which is the nonpositive initial energy, based on a concavity argument we prove the blowup 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 blowup theorem in the sense of the initial energy, \(\infty< E(0)<+\infty\).
1 Introduction
In this paper our aim is to study some nonlinear wave equation with damping and source terms in the following form:
where Î” is Laplacian operator on \(\mathbb{R}^{n}\) (\(n\geq3\)), \(u_{0}(x)\) and \(u_{1}(x)\) are real valued functions, m is a real constant (the case \(m=0\) is called the mass free case; \(m\ne 0\) the mass case), \(\rho(x)\) satisfies the following condition:

(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, [1â€“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 nonlinear power, throughout the paper we make the following assumption: for the nonlinear power \(f(s)\) there exists some constant \(\epsilon>0\) such that
for any \(s\in\mathbb{R}\), where
The nonlinear power satisfying (2) was firstly introduced for abstract wave equations with \(\rho(x)=1\) by Levine [5]. And then Cazenave [6] also considered it for KleinGordon equations.
Before going any further, we briefly introduce some results for the wave equation (1) with \(\rho(x)=\mathrm{constant}\ne 0\) (without loss of generality let \(\rho(x)=1\)), obviously it does not satisfy the assumption (H). The general nonlinear power \(f(u)\) with (2) it was firstly considered for some abstract wave equations in [5], where Levine proved the blowup result when the initial energy was negative. But mostly the results as regards the Cauchy problem of the wave equation were established for the typical form of nonlinear power as
where \(1< p<\frac{n+2}{n2}\) as \(n\geq3\) and \(1< p<\infty\) as \(n=1, 2\). Here we note that the above power satisfies the condition (2). For the power (4), the wave equations with damping term were studied by many authors. It is well known that the local solution blows up in finite time when the initial energy is negative. For global existence and nonexistence of solutions for the Cauchy problem of equation (1) with \(\rho(x)=1\), (4) and a (possibly nonlinear) damping term, we here refer to [7â€“9]. In particular, recently the wave equation with damping term was considered in [10], where Levine and Todorova showed that for arbitrarily positive initial energy there are choices of initial data such that the local solution blows up in finite time. Subsequently, Todorova and Vitillaro [11] established more precise result as regards the existence of initial values such that the corresponding solution blows up in finite time for arbitrarily high initial energy. More recently, Gazzola and Squassina [12] established sufficient conditions of initial data with arbitrarily positive initial energy such that the corresponding solution blows up in finite time for the wave equation with linear damping and (4) in the mass free case on a bounded Lipschitz subset of \(\mathbb{R}^{n}\). The first author [13] also studied the blowup result for the KleinGordon equation with arbitrarily positive initial energy. Also the authors [14] investigated the nonexistence of global solutions for some coupled KleinGordon equations with arbitrary initial energy. Also, the wave equations have been investigated with some special boundary conditions [15].
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 blowup 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 blowup 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 blowup result including the mass free case and mass case. Zhou established the blowup 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 blowup result for equation (1) with (H) and (2) when the initial energy is high. Based on a concavity argument, which was used to establish blowup of solutions to nonlinear damped wave equations (see e.g. [21, 22] and the references therein), we firstly establish the blowup result when the initial energy is nonpositive. 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 blowup 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 RemarkÂ 1) we improve the blowup result in [21]. As for the case with the arbitrarily positive initial energy, we establish the blowup 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 blowup 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 blowup result, but if \(\rho=\mathrm{constant}\ne 0\) it will affect the blowup result, that is, if \(m=0\) and \(\rho(x)=1\) then the blowup 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 wellknown 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 wellknown 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 âˆ«. The notation \(t\rightarrow T^{}\) means \(t< T\) and \(t\rightarrow T\).
As in [23], we introduce the function space \(X^{1,2}(R^{n})\), which is defined as the closure of \(C_{0}^{\infty}\) functions with respect to the energy norm \(\u\_{X^{1,2}}:=\int \nabla u(x)^{2}\,dx\), that is,
It is well known that \(X^{1,2}\) is embedded continuously in \(L^{\frac{2n}{n2}}\), which means that there exists a constant \(k>0\) such that
Following (6), we have the following inequality (see [24]).
Lemma 1
Suppose \(\rho\in L^{\frac{n}{2}}(\mathbb{R}^{n})\) and \(n\geq3\). Then there exists a constant \(\alpha>0\) such that
for every \(u\in C_{0}^{\infty}(\mathbb{R}^{n})\). Moreover, \(\alpha=k^{2}\\rho\^{1}_{\frac{n}{2}}\) where k is defined in (6).
In addition, the weighted space \(L^{2}_{\rho}(\mathbb{R}^{n})\) is defined to be the closure of \(C_{0}^{\infty}(\mathbb{R}^{n})\) functions with respect to the inner product
and norm
For the local existence of solutions of equation (1), we state the following.
Theorem 1
Under the assumption (H), for any initial data \((u_{0}, u_{1})\in X^{1,2}(\mathbb{R}^{n})\times L_{\rho}^{2}(\mathbb{R}^{n})\), if f satisfies the conditions \(f(0)=0\) and
for all \(\lambda_{1}, \lambda_{2}\in\mathbb{R}\), some constant \(c>0\), and
then there exists a unique local solution \(u(t)\) of the equation (1) on a maximal time interval \([0, T_{{\mathrm{max}}})\) satisfying
In addition, \(u(t)\) satisfies
for every \(t\in[0, T_{{\mathrm{max}}})\), where
where \(F(u)\) is defined in (3).
This result can be proved by the Banach fixed point theorem. The proof follows from the weightednorm 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 blowup result for equation (1) with (H) and (2) in the case of nonpositive initial energy.
Theorem 2
Under the assumptions (H) and (2), if the nonzero initial data \((u_{0},u_{1})\in X^{1,2}(\mathbb{R}^{n})\times L_{\rho}^{2}(\mathbb{R}^{n})\) satisfies
or
then the corresponding local solution of equation (1) blows up in finite time, \(T_{{\mathrm{max}}}<\infty\).
Remark 1
In the case \(E(0)<0\), Zhou [21] also established the blowup 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 resultÂ [21].
To state our main blowup result with arbitrarily positive initial energy, we introduce a function as follows:
Now we introduce our main blowup result for equation (1) with arbitrarily positive initial energy, as far as we know, which is the first blowup result for equation (1) with (H) on the whole space \(\mathbb{R}^{n}\).
Theorem 3
Under the assumptions (H) and (2), if the initial data \((u_{0},u_{1})\in X^{1,2}(\mathbb{R}^{n})\times L_{\rho}^{2}(\mathbb {R}^{n})\) satisfies
where Ïµ and Î± are stated in (2) and LemmaÂ 1, respectively, then the corresponding local solution of equation (1) blows up in finite time \(T_{{\mathrm {max}}}<\infty\).
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 RemarkÂ 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 \(f(u)=u^{p1}u\) 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
Let \(F(t)\in C^{2}([0,T))\) be a function satisfying that
for some \(\alpha>0\), if \(F(0)> 0\) and \(F^{\prime}(0)>0\) then \(F(t)\rightarrow+\infty\) as \(t\rightarrow T_{\mathrm{max}}\) (\(\leq F(0)/\alpha F^{\prime}(0)\)).
We next show a lemma, which plays a role in our proofs of Theorems 2 and 3.
Lemma 3
Assume that \((u_{0}, u_{1})\in X^{1,2}(\mathbb{R}^{n})\times L^{2}_{\rho}(\mathbb{R}^{n})\) satisfies
If the corresponding local solution \((u(t,x), u_{t}(t,x))\in C([0, T_{{\mathrm{max}}}), X^{1,2}(\mathbb{R}^{n})\times L^{2}_{\rho}(\mathbb {R}^{n}))\) is such that
for every \(t\in[0, T_{{\mathrm{max}}})\), then \(\u(t,\cdot)\ _{L^{2}_{\rho}}^{2}\) is strictly increasing on \([0, T_{{\mathrm{max}}})\).
Proof
Since \(u(t,x)\) is the local solution of equation (1), we easily have
for every \(t\in[0, T_{{\mathrm{max}}})\).
As a result, it follows that
By (26) and the above equation we have
for every \(t\in[0, T_{{\mathrm{max}}})\).
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 \([0, T_{{\mathrm{max}}})\).â€ƒâ–¡
Proof of TheoremÂ 2
We first define the following auxiliary function:
where the constants, \(T_{0}>0\), \(T_{1}>0\), \(\zeta>0\), will be determined later.
We then have
and
Case I: \(E(0)<0\). By (2), (14), and (15), we see that
for all \(t\in[0, T_{{\mathrm{max}}})\).
Thus, from (31) it follows that
We now let the constant Î¶ satisfy
Then it follows from (34)
Obviously, \(G^{\prime\prime}(t)>0\) on \([0, T_{{\mathrm{max}}})\). Moreover, we can take \(T_{1}>0\) sufficiently large such that
and
Thus, by (36) we see that \(G(t)>0\), \(G^{\prime}(t)>0\), and \(G^{\prime\prime}(t)>0\) for every \(t\in[0, T_{{\mathrm{max}}})\). That is, \(G(t)\) and \(G^{\prime}(t)\) are strictly increasing on \([0,T_{{\mathrm{max}}})\). Then we let
For every \(t\in[0, T_{0}]\) we obviously have
We now let \(T_{0}\) be sufficiently large and satisfy
Noting the inequalities (29), (30), and (37), we see that \(T_{0}\), defined by (43), is reasonable.
And suppose the solution \(u(t,x)\) exists on \([0, T_{0}]\), then it follows that
for every \(t\in[0,T_{0}]\)
By a simple computation we see that
for every \(s\in\mathbb{R}\) and \(t\in[0, T_{0}]\), which means that \((2B)^{2}4AC\leq0\).
Thus we see that
Since \(\frac{4+\epsilon}{4}>1\), we put \(\alpha=\frac{\epsilon}{4}\). By LemmaÂ 2 we see that
which implies that the corresponding solution \(u(t,x)\) of equation (1) blows up in finite time \(T_{{\mathrm {max}}}<\infty\).
Case II: \(E(0)=0\) and \(\int\rho(x)u_{0}(x)u_{1}(x)\,dx\geq0\). By (14) and (15) we have
for every \(t\in[0, T_{{\mathrm{max}}})\).
Noting the fact that \(\int\rho(x)F(u(t,x))\,dx\ne0\), we obtain by (2)
We then get
for every \(t\in[0, T_{{\mathrm{max}}})\).
Thus by (17) and LemmaÂ 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 \([0, T_{{\mathrm{max}}})\).
And as in (33) we also have
where the last inequality comes from LemmaÂ 1 and LemmaÂ 3.
Now we let the constant Î¶ satisfy
for the mass free case or the mass case, and the other positive constants, \(T_{0}\) and \(T_{1}\), be large such that
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 TheoremÂ 2 is completed.â€ƒâ–¡
In the following part we will address TheoremÂ 3. The next lemma is the crux to prove TheoremÂ 3.
Lemma 4
Under the assumptions on \(\rho(x)\), \(f(u)\) and \((u_{0},u_{1})\) in TheoremÂ 3, the corresponding local solution \((u(t,x), u_{t}(t,x))\in C([0, T_{{\mathrm{max}}}), X^{1,2}(\mathbb{R}^{n})\times L^{2}_{\rho}(\mathbb{R}^{n}))\) satisfies
for every \(t\in[0, T_{{\mathrm{max}}})\).
Proof
Here the proof uses a contradiction argument. We assume that (55) is not true over \([0, T_{\mathrm{max}})\), that is, there exists a time \(T>0\) such that
Case I: \(m\ne0\). Since \(I(u(t,\cdot))<0\) on \([0, T)\), by LemmaÂ 3 we see that \(\u(t,\cdot)\_{L^{2}_{\rho}}^{2}\) is strictly increasing over \([0, T)\), which implies that
for \(m\ne0\) and every \(t\in(0, T)\).
And by the continuity of \(\u(t,\cdot)\^{2}_{L^{2}_{\rho}}\) at t, we see that
On the other hand, by (14) and (15) we see that
Moreover, noting the assumption \(I(u(T,\cdot))=0\) and (2), we then have
Combining (61) and (62) we then obtain
Obviously there is a contradiction between (60) and (63). Thus we have proved that
for every \(t\in[0, T_{{\mathrm{max}}})\).
By LemmaÂ 3 we see that if \(I(u(t,\cdot))<0\) for every \(t\in[0, T_{{\mathrm{max}}})\), then \(\u(t,\cdot)\_{L^{2}_{\rho}}^{2}\) is strictly increasing on t. Thus, (64) implies that
for every \(t\in[0, T_{{\mathrm{max}}})\).
Thus the proof for Case I, \(m\ne0\), is complete.
Case II: \(m=0\). As in the argument for (60), we can also obtain
Since \(m=0\), the inequality (63) is rewritten as
By LemmaÂ 1, we see that
Thus by (66) and (68) we see that the assumption \(I(u(T,\cdot))=0\) is wrong. That is to say, \(I(u(t,\cdot))<0\) for every \(t\in[0, T_{{\mathrm{max}}})\).
Similarly, we also get
for every \(t\in[0, T_{{\mathrm{max}}})\).
Thus the proof of LemmaÂ 4 has been completed.â€ƒâ–¡
Proof of TheoremÂ 3
Here we still use the auxiliary function G, defined in (29). We have
By LemmaÂ 4, we see that
for every \(t\in[0, T_{{\mathrm{max}}})\).
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 \([0, T_{{\mathrm{max}}})\).
Obviously the inequality (32) is also valid for every \(t\in[0, T_{{\mathrm{max}}})\). Then by LemmaÂ 1, LemmaÂ 3, (56), (57), and (70) we have
for every \(t\in[0, T_{{\mathrm{max}}})\).
for every \(m\in\mathbb{R}\).
We now let Î¶ satisfy
for every \(m\in\mathbb{R}\), the other positive constants, \(T_{0}\), and \(T_{1}\), be large such that
We next let A, B, C denote the same terms as (38), (39), and (40), respectively.
We assume that the solution \(u(t)\) exists on \([0, T_{0}]\). Then we have
and
for every \(t\in[0, T_{0}]\).
In the same way as the proof of TheoremÂ 2 we have
By LemmaÂ 2 we see that there exists a finite time \(T_{{\mathrm{max}}}<\infty\) such that the corresponding solution blows up in finite time \(T_{{\mathrm{max}}}\).â€ƒâ–¡
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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.
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Wang, Y., Wang, Y. Global nonexistence for some nonlinear wave equations with damping and source terms in an inhomogeneous medium. Bound Value Probl 2017, 39 (2017). https://doi.org/10.1186/s1366101707624
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DOI: https://doi.org/10.1186/s1366101707624