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.
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
.
DOI: 10.1007/s10473-025-0608-6
[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
