Global controllability and stabilizability of Kawahara equation on a periodic domain

Previous Article
Exponential stability of a joint-leg-beam system with memory damping
MCRF Home
This Issue
Next Article
Feedback controls to ensure global solutions and asymptotic stability of Markovian switching diffusion systems

June 2015, 5(2): 335-358. doi: 10.3934/mcrf.2015.5.335

Global controllability and stabilizability of Kawahara equation on a periodic domain

Xiangqing Zhao ^1, and Bing-Yu Zhang ^2,

1.	Department of Mathematics, Zhejiang Ocean University, Zhoushan, Zhejiang 316022, China
2.	Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Oh 45221

Received January 2014 Revised July 2014 Published April 2015

Abstract
Related Papers

In this paper we study controllability and stabilizability of a class of distributed parameter control system described by the Kawahara equation posed on a periodic domain $\mathbb{T}$ with internal control acting on a sub-domain $\omega $ of $\mathbb{T}$. Earlier in [42], aided by Bourgain smoothing property of the system, we showed that the system is locally exactly controllable and exponentially stabilizable. In this paper, helped further by certain properties of propagation of compactness and regularity in Bourgain spaces for the solutions of the associated linear system, we show that the system is globally exactly controllable and globally exponentially stabilizable.

Keywords: propagation of compactness, Kawahara equation, global stabilizability, propagation of regularity, global exact controllability, Bourgain space and Bourgain smoothing..

Mathematics Subject Classification: Primary: 93B05, 93D15; Secondary: 35Q5.

Citation: Xiangqing Zhao, Bing-Yu Zhang. Global controllability and stabilizability of Kawahara equation on a periodic domain. Mathematical Control & Related Fields, 2015, 5 (2) : 335-358. doi: 10.3934/mcrf.2015.5.335

References:

[1]	L. A. Abramyan and Yu. A. Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion,, Soviet Phys. JETP, 61 (1985), 963. Google Scholar
[2]	J. L. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-Value problem for the Kortewege Vries Equation posed on a finite domain,, Communications in Partial Differetial Equations, 28 (2003), 1391. doi: 10.1081/PDE-120024373. Google Scholar
[3]	J. L. Bona, S. M. Sun and B.-Y. Zhang, Non-homogeneous boundary value problems for the Kortewege Vries and the Korteweg de Vries Burgers equations in aquarter plane,, Ann. I. H. Poincaré-AN, 25 (2008), 1145. doi: 10.1016/j.anihpc.2007.07.006. Google Scholar
[4]	J. P. Boyd, Weakly non-local solitons for capillary-gravity waves: Fifth degree Korteweg-de Vries equation,, Phys. D, 48 (1991), 129. doi: 10.1016/0167-2789(91)90056-F. Google Scholar
[5]	J. Bourgain, Fourrier transform restriction phenomena for certain lattice subsets and applications to nonlinear evlution equations, Part I: The Schrödinger equation, Part II: The KdV equation,, Geom. Funct. Anal., 3 (1993), 209. doi: 10.1007/BF01895688. Google Scholar
[6]	E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain,, SIAM J. Control Optim., 46 (2007), 877. doi: 10.1137/06065369X. Google Scholar
[7]	E. Cerpa and E. Crepeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain,, Ann. I.H. Poincaré-AN, 26 (2009), 457. doi: 10.1016/j.anihpc.2007.11.003. Google Scholar
[8]	W. Chen and Z. Guo, Global well-posedness and I-method for the fifth-order Korteweg-de Vries equation,, Amer. J. Math., 114 (2011), 121. doi: 10.1007/s11854-011-0014-y. Google Scholar
[9]	P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations,, J. Amer. Math. Soc., 1 (1988), 413. doi: 10.1090/S0894-0347-1988-0928265-0. Google Scholar
[10]	J. M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length,, J. Eur. Math. Soc., 6 (2004), 367. Google Scholar
[11]	S. B. Cui and S. P. Tao, Strichartz estimates for dispersive equations and solvability of the Kawahara equation,, J. Math. Appl., 304 (2005), 683. doi: 10.1016/j.jmaa.2004.09.049. Google Scholar
[12]	S. B. Cui, D. G. Deng and S. P. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with $L^2$ initial data,, Acta Math. Sin., 22 (2006), 1457. doi: 10.1007/s10114-005-0710-6. Google Scholar
[13]	B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Anna. Sci. Ec. Norm. Super., 36 (2003), 525. doi: 10.1016/S0012-9593(03)00021-1. Google Scholar
[14]	J. Gorsky and A. A. Himonas, Well-posedness of KdV with higher dispersion,, Math. Comput. Simul., 80 (2009), 173. doi: 10.1016/j.matcom.2009.06.007. Google Scholar
[15]	A. Grünrock, New applications of the Fourier restriction norm method,, Ph.D thesis, (2002). Google Scholar
[16]	H. Hasimoto, Water waves,, Kagaku, 40 (1970), 401. Google Scholar
[17]	H. Hirayama, Local well-posedness for the periodic higher order KdV type equations,, NoDEA Nonlinear Differential Equations, 19 (2012), 677. doi: 10.1007/s00030-011-0147-9. Google Scholar
[18]	J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves,, Physica D, 32 (1988), 253. doi: 10.1016/0167-2789(88)90054-1. Google Scholar
[19]	Y. Hu, Discrete Fourier Restriction Phenomena Associated with Some Periodic Dispersive Wave Equations,, Ph.D thesis, (2012). Google Scholar
[20]	T. Kakutani and H. Ono, Weak non-linear hydromagnetic waves in a cold collision-free plasma,, J. Phys. Soc. Japan, 26 (1969), 1305. doi: 10.1143/JPSJ.26.1305. Google Scholar
[21]	V. I. Karpman and V. Yu. Belashov, Dynamics of two-dimensional soliton in weakly dispersive media,, Phys. Lett. A, 154 (1991), 131. Google Scholar
[22]	T. Kato, Low regularity well-posedness for the periodic Kawahara equation,, , (). Google Scholar
[23]	T. Kawahara, Oscillatory solitary waves in dispersive media,, J. Phys. Soc. Japan, 33 (1972), 260. doi: 10.1143/JPSJ.33.260. Google Scholar
[24]	C. Kenig, G. Ponce and L. Vega, A biliner estimate with applications to the KdV equations,, J. Amer. Math. Soc., 9 (1996), 573. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar
[25]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,, Philos. Mag., 39 (1895), 422. doi: 10.1080/14786449508620739. Google Scholar
[26]	C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval,, ESAIM-COCV, 16 (2010), 356. doi: 10.1051/cocv/2009001. Google Scholar
[27]	C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain,, Communications in Partial Differential Equations, 35 (2010), 707. doi: 10.1080/03605300903585336. Google Scholar
[28]	J. F. Li and S. G. Shi, Local well-posedness for the dispersion generalized periodic KdV equation,, J. Math. Anal. Appl., 379 (2011), 706. doi: 10.1016/j.jmaa.2011.01.026. Google Scholar
[29]	L. Rosier and B.-Y. Zhang, Exact boundary controllability of the nonlinear Schrödinger equation,, J. Differential Equations, 246 (2009), 4129. doi: 10.1016/j.jde.2008.11.004. Google Scholar
[30]	D. L. Russell, Computational study of the Korteweg-de Vries equation with localized control action,, in Distributed Parameter Control Systems* (eds. G. Chen*, (1991), 195. Google Scholar
[31]	D. L. Russell and B.-Y. Zhang, Exact controllability and stabiability of the Korteweg-de Veies equation,, Transaction of the American Mathematical Society, 348 (1996), 3643. doi: 10.1090/S0002-9947-96-01672-8. Google Scholar
[32]	D. L. Russell and B.-Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain,, SIAM J. Control and Optimization, 31 (1993), 659. doi: 10.1137/0331030. Google Scholar
[33]	P. N. da Silva, Unique Continuation for the Kawahara Equation,, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463. doi: 10.5540/tema.2007.08.03.0463. Google Scholar
[34]	M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM J. Control, 12 (1974), 500. doi: 10.1137/0312038. Google Scholar
[35]	T. Tao, Nonlinear Dispersive Equations, Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106.,, American Mathematical Society, (2006). Google Scholar
[36]	L. Tartar, Nonlinear interpolation and regularity,, J. Funct. Anal., 9 (1972), 469. Google Scholar
[37]	C. F. Vasconcellos and P. N. da Silva, Stabilization of the linear Kawahara equation with localized damping,, Asymptotic Analysis, 58 (2008), 229. Google Scholar
[38]	C. F. Vasconcellos and P. N. da Silva, Stabilization of the Kawahara equation with a localized damping,, ESAIM: COCV, 17 (2011), 102. doi: 10.1051/cocv/2009041. Google Scholar
[39]	H. Wang, S. B. Cui and D. G. Deng, Global existence of solutions for the Kawahara equation in Sobolev spaces of negative indices,, Acta Math. Sin., 23 (2007), 1435. doi: 10.1007/s10114-007-0959-z. Google Scholar
[40]	Y. Wu and Y. S. Li, The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity,, Math. Meth. Appl. Sci., 33 (2010), 1647. doi: 10.1002/mma.1273. Google Scholar
[41]	B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation,, SIAM J. Cont. Optim., 37 (1999), 543. doi: 10.1137/S0363012997327501. Google Scholar
[42]	B.-Y. Zhang and X. Q. Zhao, Control and stabilization of the Kawahara equation on a periodic,, Communications in Information and Systems, 12 (2012), 77. doi: 10.4310/CIS.2012.v12.n1.a4. Google Scholar

show all references

References:

[1]	L. A. Abramyan and Yu. A. Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion,, Soviet Phys. JETP, 61 (1985), 963. Google Scholar
[2]	J. L. Bona, S. M. Sun and B.-Y. Zhang, A nonhomogeneous boundary-Value problem for the Kortewege Vries Equation posed on a finite domain,, Communications in Partial Differetial Equations, 28 (2003), 1391. doi: 10.1081/PDE-120024373. Google Scholar
[3]	J. L. Bona, S. M. Sun and B.-Y. Zhang, Non-homogeneous boundary value problems for the Kortewege Vries and the Korteweg de Vries Burgers equations in aquarter plane,, Ann. I. H. Poincaré-AN, 25 (2008), 1145. doi: 10.1016/j.anihpc.2007.07.006. Google Scholar
[4]	J. P. Boyd, Weakly non-local solitons for capillary-gravity waves: Fifth degree Korteweg-de Vries equation,, Phys. D, 48 (1991), 129. doi: 10.1016/0167-2789(91)90056-F. Google Scholar
[5]	J. Bourgain, Fourrier transform restriction phenomena for certain lattice subsets and applications to nonlinear evlution equations, Part I: The Schrödinger equation, Part II: The KdV equation,, Geom. Funct. Anal., 3 (1993), 209. doi: 10.1007/BF01895688. Google Scholar
[6]	E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain,, SIAM J. Control Optim., 46 (2007), 877. doi: 10.1137/06065369X. Google Scholar
[7]	E. Cerpa and E. Crepeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain,, Ann. I.H. Poincaré-AN, 26 (2009), 457. doi: 10.1016/j.anihpc.2007.11.003. Google Scholar
[8]	W. Chen and Z. Guo, Global well-posedness and I-method for the fifth-order Korteweg-de Vries equation,, Amer. J. Math., 114 (2011), 121. doi: 10.1007/s11854-011-0014-y. Google Scholar
[9]	P. Constantin and J. C. Saut, Local smoothing properties of dispersive equations,, J. Amer. Math. Soc., 1 (1988), 413. doi: 10.1090/S0894-0347-1988-0928265-0. Google Scholar
[10]	J. M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length,, J. Eur. Math. Soc., 6 (2004), 367. Google Scholar
[11]	S. B. Cui and S. P. Tao, Strichartz estimates for dispersive equations and solvability of the Kawahara equation,, J. Math. Appl., 304 (2005), 683. doi: 10.1016/j.jmaa.2004.09.049. Google Scholar
[12]	S. B. Cui, D. G. Deng and S. P. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with $L^2$ initial data,, Acta Math. Sin., 22 (2006), 1457. doi: 10.1007/s10114-005-0710-6. Google Scholar
[13]	B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Anna. Sci. Ec. Norm. Super., 36 (2003), 525. doi: 10.1016/S0012-9593(03)00021-1. Google Scholar
[14]	J. Gorsky and A. A. Himonas, Well-posedness of KdV with higher dispersion,, Math. Comput. Simul., 80 (2009), 173. doi: 10.1016/j.matcom.2009.06.007. Google Scholar
[15]	A. Grünrock, New applications of the Fourier restriction norm method,, Ph.D thesis, (2002). Google Scholar
[16]	H. Hasimoto, Water waves,, Kagaku, 40 (1970), 401. Google Scholar
[17]	H. Hirayama, Local well-posedness for the periodic higher order KdV type equations,, NoDEA Nonlinear Differential Equations, 19 (2012), 677. doi: 10.1007/s00030-011-0147-9. Google Scholar
[18]	J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves,, Physica D, 32 (1988), 253. doi: 10.1016/0167-2789(88)90054-1. Google Scholar
[19]	Y. Hu, Discrete Fourier Restriction Phenomena Associated with Some Periodic Dispersive Wave Equations,, Ph.D thesis, (2012). Google Scholar
[20]	T. Kakutani and H. Ono, Weak non-linear hydromagnetic waves in a cold collision-free plasma,, J. Phys. Soc. Japan, 26 (1969), 1305. doi: 10.1143/JPSJ.26.1305. Google Scholar
[21]	V. I. Karpman and V. Yu. Belashov, Dynamics of two-dimensional soliton in weakly dispersive media,, Phys. Lett. A, 154 (1991), 131. Google Scholar
[22]	T. Kato, Low regularity well-posedness for the periodic Kawahara equation,, , (). Google Scholar
[23]	T. Kawahara, Oscillatory solitary waves in dispersive media,, J. Phys. Soc. Japan, 33 (1972), 260. doi: 10.1143/JPSJ.33.260. Google Scholar
[24]	C. Kenig, G. Ponce and L. Vega, A biliner estimate with applications to the KdV equations,, J. Amer. Math. Soc., 9 (1996), 573. doi: 10.1090/S0894-0347-96-00200-7. Google Scholar
[25]	D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,, Philos. Mag., 39 (1895), 422. doi: 10.1080/14786449508620739. Google Scholar
[26]	C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval,, ESAIM-COCV, 16 (2010), 356. doi: 10.1051/cocv/2009001. Google Scholar
[27]	C. Laurent, L. Rosier and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain,, Communications in Partial Differential Equations, 35 (2010), 707. doi: 10.1080/03605300903585336. Google Scholar
[28]	J. F. Li and S. G. Shi, Local well-posedness for the dispersion generalized periodic KdV equation,, J. Math. Anal. Appl., 379 (2011), 706. doi: 10.1016/j.jmaa.2011.01.026. Google Scholar
[29]	L. Rosier and B.-Y. Zhang, Exact boundary controllability of the nonlinear Schrödinger equation,, J. Differential Equations, 246 (2009), 4129. doi: 10.1016/j.jde.2008.11.004. Google Scholar
[30]	D. L. Russell, Computational study of the Korteweg-de Vries equation with localized control action,, in Distributed Parameter Control Systems* (eds. G. Chen*, (1991), 195. Google Scholar
[31]	D. L. Russell and B.-Y. Zhang, Exact controllability and stabiability of the Korteweg-de Veies equation,, Transaction of the American Mathematical Society, 348 (1996), 3643. doi: 10.1090/S0002-9947-96-01672-8. Google Scholar
[32]	D. L. Russell and B.-Y. Zhang, Controllability and stabilizability of the third-order linear dispersion equation on a periodic domain,, SIAM J. Control and Optimization, 31 (1993), 659. doi: 10.1137/0331030. Google Scholar
[33]	P. N. da Silva, Unique Continuation for the Kawahara Equation,, TEMA Tend. Mat. Apl. Comput., 8 (2007), 463. doi: 10.5540/tema.2007.08.03.0463. Google Scholar
[34]	M. Slemrod, A note on complete controllability and stabilizability for linear control systems in Hilbert space,, SIAM J. Control, 12 (1974), 500. doi: 10.1137/0312038. Google Scholar
[35]	T. Tao, Nonlinear Dispersive Equations, Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106.,, American Mathematical Society, (2006). Google Scholar
[36]	L. Tartar, Nonlinear interpolation and regularity,, J. Funct. Anal., 9 (1972), 469. Google Scholar
[37]	C. F. Vasconcellos and P. N. da Silva, Stabilization of the linear Kawahara equation with localized damping,, Asymptotic Analysis, 58 (2008), 229. Google Scholar
[38]	C. F. Vasconcellos and P. N. da Silva, Stabilization of the Kawahara equation with a localized damping,, ESAIM: COCV, 17 (2011), 102. doi: 10.1051/cocv/2009041. Google Scholar
[39]	H. Wang, S. B. Cui and D. G. Deng, Global existence of solutions for the Kawahara equation in Sobolev spaces of negative indices,, Acta Math. Sin., 23 (2007), 1435. doi: 10.1007/s10114-007-0959-z. Google Scholar
[40]	Y. Wu and Y. S. Li, The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity,, Math. Meth. Appl. Sci., 33 (2010), 1647. doi: 10.1002/mma.1273. Google Scholar
[41]	B.-Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation,, SIAM J. Cont. Optim., 37 (1999), 543. doi: 10.1137/S0363012997327501. Google Scholar
[42]	B.-Y. Zhang and X. Q. Zhao, Control and stabilization of the Kawahara equation on a periodic,, Communications in Information and Systems, 12 (2012), 77. doi: 10.4310/CIS.2012.v12.n1.a4. Google Scholar

[1]	Irena Lasiecka, Roberto Triggiani. Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument. Conference Publications, 2005, 2005 (Special) : 556-565. doi: 10.3934/proc.2005.2005.556
[2]	Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321
[3]	Peng Gao. Global exact controllability to the trajectories of the Kuramoto-Sivashinsky equation. Evolution Equations & Control Theory, 2019, 0 (0) : 1-11. doi: 10.3934/eect.2020002
[4]	Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088
[5]	Bo Su and Martin Burger. Global weak solutions of non-isothermal front propagation problem. Electronic Research Announcements, 2007, 13: 46-52.
[6]	Piermarco Cannarsa, Marco Mazzola, Carlo Sinestrari. Global propagation of singularities for time dependent Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4225-4239. doi: 10.3934/dcds.2015.35.4225
[7]	Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419
[8]	Emmanuel Hebey and Frederic Robert. Compactness and global estimates for the geometric Paneitz equation in high dimensions. Electronic Research Announcements, 2004, 10: 135-141.
[9]	Lorenzo Brasco, Marco Squassina, Yang Yang. Global compactness results for nonlocal problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 391-424. doi: 10.3934/dcdss.2018022
[10]	Hua Chen, Wei-Xi Li, Chao-Jiang Xu. Propagation of Gevrey regularity for solutions of Landau equations. Kinetic & Related Models, 2008, 1 (3) : 355-368. doi: 10.3934/krm.2008.1.355
[11]	Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[12]	Z.Y. Wu, H.W.J. Lee, F.S. Bai, L.S. Zhang. Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization. Journal of Industrial & Management Optimization, 2005, 1 (4) : 533-547. doi: 10.3934/jimo.2005.1.533
[13]	C.B. Muratov. A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 867-892. doi: 10.3934/dcdsb.2004.4.867
[14]	Alexander S. Bratus, Vladimir P. Posvyanskii, Artem S. Novozhilov. A note on the replicator equation with explicit space and global regulation. Mathematical Biosciences & Engineering, 2011, 8 (3) : 659-676. doi: 10.3934/mbe.2011.8.659
[15]	Jingqun Wang, Lixin Tian, Weiwei Guo. Global exact controllability and asympotic stabilization of the periodic two-component $\mu\rho$-Hunter-Saxton system. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2129-2148. doi: 10.3934/dcdss.2016088
[16]	Manil T. Mohan. On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations. Evolution Equations & Control Theory, 2019, 0 (0) : 1-39. doi: 10.3934/eect.2020007
[17]	Yong Zhou, Zhengguang Guo. Blow up and propagation speed of solutions to the DGH equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 657-670. doi: 10.3934/dcdsb.2009.12.657
[18]	Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.
[19]	P. M. Jordan, Louis Fishman. Phase space and path integral approach to wave propagation modeling. Conference Publications, 2001, 2001 (Special) : 199-210. doi: 10.3934/proc.2001.2001.199
[20]	Dongbing Zha, Yi Zhou. The lifespan for quasilinear wave equations with multiple propagation speeds in four space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1167-1186. doi: 10.3934/cpaa.2014.13.1167

2018 Impact Factor: 1.292

American Institute of Mathematical Sciences