
Previous Article
Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth
 DCDS Home
 This Issue

Next Article
Elliptic islands on the elliptical stadium
Eigenfunction expansion method and the longtime asymptotics for the damped Boussinesq equation
1.  Department of Mathematics, The University of Texas at Austin, Austin, TX 787121082, United States 
$u_{t t}  2b\Delta u_t = \alpha \Delta^2 u+ \Delta u + \beta\Delta(u^2)$
in a unit ball $B$. Homogeneous boundary conditions and small initial data are examined. The existence of mild globalintime solutions is established in the space $C^0([0,\infty), H^s_0(B)), s < 3/2$, and the solutions are constructed in the form of the expansion in the eigenfunctions of the Laplace operator in $B$. For $ 3/2 +\varepsilon \leq s <3/2$, where $\varepsilon > 0$ is small, the uniqueness is proved. The secondorder longtime asymptotics is calculated which is essentially nonlinear and shows the nonlinear mode multiplication.
[1] 
George J. Bautista, Ademir F. Pazoto. Decay of solutions for a dissipative higherorder Boussinesq system on a periodic domain. Communications on Pure & Applied Analysis, 2020, 19 (2) : 747769. doi: 10.3934/cpaa.2020035 
[2] 
Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higherorder setvalued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223255. doi: 10.3934/mcrf.2019012 
[3] 
JeanPaul Chehab, Pierre Garnier, Youcef Mammeri. Longtime behavior of solutions of a BBM equation with generalized damping. Discrete & Continuous Dynamical Systems  B, 2015, 20 (7) : 18971915. doi: 10.3934/dcdsb.2015.20.1897 
[4] 
Yihong Du, Yoshio Yamada. On the longtime limit of positive solutions to the degenerate logistic equation. Discrete & Continuous Dynamical Systems  A, 2009, 25 (1) : 123132. doi: 10.3934/dcds.2009.25.123 
[5] 
Min Chen, Olivier Goubet. Longtime asymptotic behavior of dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems  A, 2007, 17 (3) : 509528. doi: 10.3934/dcds.2007.17.509 
[6] 
Mamoru Okamoto. Asymptotic behavior of solutions to a higherorder KdVtype equation with critical nonlinearity. Evolution Equations & Control Theory, 2019, 8 (3) : 567601. doi: 10.3934/eect.2019027 
[7] 
Peter V. Gordon, Cyrill B. Muratov. Selfsimilarity and longtime behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks & Heterogeneous Media, 2012, 7 (4) : 767780. doi: 10.3934/nhm.2012.7.767 
[8] 
Belkacem SaidHouari. Longtime behavior of solutions of the generalized Kortewegde Vries equation. Discrete & Continuous Dynamical Systems  B, 2016, 21 (1) : 245252. doi: 10.3934/dcdsb.2016.21.245 
[9] 
Josef Diblík. Longtime behavior of positive solutions of a differential equation with statedependent delay. Discrete & Continuous Dynamical Systems  S, 2020, 13 (1) : 3146. doi: 10.3934/dcdss.2020002 
[10] 
Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higherorder μCamassaHolm equation. Discrete & Continuous Dynamical Systems  A, 2018, 38 (8) : 41634187. doi: 10.3934/dcds.2018181 
[11] 
David F. Parker. Higherorder shallow water equations and the CamassaHolm equation. Discrete & Continuous Dynamical Systems  B, 2007, 7 (3) : 629641. doi: 10.3934/dcdsb.2007.7.629 
[12] 
Min Zhu. On the higherorder bfamily equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems  A, 2014, 34 (7) : 30133024. doi: 10.3934/dcds.2014.34.3013 
[13] 
Robert Jankowski, Barbara Łupińska, Magdalena NockowskaRosiak, Ewa Schmeidel. Monotonic solutions of a higherorder neutral difference system. Discrete & Continuous Dynamical Systems  B, 2018, 23 (1) : 253261. doi: 10.3934/dcdsb.2018017 
[14] 
Aliang Xia, Jianfu Yang. Normalized solutions of higherorder Schrödinger equations. Discrete & Continuous Dynamical Systems  A, 2019, 39 (1) : 447462. doi: 10.3934/dcds.2019018 
[15] 
Min Chen, Olivier Goubet. Longtime asymptotic behavior of twodimensional dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems  S, 2009, 2 (1) : 3753. doi: 10.3934/dcdss.2009.2.37 
[16] 
Vladimir AnguloCastillo, Lucas C. F. Ferreira. Longtime solvability in Besov spaces for the inviscid 3DBoussinesqCoriolis equations. Discrete & Continuous Dynamical Systems  B, 2017, 22 (11) : 00. doi: 10.3934/dcdsb.2020112 
[17] 
Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higherorder wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems  S, 2017, 10 (5) : 11751185. doi: 10.3934/dcdss.2017064 
[18] 
Annalisa Iuorio, Stefano Melchionna. Longtime behavior of a nonlocal CahnHilliard equation with reaction. Discrete & Continuous Dynamical Systems  A, 2018, 38 (8) : 37653788. doi: 10.3934/dcds.2018163 
[19] 
Pelin G. Geredeli, Azer Khanmamedov. Longtime dynamics of the parabolic $p$Laplacian equation. Communications on Pure & Applied Analysis, 2013, 12 (2) : 735754. doi: 10.3934/cpaa.2013.12.735 
[20] 
Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic & Related Models, 2014, 7 (1) : 79108. doi: 10.3934/krm.2014.7.79 
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]