| This dissertation is concerned with the existence of branches.;equations of the form (*) u(,tt) - (DELTA)u + (alpha)u(,t) - mu + F(x,t;u,Du,D('2)u) = 0.;for (x,t) (ELEM) (OMEGA) x {0,(tau)}, (OMEGA) (L-HOOK EQ) R('n) bounded Du = ,;D('2)u = , u((.),t) (VBAR)(,(PAR-DIFF)(OMEGA)) = 0, u(x,t+ (tau)) = u(x,t) and.;of periodic solutions of genuinely nonlinear dissipative wave.;F(x,t,(.),(.),(.)) is smooth in its arguments, satisfying the conditions F(x,t;0,0,0) = dF(x,t;0,0,0) = 0. Considering m as a parameter, and taking the Frechet derivative of (*) at the zero solution, the linearized operator has nontrivial kernel and corange for m = (lamda)(,1), where -(lamda)(,k) are eigenvalues of the Dirichlet problem for (OMEGA). This suggests a bifurcation theoretic approach to an existence theory. However, best estimates on solutions of the linearized equation are of the form.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;where Lv denotes v(,tt) - (DELTA)v + (alpha)v(,t). The extra time derivative appearing on the right hand side is the manifestation of loss of derivatives for solutions of hyperbolic equations. These estimates are not sufficient to apply the 'soft' implicit function theorem to the first bifurcation equation of the usual Lyapounov-Schmidt decomposition. In this paper the Nash-Moser technique is employed instead, overcoming the loss of derivatives, and reducing the problem to a simple finite dimensional problem.;Theorem. A family of nontrivial smooth solutions (u,(x,t;(sigma)), m((sigma))).;(ELEM) H('2) x (//R) of the equation (*) bifurcates from m(0) = -(lamda)(,1). These solutions are of the form u = (sigma)((phi)(,1) + w((sigma))), where w (PERP) (phi)(,1). Solutions are smooth in (sigma), and intersect transversally the branch of trivial solutions u(x,t) (TBOND) 0.;Most of the work in this paper is to obtain existence and regularity estimates for solutions of the projected equation (*) linearized around small but nonzero approximate solutions.;Theorem. Denote.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;Suppose that g and a(,ij) are (tau)-periodic in time, (VBAR)(VBAR)a(,ij)(VBAR)(VBAR)(,2) + (VBAR)m-(lamda)(,1)(VBAR) < (delta), and.;P represents orthogonal projection onto (phi)(,1)('(PERP)). Then if g (PERP) (phi)(,1) there.;is a unique solution v (PERP) (phi)(,1) such that.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;In this problem two things are hampering the invertibility of the nonlinear operator in a neighborhood of the zero solution; the range of the linearized operator is not closed, and the range had nontrivial codimension. The first of these difficulties is often characterized by a loss of derivatives for solutions of the linearized equations. It has been possible to overcome this in many cases using Nash-Moser type iteration schemes. The second of these difficulties is treated classically with the Lyapounov-Schmidt decomposition. Of main interest in this dissertation is that both situations occur, and a combination of the two techniques can be used to prove the existence of smooth solutions and to describe the solution sets. |
