# American Institute of Mathematical Sciences

March  2021, 14(3): 1133-1143. 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, 2021, 14 (3) : 1133-1143. 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, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 [2] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [3] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [4] Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 [5] Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318 [6] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [7] Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039 [8] George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003 [9] Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354 [10] Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015 [11] Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466 [12] Xinyu Mei, Yangmin Xiong, Chunyou Sun. Pullback attractor for a weakly damped wave equation with sup-cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 569-600. doi: 10.3934/dcds.2020270 [13] Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390 [14] Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 [15] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [16] Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243 [17] Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020355 [18] Ludovick Gagnon, José M. Urquiza. Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation. Evolution Equations & Control Theory, 2021, 10 (1) : 129-153. doi: 10.3934/eect.2020054 [19] Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $\mathbb{R}^n_+$. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033 [20] Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

2019 Impact Factor: 1.233

Article outline