Abstract
In this paper, we discuss the blow-up and lifespan phenomenon for the following wave equation with variable coefficient:
with small initial data, where \(a(x)>0\), \(Du=(u_{x_{0}},u_{x_{1}},\ldots ,u_{x_{n}})\) and \(D_{x}Du=(u_{x_{k}x_{l}}, k,l=0,1,\ldots ,n, k+l\geq 1)\).
Then we find a new phenomenon. The Cauchy problem
is globally well-posed for small initial data, while for the combined nonlinearities
with small initial data will blow up in finite time. Moreover, we obtain the lifespan results for the above problems.
Similar content being viewed by others
1 Introduction and main results
1.1 Introduction
The blow-up results concerning the semilinear wave equation
were firstly studied by John [8] when \(n=3\). More precisely, he showed that the semilinear wave equation (1) has the global solutions if \(p>1+\sqrt{2}\) and initial data are sufficiently small. Meanwhile, he proved the finite time blow-up of solutions if \(p<1+\sqrt{2}\) and the initial data are not both identically zero. Then Strauss conjectured, when \(n\geq 2\), the existence or nonexistence of global solutions to equation (1) for \(p\in (p_{c}(n),\infty )\) or \(p\in (1,p_{c}(n)]\), where \(p_{c}(n)\) is the positive root of the quadratic equation
After that, there has been much work concerning this conjecture. We give a brief summary here. To see the global existence of solutions to (3), one can refer to Glassey [3] for \(n=2\), Lindblad and Sogge [14] for \(n\leq 8\) for \(n\geq 4\); Georgiev, Lindblad, and Sogge [2] for \(n\geq 4\) and \(p_{c}< p\leq \frac{n+3}{n-1}\). To see the finite time blow-up of solutions to (3), one can see Glassey [4] for \(n=2\) and Sideris [15] for \(n\geq 4\), Yordanov and Zhang [21] and Zhou [23] for \(n\geq 4\), Takamura and Wakasa [18] and Zhou and Han [25] for \(n\geq 2\) and the sharp upper bound of the lifespan of the solution by using a different method, respectively. The lifespan \(T(\epsilon )\) of the solutions of (1) is the largest value such that solutions exist for \(x\in \mathbb{R}^{n}\), \(0\leq t< T(\epsilon )\). To the best knowledge of the authors, there is little work concerning the analog of the Strauss conjecture on cosmological spacetimes except the work of Lindblad et al. [13]. They showed the global existence of solutions for the semilinear wave equation on Kerr black hole backgrounds. Zhou and Han [24] first obtained a blow-up result on semilinear wave equations with variable coefficients and boundary. Lai and Zhou [9, 10] obtained finite time blow-up result for nonlinear wave equations in exterior domains. Yan [19] verified this conjecture on blow-up result for semilinear wave equation in de Sitter spacetimes. After that, Li, Li, and Yan [11] gave the blow-up results of semilinear damped wave equation in de Sitter spacetimes. We refer the reader to [1, 5, 12, 22] for more related results. In this paper, one of our main results is to study the blow-up result on quasilinear wave equations with variable coefficients.
Another problem concerns the blow-up solution of nonlinear wave equation with exponential type nonlinearity. The global existence of initial value problem for the nonlinear wave equation with exponential type nonlinearity
was studied by Ibrahim, Majdoub, and Masmoudi [6]. They showed that if the initial energy is small, then the nonlinear wave equation with exponential type nonlinearity is globally well-posed. Here α is a positive constant in \((0,4\pi ]\). A scattering problem in the energy space for Klein–Gordon equations with nonlinearity of exponential growth in two space dimensions was studied in [7]. Struwe [16] established the global well-posedness of solutions to the Cauchy problem for the wave equations with exponential nonlinearities in the super-critical regime of large energies for smooth and radially symmetric data. Then, he [17] showed that the Cauchy problem for wave equations with critical exponential nonlinearities in two space dimensions is globally well-posed for arbitrary smooth initial data.
1.2 Main results
In this paper, we consider the following Cauchy problem with small initial data in \(n\geq 2\) space dimensions:
where \((t,x)\in \mathbb{R}^{+}\times \mathbb{R}^{n}\), \(a(x)\) is a positive smooth function,
\(f(x)\), \(g(x)\in \mathbb{C}_{0}^{\infty }(\mathbb{R}^{n})\), ϵ is a small parameter, \(\lambda _{k}\) (\(k=0,1,2,3\)) are nonnegative constants, \(p_{k}>1\). Here, for simplicity of notations, we write \(x_{0}=t\).
We assume that compactly supported nonnegative data f and g satisfy
Here we give one of our main results.
Theorem 1
Letf, gbe smooth functions with compact support\(f,g\in \mathbb{C}_{0}^{\infty }\)and satisfy (5), space dimensions\(n\geq 2\). Assume that problem (3) has a solution\((u,u_{t})\in \mathbb{C}([0,T),\mathbb{H}^{1}(\mathbb{R}^{n})\times \mathbb{L}^{r}(\mathbb{R}^{n}))\), \(a(x)>0\)and\(\frac{\triangle a(x)}{a(x)}\in (0,C(1+|x|^{2+\delta })^{-1})\)is local Hölder continuous, where\(r=\max \{2,p_{0}\}\)such that
and the index\(p_{0}>1\), \(p_{2}>1\), \(p_{3}>1\)and\(p_{1}\)satisfies
Then the solution\(u(t,x)\)will blow up in finite time, that is, \(T<\infty \). Moreover, we have the following estimates for the lifespan\(T(\epsilon )\)of solutions of (3): there exists a positive constantC, which is independent ofϵ, such that
where\(C_{0}\)is a positive constant which is independent ofϵ.
Secondly, we consider the following problem:
We assume that compactly supported nonnegative data f and g satisfy
Theorem 2
Letgbe a smooth function with compact support\(g\in \mathbb{C}_{0}^{\infty }(\mathbb{R}^{2})\)and satisfy (7). Assume that problem (6) has a solution\((u,u_{t})\in \mathbb{C}([0,T),\mathbb{H}^{1}(\mathbb{R}^{2})\times \mathbb{L}^{2}(\mathbb{R}^{2}))\)such that
Then the solution\(u(t,x)\)will blow up in finite time, that is, \(T<\infty \). Moreover, we have the following estimates for the lifespan\(T(\epsilon )\)of solutions of (6): there exists a positive constantC, which is independent ofϵ, such that
where\(C_{0}\)is a positive constant which is independent ofϵ.
The organization of this paper is as follows. In Sect. 2, we recall some blow-up criteria on ODEs. Section 3 is devoted to proving the finite time blow-up of solutions for the quasilinear wave equation (3) with variable coefficients. In the last section, the proof of Theorem 2 is given.
2 Preliminaries
This section recalls some blow-up results for ordinary differential inequality. The first relevant result on ODE was established by Sideris [15]. The following blow-up result can be found in [18, 21] as Lemma 2.1.
Lemma 1
([18])
Let\(p > 1\), \(a>0\), and\((p-1)a=q-2\). Assume that\(G\in \mathbb{C}^{2}([0, T ))\)satisfies
whereK, \(T_{0}\), A, andRdenote positive constants with\(T_{0}\geq R\). ThenTmust satisfy\(T\leq 2T_{1}\)provided that\(K\geq K_{0}\), where
with arbitrarily chosenδsatisfying\(0<\delta <\frac{p-1}{2}\)and a fixed positive constantB.
A more general blow-up result was given in [19]. One can see Lemma 2.2 in [19] for more details on the proof.
Lemma 2
([19])
Let\(p > 1\). Assume that\(G\in \mathbb{C}^{2}([0, T ))\)satisfies
whereK, \(T_{0}\), A, andRdenote positive constants with\(T_{0}\geq R\), \(a(t)\)and\(b(t)\)are positive strictly increasing smooth functions, and\(b^{-\frac{1}{2}}(t+R)a^{\frac{p-1}{2}-\delta }(t)\)is a strictly decreasing smooth function for\(t>0\), and there exist fixed\(t_{0}\geq 2T_{1}\)and a positive constantK̃such that
ThenTmust satisfy\(T\leq 2T_{1}\)provided that\(K\geq K_{0}\), where
with arbitrarily chosenδsatisfying\(0<\delta <\frac{p-1}{2}\).
Now we have a new blow-up result.
Lemma 3
Let\(p > 1\)and\(b_{1}-a_{1}(p-1)=2\). Assume that\(G\in \mathbb{C}^{2}([0, T ))\)satisfies
whereK, \(T_{0}\), A, andRdenote positive constants with\(T_{0}\geq R\).
ThenTmust satisfy\(T\leq 2T_{1}\)provided that\(K\geq K_{0}\), where
for arbitrarily chosenδsatisfying\(0<\delta <\frac{p-1}{2}\), and a positive constant\(\tilde{K}\geq a_{1}\delta e^{-\frac{1}{2}}\).
Proof
We verify condition (11). Let \(t_{0}=2T_{1}\). Then direct computation shows that
with \(\tilde{K}\geq a_{1}\delta e^{-\frac{1}{2}}\). This completes the proof. □
It follows from Yordanov and Zhang [20] that we introduce \(\phi _{0}(x)\in \mathbb{C}^{2}(\mathbb{R}^{n})\) and
which are solutions of
respectively. It is easy to see that \(\phi _{0}(x)\neq \phi _{1}(x)\).
Then one can verify \(\phi _{0}(x)\) and \(\phi _{1}(x)\) (see [26]) such that
where C is a positive constant.
Moreover, we introduce a test function
It is easy to see
3 Proof of Theorem 1
Rewrite the variable wave equation (3) as
with the initial data \((u_{0},u_{1})\) satisfying (5), where F takes the form of (4).
Define
Since \(0<\frac{\triangle a(x)}{a(x)}<C(1+|x|^{2+\delta })^{-1}\) is local Hölder continuous, we derive
So multiplying (20) both sides by \(\phi _{0}(x)\) and using (21), we have
where the last inequality is derived by noticing \(F(|u(t,x)|,|\nabla u(t,x)|, |\triangle u(t,x)|)>0\) from (4).
Similarly, by (4) and (17), we obtain
By the Hölder inequality, we have
Thus it follows from (22) that
which implies that
Integrating (24) over \([0,t]\), we get
On the other hand, it follows from (22) that \(G'(t)=\int _{\mathbb{R}^{n}}\partial _{t}u(t,x)\phi _{0}(x)\,dx\) is an increasing function for \(t\geq 0\). Since \(g(x)\geq 0\) in (5), \(G(t)\) is also an increasing function for \(t\geq 0\). By \(f(x)\geq 0\) in (5), we know that \(G(t)>0\). Thus it follows from (25) that
On the other hand, by the Hölder inequality, we derive
which combining with (17) gives that
Then by (23) we obtain
where \(C_{\lambda }\) is a positive constant which depends on \(\lambda _{1}\).
Let
Next we apply Lemma 2 to prove our result. Let \(0<\delta <\min \{\frac{1}{n},\frac{p_{1}-1}{2}\}\). It follows from (26)–(27) that (8)–(10) hold. The rest is to verify conditions (11) and (12). Substituting (28) into (11), direct computation shows that if we take \(p_{0}>1\) and
then (11) holds.
Taking a fixed positive constant K̃ such that
then (12) holds.
Thus \(G(t)\) will blow up in finite time, then the solutions to problem (3) will blow up in finite time. At last, we estimate the lifespan result. Since \(G''(t)\geq 0\) and \(G'(0)\geq 0\), \(G(t)\) is an increasing smooth function. So it holds
It follows from (27) that
Furthermore, we have
where \(C_{0}\) is a positive constant which is independent of ϵ. We complete the proof of Theorem 1.
4 Proof of Theorem 2
Then we consider the approximation equation of (6)
where \((t,x)\in \mathbb{R}^{+}\times \mathbf{R}^{2}\) and \(m\in \mathbb{N}\).
By the local existence of classical solutions, the solution to Cauchy problem (29) can be approximated by Picard iteration. Set \(u^{(0)}\equiv 0\). Then \(u_{t}^{(0)}\equiv 0\). So by the positivity of the fundamental solution of the wave operator in two space dimensions, we can prove that \(u^{(m)}(t,x)\) is a series of approximate solutions to (29) by induction. Let \(m\longrightarrow \infty \), we conclude that \(u(t,x)\) is a nonnegative solution to (6).
Let \(r=|x|\) and \(\mathcal{G}(r)=\frac{1}{2}r^{\frac{1}{2}}g(r)\), \(x\in \mathbb{R}^{2}\). The radial symmetric form of equation (6) is
with initial data
Using (31) and D’Alembert’s formula, for \(r>t\), we have
Differentiating (32) with respect to t, we get
For \(t\geq \frac{1}{2}\) and \(\frac{1}{4}< r-t\leq \frac{3}{4}\), by the form of \(\mathcal{G}\), it follows from (33) that
Let
Note that \(e^{u}>u\) for \(u>0\). By (34), direct computation shows that
and
which implies that
Since \(ue^{u^{2}}\) is a positive function, by the Hölder inequality, we derive
which combining with (35) gives that
Let
It is easy to see that (16) holds for \(0<\delta <\frac{1}{3}\). Applying Lemma 1 to \(G(t)\), we know that \(G(t)\) will blow up in finite time, then the solutions to problem (6) will blow up in finite time. Furthermore, we have \(T(\epsilon )\leq C_{0}\epsilon ^{-2}\), where \(C_{0}>0\) is independent of ϵ. We complete the proof of Theorem 2.
5 Conclusion
In our paper, we show blow-up phenomena for two kinds of nonlinear wave equations, i.e., nonlinear wave equation with with variable coefficient and wave equation with exponential type nonlinearity. The importance of our results is that if the null condition or weak null condition cannot be satisfied, a perturbation of quasilinear term can destroy the global well-posedness.
References
Cooper, S., Savostianov, A.: Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations. Adv. Nonlinear Anal. 9, 745–787 (2020)
Georgiev, V., Lindblad, H., Sogge, C.D.: Weighted Strichartz estimates and global existence for semilinear wave equations. Am. J. Math. 119, 1291–1319 (1997)
Glassey, R.T.: Finite time blow up for solutions of nonlinear wave equations. Math. Z. 177, 323–340 (1981)
Glassey, R.T.: Existence in the large for \(\square u=|u|^{p}\) in two space dimensions. Math. Z. 178, 233–261 (1981)
Goubet, O., Manoubi, I.: Theoretical analysis of a water wave model with a nonlocal viscous dispersive term using the diffusive approach. Adv. Nonlinear Anal. 8, 253–266 (2019)
Ibrahim, S., Majdoub, M., Masmoudi, N.: Global solutions for a semilinear, two-dimensional Klein–Gordon equation with exponential-type nonlinearity. Commun. Pure Appl. Math. 59, 1639–1658 (2006)
Ibrahim, S., Majdoub, M., Masmoudi, N., Nakanishi, K.: Scattering for the two-dimensional energy-critical wave equation. Duke Math. J. 150, 287–329 (2009)
John, F.: Blow up of solutions of nonlinear wave equations in three space dimensions. Manuscr. Math. 28, 235–268 (1979)
Lai, N.A., Zhou, Y.: Finite time blow up to critical semilinear wave equation outside the ball in 3-D. Nonlinear Anal. 125, 550–560 (2015)
Lai, N.A., Zhou, Y.: Nonexistence of global solutions to critical semilinear wave equations in exterior domain in high dimensions. Nonlinear Anal. 143, 89–104 (2016)
Li, H., Li, X., Yan, W.: Lifespan of solutions to semilinear damping wave equations in de Sitter spacetime. Nonlinear Anal. 195, 111735 (2020)
Lian, W., Xu, R.Z.: Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. 9, 613–632 (2020)
Lindblad, H., Metcalfe, J., Sogge, M., Tohaneanu, M., Wang, C.: The Strauss conjecture on Kerr black hole backgrounds. Math. Ann. 359, 637–661 (2014)
Lindblad, H., Sogge, C.D.: Long time existence for small amplitude semilinear wave equations. Am. J. Math. 118, 1047–1135 (1996)
Sideris, T.C.: Nonexistence of global solutions to semilinear wave equations in high dimensions. J. Differ. Equ. 52, 378–406 (1984)
Struwe, M.: Global well-posedness of the Cauchy problem for a super-critical nonlinear wave equation in 2 space dimensions. Math. Ann. 350, 707–719 (2011)
Struwe, M.: The critical nonlinear wave equation in two space dimensions. J. Eur. Math. Soc. 15, 1805–1823 (2013)
Takamura, H., Wakasa, K.: The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions. J. Differ. Equ. 251, 1157–1171 (2011)
Yan, W.P.: Lifespan of solutions to wave equations on de Sitter spacetime. Proc. R. Soc. Edinb. A 148, 1313–1330 (2018)
Yordanov, B., Zhang, Q.S.: Finite time blowup for wave equations with a potential. SIAM J. Math. Anal. 36, 1426–1433 (2005)
Yordanov, B., Zhang, Q.S.: Finite time blow up for critical wave equations in high dimensions. J. Funct. Anal. 231, 361–374 (2006)
Zhao, X., Yan, W.: Existence of standing waves for quasi-linear Schrödinger equations on \(T^{n}\). Adv. Nonlinear Anal. 9, 978–993 (2020)
Zhou, Y.: Blow up of solutions to semilinear wave equations with critical exponent in high dimensions. Chin. Ann. Math. 28B, 205–212 (2007)
Zhou, Y., Han, W.: Blow up of solutions to semilinear wave equations with variable coefficients and boundary. J. Math. Anal. Appl. 374, 585–601 (2011)
Zhou, Y., Han, W.: Life-span of solutions to critical semilinear wave equations. Commun. Partial Differ. Equ. 39, 439–451 (2014)
Zhou, Y., Han, W.: Blow up for some semilinear wave equations in multi-space dimensions. Commun. Partial Differ. Equ. 39, 651–665 (2014)
Acknowledgements
The authors express their sincere thanks to the editors and anonymous referees for very careful reading and for providing many valuable comments and suggestions which led to improvement of this paper.
Availability of data and materials
Not applicable.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
The authors contributed equally to this paper. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Ethics approval and consent to participate
We agree.
Competing interests
The authors declare that no competing interests exist.
Consent for publication
We agree.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Li, H., Sun, J. & Zhao, X. Blow-up analysis for two kinds of nonlinear wave equations. Bound Value Probl 2020, 59 (2020). https://doi.org/10.1186/s13661-020-01357-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-020-01357-w