The initial-boundary value problem for some nonlocal nonlinear wave type problem

Official URL: http://risc01.sabanciuniv.edu/record=b2325796 (Table of contents)

In this doctoral thesis we study some nonlinear nonlocal wave type equations. We consider three related problems which have connections with the study of some physical models in the theory of nonlocal elasticity. The nonlocality term is introduced via a convolution type integral operator L with kernel ; de ned on a bounded domain Ωc Rn as Lav(x) = IΩ a(x-y)v(y)dy: The rst problem is an initial value problem for the nonlocal nonlinear integro-partial di erential equation given by, t - △u= Lag(u) x ∈Ω, t >0 For this problem rstly local well-posedness is studied. Further analysis of the solution, like global existence and nite time blow-up, are investigated by various approaches such as assumptions of various smoothness and growth conditions on the nonlinearity. The second problem included in the thesis is the initial boundary value problem for some nonlocal nonlinear wave type equation given by the equation, utt - △u= Lag(u) x ∈Ω, t >0. (0.0.1) u=0 x ∈Ω, t>0. Local well-posedness of this problem is studied in proper Banach space settings. The third problem is variant form of the second problem and is given by the equation, utt - △u= Lau + g(u) x ∈Ω, t >0. (0.0.2) u=0 x ∈Ω, t>0. Unlike the previous problem, here the nonlocality and nonlinearity are expressed with separate terms. While we have imposed general assumptions on g(u) such as that it should be su ciently smooth and g(0) = 0, in the last two problems we have used power type nonlinear function of the form g(u) = ∣ u∣ p-1u, p > 1. The symmetry of the integral operator involved in the last problem enables us to de ne an explicit energy, which is a conserved quantity. For further analysis of solutions of the third problem, we have used the method of Nehari Manifold. Functionals like the total energy, the potential energy and the Nehari functional associated to the equation are de ned and the potential well depth is obtained in terms these functionals. The two subsets of the initial value space, namely the stable set and the unstable set that are invariant under the ow of the solution are obtained accordingly. Based on the initial energy and the sets where the initial data are located in, the blow-up or the global existence conditions for solutions is analysed.