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Nonlinear second order equations with applications to partial differential equations. (English) Zbl 0572.34004
In this paper we study the Cauchy problem for the abstract second order (in time) semilinear differential equation \(u''(t)+Au'(t)+Bu(t)=f(t,u(t))\) where A and B are linear operators in a Banach space. Then we use the abstract results that we obtained together with energy estimates and the center manifold theorem to study in concrete cases, global existence, stability and bifurcation of solutions of certain parabolic and hyperbolic equations.

MSC:
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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[1] Berger, M, On von karmán’s equations and the buckling of a thin elastic plate. I. the clamped plate, Comm. pure appl. math., 20, 687-719, (1967) · Zbl 0162.56405
[2] Chadam, J; Chadam, J, Asymptotic for u = m2u + G(x, t, u, ux, ut) II, Ann. scuola norm. sup. Pisa, Ann. scuola norm. sup. Pisa, 26, 65-95, (1972)
[3] Dunford, N; Schwartz, J.J, Linear operators, (1963), Interscience New York, Part III
[4] Henry, D, Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001
[5] Holmes, P; Marsden, J, Bifurcation to divergence and flutter in flow induced oscillations: an infinite dimensional analysis, Automatica, 14, 367-384, (1978) · Zbl 0385.93028
[6] Kato, T, Perturbation theory for linear operators, (1966), Springer-Verlag New York · Zbl 0148.12601
[7] Lions, J.L; Magenes, E, ()
[8] Ponce, G, Long time stability of solutions of nonlinear evolution equations, ()
[9] Reed, M, Abstract non-linear wave equations, () · Zbl 0317.35002
[10] Sandefur, J, Existence and uniqueness of solutions of second order nonlinear differential equations, SIAM J. math. anal., 14, 477-487, (1983) · Zbl 0513.34069
[11] \scD. Schaeffer and M. Golubitsky, Boundary conditions and mode jumping in the buckling of a rectangular plate, to appear. · Zbl 0414.73036
[12] Showalter, R, Hilbert space methods for partial differential equations, (1977), Pitman San Francisco · Zbl 0364.35001
[13] Strauss, W, Decay and asymptotic for u = f(u), J. funct. anal., 2, 409-457, (1968) · Zbl 0182.13602
[14] Webb, G.F, Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. math., 32, 631-643, (1980) · Zbl 0414.35046
[15] Webb, G.F, A bifurcation problem for a nonlinear hyperbolic partial differential equation, SIAM J. math. anal., 10, 922-932, (1979) · Zbl 0424.35051
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