THE GLOBAL SOLUTION AND BLOWUP OF A EQUATION MODELED FROM THE WATER WAVE PROBLEM WITH CRITICAL GROWTH

doi:10.1007/s10473-025-0608-6

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (6): 2510-2535.doi: 10.1007/s10473-025-0608-6

THE GLOBAL SOLUTION AND BLOWUP OF A EQUATION MODELED FROM THE WATER WAVE PROBLEM WITH CRITICAL GROWTH

Zhong TAN^1,2, Yiying WANG^1,*

1. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China;
2. Shenzhen Research Institute of Xiamen University, Shenzhen 518057, China

收稿日期:2025-02-20 修回日期:2025-04-11 出版日期:2025-11-25 发布日期:2025-11-14

THE GLOBAL SOLUTION AND BLOWUP OF A EQUATION MODELED FROM THE WATER WAVE PROBLEM WITH CRITICAL GROWTH

Zhong TAN^1,2, Yiying WANG^1,*

1. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China;
2. Shenzhen Research Institute of Xiamen University, Shenzhen 518057, China

Received:2025-02-20 Revised:2025-04-11 Online:2025-11-25 Published:2025-11-14
Contact: ^*Yiying WANG, E-mail: elainemath@163.com
About author:Zhong TAN, E-mail: tan85@xmu.edu.cn
Supported by:
NSFC (12071391, 12231016) and the Guangdong Basic and Applied Basic Research Foundation (2022A1515010860).

摘要/Abstract

摘要： In this article, we study the water wave problem with critical growth. We mainly concern with the blowup and asymptotic estimates of the global solution. First, we prove the blow up and decay estimates of the solution with low-energy initial value. Next, we prove the regularity of the global solution with low-energy initial value. In the last part, we study the concentration phenomenon of the global solution no matter with low energy or not by the method of concentration compactness principle.

关键词: critical Sobolev exponent, blow up, regularity, long time asymptotic behaviour

Abstract: In this article, we study the water wave problem with critical growth. We mainly concern with the blowup and asymptotic estimates of the global solution. First, we prove the blow up and decay estimates of the solution with low-energy initial value. Next, we prove the regularity of the global solution with low-energy initial value. In the last part, we study the concentration phenomenon of the global solution no matter with low energy or not by the method of concentration compactness principle.

Key words: critical Sobolev exponent, blow up, regularity, long time asymptotic behaviour

中图分类号:

35B33

Zhong TAN, Yiying WANG. THE GLOBAL SOLUTION AND BLOWUP OF A EQUATION MODELED FROM THE WATER WAVE PROBLEM WITH CRITICAL GROWTH[J]. 数学物理学报(英文版), 2025, 45(6): 2510-2535.

Zhong TAN, Yiying WANG. THE GLOBAL SOLUTION AND BLOWUP OF A EQUATION MODELED FROM THE WATER WAVE PROBLEM WITH CRITICAL GROWTH[J]. Acta mathematica scientia,Series B, 2025, 45(6): 2510-2535.

参考文献

[1] Ambrosetti A, Malchiodi A.Nonlinear Analysis and Semilinear Elliptic Problems. Cambridge: Cambridge University Press, 2007
[2] Barrios B, Colorado E, de Pablo A, Sánchez U. On some critical problems for the fractional Laplacian operator. J Differential Equations,2012, 252(11): 6133-6162
[3] Brändle C, Colorado E, de Pablo A, Sánchez U. A concave-convex elliptic problem involving the fractional Laplacian. Proc Roy Soc Edinburgh Sect A,2013, 143(1): 39-71
[4] Brézis H, Lieb E. A relation between pointwise convergence of functions and convergence of functionals. Proc Amer Math Soc,1983, 88(3): 486-490
[5] Cabré X, Tan J G. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv Math,2010, 224(5): 2052-2093
[6] Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Comm Partial Differential Equations, 2007, 32(7-9): 1245-1260
[7] Capella A, Dávila J, Dupaigne L, Sire Y. Regularity of radial extremal solutions for some non-local semilinear equations. Comm Partial Differential Equations,2011, 36(8): 1353-1384
[8] Chen W X, Li Y, Ma P. The Fractional Laplacian.Hackensack, NJ: World Scientiffc Publishing, 2020
[9] Chen W X, Qi S J. Direct methods on fractional equations. Discrete Contin. Dyn Syst, 2019, 39(3): 1269-1310
[10] Chen W X, Zhu J Y. Indeffnite fractional elliptic problem and Liouville theorems. J Differential Equations, 2016, 260(5): 4758-4785
[11] Llave R, Valdinoci E. Symmetry for a Dirichlet-Neumann problem arising in water waves. Math Res Lett, 2009, 16(5): 909-918
[12] Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136(5): 521-573
[13] Dipierro S, Miraglio P, Valdinoci E.Symmetry results for the solutions of a partial differential equation arising in water waves//2018 MATRIX Annals. Cham: Springer, 2020: 229-248
[14] Du S Z. On partial regularity of the borderline solution of semilinear parabolic equation with critical growth. Adv Differential Equations, 2013, 18(1/2): 147-177
[15] Fang F, Tan Z. Heat ffow for Dirichlet-to-Neumann operator with critical growth. Adv Math, 2018, 328: 217-247
[16] Gilbarg D, Trudinger Neil S.Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 1977
[17] Ladyženskaja O A, Solonnikov V A, Ural ceva N N. Linear and Quasilinear Equations of Parabolic Type. Providence, RI: American Mathematical Society, 1968
[18] Lions P L. The concentration-compactness principle in the calculus of variations. The locally compact case. I. Ann Inst H Poincaré Anal Non Linéaire,1984, 1(2): 109-145
[19] Lions P L. The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann Inst H Poincaré Anal Non Linéaire,1984, 1(4): 223-283
[20] Lions P L. The concentration-compactness principle in the calculus of variations. The limit case. I. Rev Mat Iberoamericana, 1985, 1(1): 145-201
[21] Lions P L. The concentration-compactness principle in the calculus of variations. The limit case. II. Rev Mat Iberoamericana, 1985, 1(2): 45-121
[22] Miraglio P, Valdinoci E. Energy asymptotics of a Dirichlet to Neumann problem related to water waves. Nonlinearity, 2020, 33(11): 5997-6025
[23] Payne L E, Sattinger D H. Saddle points and instability of nonlinear hyperbolic equations. Israel J Math, 1975, 22(3/4): 273-303
[24] Struwe M. A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math Z, 1984, 187(4): 511-517
[25] Tan J G. The Brezis-Nirenberg type problem involving the square root of the Laplacian. Calc Var Partial Differential Equations, 2011, 42(1/2): 21-41
[26] Tan J G. Positive solutions for non local elliptic problems. Discrete Contin Dyn Syst, 2013, 33(2): 837-859
[27] Tan Z. Global solution and blowup of semilinear heat equation with critical Sobolev exponent. Comm Partial Differential Equations, 2001, 26(3/4): 717-741
[28] Weissler Fred B. Local existence and nonexistence for semilinear parabolic equations in $L^p$. Indiana Univ Math J, 1980, 29(1): 79-102
[29] Xie M H, Tan Z. The global solution and blowup of a spatiotemporal EIT problem with a dynamical boundary condition. Acta Math Sci, 2023, 43B(4): 1881-1914

THE GLOBAL SOLUTION AND BLOWUP OF A EQUATION MODELED FROM THE WATER WAVE PROBLEM WITH CRITICAL GROWTH

THE GLOBAL SOLUTION AND BLOWUP OF A EQUATION MODELED FROM THE WATER WAVE PROBLEM WITH CRITICAL GROWTH

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	Huanyuan LI, Junhao ZHANG, Huijiang ZHAO, Jialing ZHU. ONE-DIMENSIONAL COMPRESSIBLE RADIATION HYDRODYNAMICS MODEL WITH DENSITY-DEPENDENT VISCOSITY AND THERMAL CONDUCTIVITY[J]. 数学物理学报(英文版), 2025, 45(6): 2421-2446.
[2]	Yifei WU, Zhibo YANG, Qi ZHOU. WELL-POSEDNESS OF THE DISCRETE NONLINEAR SCHRÖDINGER EQUATIONS AND THE KLEIN-GORDON EQUATIONS[J]. 数学物理学报(英文版), 2025, 45(6): 2447-2477.
[3]	Siyan GUO, Jiangbo HAN, Runzhang XU. WELL-POSEDNESS OF 2-D HYPERBOLIC VISCOUS CAHN-HILLIARD EQUATION[J]. 数学物理学报(英文版), 2025, 45(4): 1438-1470.
[4]	Di WU. ${C^{1, 1}}$ REGULARITY FOR SOLUTIONS TO THE DEGENERATE DUAL ORLICZ-MINKOWSKI PROBLEM[J]. 数学物理学报(英文版), 2025, 45(2): 327-337.
[5]	Lihua TAN, Yingzhe FAN. LOW-REGULARITY SOLUTIONS TO FOKKER-PLANCK-TYPE SYSTEMS IN THE WHOLE SPACE[J]. 数学物理学报(英文版), 2024, 44(6): 2361-2390.
[6]	Jun Chen, Xuemei Deng. ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH DISCONTINUOUS COEFFICIENTS IN DOMAINS WITH CORNERS^*[J]. 数学物理学报(英文版), 2024, 44(5): 1903-1915.
[7]	Cui NING, Chenxi HAO, Yaohong WANG. A LOW-REGULARITY FOURIER INTEGRATOR FOR THE DAVEY-STEWARTSON II SYSTEM WITH ALMOST MASS CONSERVATION[J]. 数学物理学报(英文版), 2024, 44(4): 1536-1549.
[8]	Lvqiao LIU, Juan ZENG. THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARD POTENTIAL[J]. 数学物理学报(英文版), 2024, 44(2): 455-473.
[9]	Changlin XIANG, Gaofeng ZHENG. SHARP MORREY REGULARITY THEORY FOR A FOURTH ORDER GEOMETRICAL EQUATION[J]. 数学物理学报(英文版), 2024, 44(2): 420-430.
[10]	Yu DONG, Yaofang HUANG, Li LI, Qing LU. THE REGULARITY CRITERIA OF WEAK SOLUTIONS TO 3D AXISYMMETRIC INCOMPRESSIBLE BOUSSINESQ EQUATIONS^*[J]. 数学物理学报(英文版), 2023, 43(6): 2387-2397.
[11]	Renhai WANG, Boling GUO, Daiwen HUANG. THEORETICAL RESULTS ON THE EXISTENCE, REGULARITY AND ASYMPTOTIC STABILITY OF ENHANCED PULLBACK ATTRACTORS: APPLICATIONS TO 3D PRIMITIVE EQUATIONS^*[J]. 数学物理学报(英文版), 2023, 43(6): 2493-2518.
[12]	Khalid BOUABID, Rachid ECHARGHAOUI, Mohssine EL MANSOUR. TWO DISJOINT AND INFINITE SETS OF SOLUTIONS FOR AN ELLIPTIC EQUATION INVOLVING CRITICAL HARDY-SOBOLEV EXPONENTS^*[J]. 数学物理学报(英文版), 2023, 43(5): 2061-2074.
[13]	Haitao WANG, Xiongtao ZHANG. THE REGULARITY AND UNIQUENESS OF A GLOBAL SOLUTION TO THE ISENTROPIC NAVIER-STOKES EQUATION WITH ROUGH INITIAL DATA^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1675-1716.
[14]	Bo Chen, Zhengmao Chen, Junhui Xie. PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION^*[J]. 数学物理学报(英文版), 2023, 43(3): 1161-1174.
[15]	Hongxia Lin, Sen Liu, Heng Zhang, Ru Bai. GLOBAL REGULARITY OF 2D INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID EQUATIONS WITH PARTIAL VISCOSITY^*[J]. 数学物理学报(英文版), 2023, 43(3): 1275-1300.