# 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  March 2021 Early access  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 and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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