ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM*

doi:10.1007/s10473-024-0508-1

Abstract

Abstract: This paper is devoted to the Cauchy problem for the generalized damped Boussinesq equation with a nonlinear source term in the natural energy space. With the help of linear time-space estimates, we establish the local existence and uniqueness of solutions by means of the contraction mapping principle. The global existence and blow-up of the solutions at both subcritical and critical initial energy levels are obtained. Moreover, we construct the sufficient conditions of finite time blow-up of the solutions with arbitrary positive initial energy.

Key words: damped Boussinesq equation, Cauchy problem, global solutions, blow-up

CLC Number:

35L30

Xiao SU, Shubin WANG. ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM^*[J].Acta mathematica scientia,Series B, 2024, 44(5): 1766-1786.

Trendmd

References

[1] Boussinesq J. Theorie des ondes et de remous qui se propagent le long d'un canal rectangulaire horizontal en communiquant au liqude contene dans ce cannal des vitesses sensiblement pareilles de la surface au foud. J Math Pures Appl, 1872, 217: 55-108
[2] Angulo J, Scialom M. Improved blow-up of solutions of a generalized Boussinesq equation. Comput Appl Math, 1999, 18: 333-341
[3] Bona J L, Sachs R L. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Comm Math Phys, 1988, 118: 15-29
[4] Farah L G. Local solutions in Sobolev spaces with negative indices for the ``good'' Boussinesq equation. Commun Part Diff Eq, 2009, 34: 52-73
[5] Hu Q Y, Zhang H W, Liu G W. Global existence and exponential growth of solution for the logarithmic Boussinesq-type equation. J Math Anal Appl, 2016, 436: 990-1001
[6] Liu Y. Instability and blow-up of solutions to a generalized Boussinesq equation. SIAM J Math Anal, 1995, 26: 1527-1546
[7] Liu Y. Decay and scattering of small solutions of a generalized Boussinesq equation. J Funct Anal, 1997, 147: 51-68
[8] Chen J, Guo B L, Shao J. Well-posedness and scattering for the generalized Boussinesq equation. SIAM J Math Anal, 2023, 55: 133-161
[9] Guenther R B, Lee J W.Partial Differential Equations of Mathematical Physics and Integral Equations. New Jersey: Prentice Hall, 1988
[10] Varlamov V. On the Cauchy problem for the damped Boussinesq equation. Differ Integral Equ, 1996, 9: 619-634
[11] Varlamov V. On spatially periodic solutions of the damped Boussinesq equation. Differ Integral Equa, 1997, 10: 1197-1211
[12] Varlamov V. Existence and uniqueness of a solution to the Cauchy problem for the damped Boussinesq equation. Math Meth Appli Sci, 1996, 19: 639-649
[13] Xu R Z, Luo Y, Shen J, Huang S. Global existence and blow up for damped generalized Boussinesq equation. Acta Math Appl Sin-Engl, 2017, 33: 251-262
[14] Liu M, Wang W K. Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Commun Pure Appl Anal, 2013, 13: 1203-1222
[15] Chen W H, Dao T A. The Cauchy problem for the nonlinear viscous Boussinesq equation in the framework. J Differ Equations, 2022, 320: 558-597
[16] Liu G W, Wang W K. Inviscid limit for the damped Boussinesq equation. J Differ Equations, 2019, 267: 5521-5542
[17] Liu G W, Wang W K. Decay estimates for a dissipative-dispersive linear semigroup and application to the viscous Boussinesq equation. J Funct Anal, 2020, 278: 108413
[18] Su X, Wang S B. Optimal decay rates and small global solutions to the dissipative Boussinesq equation. Math Method Appl Sci, 2020, 43: 174-198
[19] Wang Y. Asymptotic decay estimate of solutions to the generalized damped Bq equation. J Inequal Appl2013, 2013: Art 323
[20] Christov C I, Velarde M G. Evolution and interactions of solitary wave (solitons) in nonlinear dissipative systems. Phys Scripta, 1994, T55: 101-106
[21] Ding H, Zhou J. Well-posedness of solutions for the dissipative Boussinesq equation with logarithmic nonlinearity. Nonlinear Anal-RWA, 2022, 67: Art 103587
[22] Wang S B, Su X. Global existence and nonexistence of the initial-boundary value problem for the dissipative Boussinesq equation. Nonlinear Anal, 2016, 134: 164-188
[23] Wang S B, Su X. The Cauchy problem for the dissipative Boussinesq equation. Nonlinear Anal-RWA, 2019, 45: 116-141
[24] Ginibre J, Velo G. The global Caucby problem for the non-linear Klein-Gordon equation. Math Z, 1985, 189: 487-505
[25] Wang S B, Xu G X. The Cauchy problem for the Rosenau equation. Nonlinear Anal, 2009, 71: 456-466
[26] Xu R Z, Liu Y C. Asymptotic behavior of solutions for initial-boumdary value problems for strongly damped nonlinear wave equations. Nonlinear Anal, 2008, 69: 2492-2495
[27] Xu R Z, Liu Y C. Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations. J Math Anal Appl, 2009, 359: 739-751
[28] Chen H, Liu G W. Global existence, uniform decay and exponential growth for a class of semi-linear wave equation with strong damping. Acta Math Sci, 2013, 33B: 41-58
[29] Sattinger D H. On global solution of nonlinear hyperbolic equations. Arch Rat Mech Anal, 1968, 30: 148-172
[30] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. ISR J Math, 1975, 22: 273-303
[31] Lian W, Rădulescu V D, Xu R Z, et al. Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations. Adv Calc Var, 2021, 14: 589-611
[32] Adams R A. Sobolev Spaces.New York: Academic Press, 1975
[33] Li Tatsien, Zhou Y. Nonlinear Wave Equations.Berlin: Springer-Verlag, 2017
[34] Koch H, Lasiecka I. Hadamard wellposedness of weak solutions in nonlinear dynamic elasticity-full Von Karman systems// Lorenzi A, Ruf B, eds. Evolution Equations, Semigroups and Functional Analysis 50. Switerland: Birkhäuser Verlag Basel, 2002: 197-211
[35] Levine H A. Some additional remarks on the nonexistence of global solutions to nonlinear equations. SIAM J Math Anal, 1974, 5: 138-146
[36] Levine H A. Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+f(u)$. Trans Amer Math Soc, 1974, 192: 1-21

ON THE CAUCHY PROBLEM FOR THE GENERALIZED BOUSSINESQ EQUATION WITH A DAMPED TERM^*

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Yiting WU. BLOW-UP CONDITIONS FOR A SEMILINEAR PARABOLIC SYSTEM ON LOCALLY FINITE GRAPHS [J]. Acta mathematica scientia,Series B, 2024, 44(2): 609-631.
[2]	Shijin Ding, Yinghua Li, Yu Wang. GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY^* [J]. Acta mathematica scientia,Series B, 2024, 44(1): 195-214.
[3]	Qianqian BAI, Xiaoguang LI, Li ZHANG. BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗ [J]. Acta mathematica scientia,Series B, 2023, 43(4): 1841-1864.
[4]	Qianqian BAI, Xiaoguang LI, Li ZHANG. BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR ÖDINGER EQUATIONS IN THE RADIAL CASE^∗ [J]. Acta mathematica scientia,Series B, 2023, 43(4): 1852-1864.
[5]	Xuemei Zhang, Shikun Kan. SUFFICIENT AND NECESSARY CONDITIONS ON THE EXISTENCE AND ESTIMATES OF BOUNDARY BLOW-UP SOLUTIONS FOR SINGULAR p-LAPLACIAN EQUATIONS^* [J]. Acta mathematica scientia,Series B, 2023, 43(3): 1175-1194.
[6]	Mingxuan Zhu, Zaihong Jiang. THE CAUCHY PROBLEM FOR THE CAMASSA-HOLM-NOVIKOV EQUATION^* [J]. Acta mathematica scientia,Series B, 2023, 43(2): 736-750.
[7]	Changxing MIAO, Junyong ZHANG, Jiqiang ZHENG. A NONLINEAR SCHRÖDINGER EQUATION WITH COULOMB POTENTIAL [J]. Acta mathematica scientia,Series B, 2022, 42(6): 2230-2256.
[8]	Chao Jiang, Zuhan Liu, Ling Zhou. BLOW-UP IN A FRACTIONAL LAPLACIAN MUTUALISTIC MODEL WITH NEUMANN BOUNDARY CONDITIONS [J]. Acta mathematica scientia,Series B, 2022, 42(5): 1809-1816.
[9]	Guotao WANG, Zedong YANG, Jiafa XU, Lihong ZHANG. THE EXISTENCE AND BLOW-UP OF THE RADIAL SOLUTIONS OF A ${(k_{1},k_{2})}$-HESSIAN SYSTEM INVOLVING A NONLINEAR OPERATOR AND GRADIENT [J]. Acta mathematica scientia,Series B, 2022, 42(4): 1414-1426.
[10]	Ning-An LAI, Wei XIANG, Yi ZHOU. GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW [J]. Acta mathematica scientia,Series B, 2022, 42(3): 887-902.
[11]	Banghe LI. UNDERSTANDING SCHUBERT'S BOOK (II) [J]. Acta mathematica scientia,Series B, 2022, 42(1): 1-48.
[12]	Jianwei DONG, Manwai YUEN. SOME SPECIAL SELF-SIMILAR SOLUTIONS FOR A MODEL OF INVISCID LIQUID-GAS TWO-PHASE FLOW [J]. Acta mathematica scientia,Series B, 2021, 41(1): 114-126.
[13]	Hua CHEN, Nian LIU. ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS [J]. Acta mathematica scientia,Series B, 2021, 41(1): 257-282.
[14]	Yongkai LIAO. REMARKS ON THE CAUCHY PROBLEM OF THE ONE-DIMENSIONAL VISCOUS RADIATIVE AND REACTIVE GAS [J]. Acta mathematica scientia,Series B, 2020, 40(4): 1020-1034.
[15]	Zongguang LI, Rui LIU. BLOW-UP SOLUTIONS FOR A CASE OF b-FAMILY EQUATIONS [J]. Acta mathematica scientia,Series B, 2020, 40(4): 910-920.