doi: 10.3934/dcdss.2020388

Convergence of a blow-up curve for a semilinear wave equation

National Institute of Technology, Ibaraki College, 866 Nakane, Hitachinaka-shi, Ibaraki-ken 312-8508, Japan

Received  January 2019 Revised  February 2020 Published  June 2020

Fund Project: This work was supported by JSPS Grant-in-Aid for Early-Career Scientists, 18K13447

We consider a blow-up phenomenon for $ { \partial_t^2 u_ \varepsilon} $ $ {- \varepsilon^2 \partial_x^2u_ \varepsilon } $ $ { = F(\partial_t u_ \varepsilon)}. $ The derivative of the solution $ \partial_t u_ \varepsilon $ blows-up on a curve $ t = T_ \varepsilon(x) $ if we impose some conditions on the initial values and the nonlinear term $ F $. We call $ T_ \varepsilon $ blow-up curve for $ { \partial_t^2 u_ \varepsilon} $ $ {- \varepsilon^2 \partial_x^2u_ \varepsilon } $ $ { = F(\partial_t u_ \varepsilon)}. $ In the same way, we consider the blow-up curve $ t = \tilde{T}(x) $ for $ {\partial_t^2 u} $ $ = $ $ {F(\partial_t u)}. $ The purpose of this paper is to show that, for each $ x $, $ T_ \varepsilon(x) $ converges to $ \tilde{T}(x) $ as $ \varepsilon\rightarrow 0. $

Citation: Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020388
References:
[1]

H. Bellout and A. Friedman, Blow-up estimates for a nonlinear hyperbolic heat equation, SIAM J. Math. Anal., 20 (1989), 354-366.  doi: 10.1137/0520022.  Google Scholar

[2]

L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc., 297 (1986), 223-241.  doi: 10.1090/S0002-9947-1986-0849476-3.  Google Scholar

[3]

A. Friedman and L. Oswald, The blow-up surface for nonlinear wave equations with small spatial velocity, Trans. Amer. Math. Soc., 308 (1988), 349-367.  doi: 10.1090/S0002-9947-1988-0946448-7.  Google Scholar

[4]

P. Godin, The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension. I, Calc. Var. Partial Differential Equations, 13 (2001), 69-95.  doi: 10.1007/PL00009924.  Google Scholar

[5]

M. A. Hamza and H. Zaag, Blow-up behavior for the Klein-Gordon and other perturbed semilinear wave equations, Bull. Sci. Math., 137 (2013), 1087-1109.  doi: 10.1016/j.bulsci.2013.05.004.  Google Scholar

[6]

F. Merle and H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal., 253 (2007), 43-121.  doi: 10.1016/j.jfa.2007.03.007.  Google Scholar

[7]

F. Merle and H. Zaag, Openness of the set of non-characteristic points and regularity of the blow-up curve for the 1D semilinear wave equation, Comm. Math. Phys., 282 (2008), 55-86.  doi: 10.1007/s00220-008-0532-3.  Google Scholar

[8]

F. Merle and H. Zaag, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math., 134 (2012), 581-648.  doi: 10.1353/ajm.2012.0021.  Google Scholar

[9]

T. Nakagawa, Blowing up of a finite difference solution to $u_t = u_xx + u^2$, Appl. Math. Optim., 2 (1975/76), 337-350.  doi: 10.1007/BF01448176.  Google Scholar

[10]

M. Ohta and H. Takamura, Remarks on the blow-up boundaries and rates for nonlinear wave equations, Nonlinear Anal., 33 (1998), 693-698.  doi: 10.1016/S0362-546X(97)00670-6.  Google Scholar

[11]

N. Saito and T. Sasaki, Blow-up of finite-difference solutions to nonlinear wave equations, J.Math.Sci. Univ. Tokyo, 23 (2016), 349-380.   Google Scholar

[12]

T. Sasaki, Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity, Advances in Differential Equations, 23 (2018), 373-408.   Google Scholar

[13]

H. Uesaka, The blow-up boundary for a system of semilinear wave equations, Further Progress in Analysis, World Sci. Publ., Hackensack, NJ, (2009), 845–853. doi: 10.1142/9789812837332_0081.  Google Scholar

show all references

References:
[1]

H. Bellout and A. Friedman, Blow-up estimates for a nonlinear hyperbolic heat equation, SIAM J. Math. Anal., 20 (1989), 354-366.  doi: 10.1137/0520022.  Google Scholar

[2]

L. A. Caffarelli and A. Friedman, The blow-up boundary for nonlinear wave equations, Trans. Amer. Math. Soc., 297 (1986), 223-241.  doi: 10.1090/S0002-9947-1986-0849476-3.  Google Scholar

[3]

A. Friedman and L. Oswald, The blow-up surface for nonlinear wave equations with small spatial velocity, Trans. Amer. Math. Soc., 308 (1988), 349-367.  doi: 10.1090/S0002-9947-1988-0946448-7.  Google Scholar

[4]

P. Godin, The blow-up curve of solutions of mixed problems for semilinear wave equations with exponential nonlinearities in one space dimension. I, Calc. Var. Partial Differential Equations, 13 (2001), 69-95.  doi: 10.1007/PL00009924.  Google Scholar

[5]

M. A. Hamza and H. Zaag, Blow-up behavior for the Klein-Gordon and other perturbed semilinear wave equations, Bull. Sci. Math., 137 (2013), 1087-1109.  doi: 10.1016/j.bulsci.2013.05.004.  Google Scholar

[6]

F. Merle and H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal., 253 (2007), 43-121.  doi: 10.1016/j.jfa.2007.03.007.  Google Scholar

[7]

F. Merle and H. Zaag, Openness of the set of non-characteristic points and regularity of the blow-up curve for the 1D semilinear wave equation, Comm. Math. Phys., 282 (2008), 55-86.  doi: 10.1007/s00220-008-0532-3.  Google Scholar

[8]

F. Merle and H. Zaag, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math., 134 (2012), 581-648.  doi: 10.1353/ajm.2012.0021.  Google Scholar

[9]

T. Nakagawa, Blowing up of a finite difference solution to $u_t = u_xx + u^2$, Appl. Math. Optim., 2 (1975/76), 337-350.  doi: 10.1007/BF01448176.  Google Scholar

[10]

M. Ohta and H. Takamura, Remarks on the blow-up boundaries and rates for nonlinear wave equations, Nonlinear Anal., 33 (1998), 693-698.  doi: 10.1016/S0362-546X(97)00670-6.  Google Scholar

[11]

N. Saito and T. Sasaki, Blow-up of finite-difference solutions to nonlinear wave equations, J.Math.Sci. Univ. Tokyo, 23 (2016), 349-380.   Google Scholar

[12]

T. Sasaki, Regularity and singularity of the blow-up curve for a wave equation with a derivative nonlinearity, Advances in Differential Equations, 23 (2018), 373-408.   Google Scholar

[13]

H. Uesaka, The blow-up boundary for a system of semilinear wave equations, Further Progress in Analysis, World Sci. Publ., Hackensack, NJ, (2009), 845–853. doi: 10.1142/9789812837332_0081.  Google Scholar

[1]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020391

[2]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280

[3]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[4]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108

[5]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006

[6]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1

[7]

Christian Klein, Benson Muite, Kristelle Roidot. Numerical study of blow-up in the Davey-Stewartson system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1361-1387. doi: 10.3934/dcdsb.2013.18.1361

[8]

Mohamed-Ali Hamza, Hatem Zaag. Blow-up results for semilinear wave equations in the superconformal case. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2315-2329. doi: 10.3934/dcdsb.2013.18.2315

[9]

Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669

[10]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[11]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089

[12]

Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639

[13]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

[14]

Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617

[15]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069

[16]

István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134

[17]

Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072

[18]

Lili Du, Zheng-An Yao. Localization of blow-up points for a nonlinear nonlocal porous medium equation. Communications on Pure & Applied Analysis, 2007, 6 (1) : 183-190. doi: 10.3934/cpaa.2007.6.183

[19]

Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11

[20]

Christian Klein, Ralf Peter. Numerical study of blow-up in solutions to generalized Kadomtsev-Petviashvili equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1689-1717. doi: 10.3934/dcdsb.2014.19.1689

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (14)
  • HTML views (42)
  • Cited by (0)

Other articles
by authors

[Back to Top]