Moduli for heteroclinic connections involving saddle-foci and periodic solutions
Asymptotics in shallow water waves
July 2015, 35(7): 3133-3154. doi: 10.3934/dcds.2015.35.3133
Variational analysis of semilinear plate equation with free boundary conditions
Andrzej Nowakowski 1,
University of Lodz, Faculty of Math & Computer Sciences, Banacha 22, 90-238 Lodz
Received September 2013 Revised November 2014 Published January 2015
We present a variational analysis for the semilinear equation of the vibrating plate $x_{t t}(t,y)+\Delta ^{2}x(t,y)+l(t,y,x(t,y))=0$ in a bounded domain and a free nonlinear boundary condition $\Delta x(t,y)=H_{x}(t,y,x(t,y))-Q_{x}(t,y,x(t,y))$. In this context new dual variational methods are developed. Applying a variational approach we discuss a stability of solutions with respect to initial conditions.
Keywords: Dual variational method, nonlinear force, free boundary conditions, nonlinerity on the boundary., semi-linear plate equation.
Mathematics Subject Classification: Primary: 35L05; Secondary: 35L2.
Citation: Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 3133-3154. doi: 10.3934/dcds.2015.35.3133
C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calc. Var. Partial Differential Equations, 34 (2009), 377-411. doi: 10.1007/s00526-008-0188-z. Google Scholar
V. Barbu, I. Lasiecka and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Trans. Amer. Math. Soc., 357 (2005), 2571-2611. doi: 10.1090/S0002-9947-05-03880-8. Google Scholar
L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683. doi: 10.1016/j.jde.2010.03.009. Google Scholar
L. Bociu, M. Rammaha and D. Toundykov, On a wave equation with supercritical interior and boundary sources and damping terms, Mathematische Nachrichten, 284 (2011), 2032-2064. doi: 10.1002/mana.200910182. Google Scholar
L. Bociu, M. Rammaha and D. Toundykov, Wave equations with super-critical interior and boundary nonlinearities, Mathematics and Computers in Simulation, 82 (2012), 1017-1029. doi: 10.1016/j.matcom.2011.04.006. Google Scholar
V. L. Carbone, M. J. D. Nascimento, K. Schiabel-Silva and R. P. Silva, Pullback attractors for a singularly nonautonomous plate equation, Electronic Journal of Differential Equations, (2011), 1-13. Google Scholar
M. M. Cavalcanti, V. N. D. Cavalcanti, J. S. P. Filho and J. A. Soriano, Existence and uniform decay of solutions of a parabolic-hyperbolic equation with nonlinear boundary damping and boundary source term, Comm. Anal. Geom., 10 (2002), 451-466. Google Scholar
M. M. Cavalcanti, V. N. D. Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158. doi: 10.1016/j.jde.2004.04.011. Google Scholar
V. V. Chepyzhov, M. I. Vishik and S. V. Zelik, Strong trajectory attractors for dissipative Euler equations, J. Math. Pures Appl., 96 (2011), 395-407. doi: 10.1016/j.matpur.2011.04.007. Google Scholar
I. Chueshov, Convergence of solutions of von Karman evolution equations to equilibria, Appl. Anal., 91 (2012), 1699-1715. doi: 10.1080/00036811.2011.577930. Google Scholar
I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary-interior damping, J. Differential Equations, 233 (2007), 42-86. doi: 10.1016/j.jde.2006.09.019. Google Scholar
I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with non-linear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912. Google Scholar
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, 2010. doi: 10.1007/978-0-387-87712-9. Google Scholar
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies Math. Appl. I, Noth-Holland Publ. Co., Amsterdam-Oxford, 1976. Google Scholar
A. Favini, I. Lasiecka, M. A. Horn and D. Tataru, Global existence, uniqueness and regularity of solutions to a von Karman system with nonlinear boundary dissipation, Differential Integral Equations, 9 (1996), 267-294. Google Scholar
J. R. Kang, Global attractor for an extensible beam equation with localized nonlinear damping and linear memory, Math. Methods Appl. Sci., 34 (2011), 1430-1439. doi: 10.1002/mma.1450. Google Scholar
J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, 1989. doi: 10.1137/1.9781611970821. Google Scholar
I. Lasiecka, Mathematical Control Theory of Coupled PDE's, CMBS-NSF Lecture Notes, SIAM, 2002. doi: 10.1137/1.9780898717099. Google Scholar
I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. Google Scholar
H. A. Levine, Instability and nonexistence of global solutions of nonlinear wave equations of the form $Pu_{t t}$ = $Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21. Google Scholar
J.-L. Lions, Quelques Méthodes de Résolution de Problémes aux Limites non Linéaires, Dunod, Paris, 1969. Google Scholar
A. Nowakowski and D. O'Regan, Periodic solutions for forced vibrations of beam equation with nonmonotone nonlinearities,, Multidimensional case, (). Google Scholar
A. Nowakowski, Solvability and stability of a semilinear wave equation with nonlinear boundary conditions, Nonlinear Anal., 73 (2010), 1495-1514. doi: 10.1016/j.na.2010.04.035. Google Scholar
R. Parreira da Silva, Lower semicontinuity of pullback attractors for a singularly nonautonomous plate equation, Electronic Journal of Differential Equations, (2012), 1-8. Google Scholar
M. A. Rammaha, The influence of damping and source terms on solutions of nonlinear wave equations, Bol. Soc. Parana. Mat., 25 (2007), 77-90. doi: 10.5269/bspm.v25i1-2.7427. Google Scholar
H. Tsutsumi, On solutions of semilinear differential equations in a Hilbert space, Math. Japonicea, 17 (1972), 173-193. Google Scholar
E. Vitillaro, A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395. doi: 10.1017/S0017089502030045. Google Scholar
B. Yordanov and Q. S. Zhang, Finite-time blowup for wave equations with a potential, SIAM J. Math. Anal., 36 (2005), 1426-1433. doi: 10.1137/S0036141004440198. Google Scholar
Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083
Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations & Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016
Bruno Fornet, O. Guès. Penalization approach to semi-linear symmetric hyperbolic problems with dissipative boundary conditions. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 827-845. doi: 10.3934/dcds.2009.23.827
Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761
Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493
Út V. Lê. Contraction-Galerkin method for a semi-linear wave equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 141-160. doi: 10.3934/cpaa.2010.9.141
Andrzej Nowakowski. Variational approach to stability of semilinear wave equation with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2603-2616. doi: 10.3934/dcdsb.2014.19.2603
Scott W. Hansen, Oleg Yu Imanuvilov. Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions. Mathematical Control & Related Fields, 2011, 1 (2) : 189-230. doi: 10.3934/mcrf.2011.1.189
Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037
Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete & Continuous Dynamical Systems, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072
Muhammad I. Mustafa. Viscoelastic plate equation with boundary feedback. Evolution Equations & Control Theory, 2017, 6 (2) : 261-276. doi: 10.3934/eect.2017014
Baowei Feng, Abdelaziz Soufyane. New general decay results for a von Karman plate equation with memory-type boundary conditions. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1757-1774. doi: 10.3934/dcds.2020092
Irena Lasiecka, Roberto Triggiani. A sharp trace result on a thermo-elastic plate equation with coupled hinged/Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 585-598. doi: 10.3934/dcds.1999.5.585
Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001
Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021028
Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051
Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems & Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51
Meng Qu, Jiayan Wu, Ting Zhang. Sliding method for the semi-linear elliptic equations involving the uniformly elliptic nonlocal operators. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2285-2300. doi: 10.3934/dcds.2020362
Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141
Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025
Andrzej Nowakowski