The theory of invariant manifolds and foliations providesindispensable tools for the study of dynamics of nonlinear systemsin finite or infinite dimensional space. As is the case here, invariant manifoldscan be used to capture complex dynamics and the long term behaviorof solutions and to reduce high dimensional problems to the analysisof lower dimensional structures. Invariant manifolds with invariantfoliations provide a coordinate system in which systems ofdifferential equations may be decoupled and normal forms derived.These play an important role in the study of structural stability ofdynamical systems or, when a degeneracy occurs, in understanding thenature of bifurcations. This thesis is devoted to the study of the construction of invariant manifolds of solutions with certain spatial structures to some nonlinear parabolic partial differential equations. I approach these problems in two steps: the first step is to construct a manifold of states that is approximately invariant, the second step is to show the existence of a truly invariant manifold of these states near the approximately invariant one, and to determine the dynamics on this manifold. Since this approach may be applied to many different systems, I also develop it in an abstract or general way, extending earlier results of \cite{BLZ1}.My thesis consists of two projects, in the first project, we consider the two-dimensional mass-conserving Allen-Cahn Equation,\begin{equation}\begin{cases}\phi_{t}(x,t)= \epi^{2}\Delta\phi(x,t)-f(\phi(x,t))+\fint_{\Omega}f(\phi(\cdot,t)), \;&x\in\Omega, \;t>0,\\\partial_{n}\phi(x,t)=0, &x\in\partial\Omega, \;t>0,\end{cases}\end{equation}where $\Omega\subset\mathbb{R}^{2}$ is a fixed bounded domain with smooth boundary $\partial\Omega$, $\partial_{n}$ is the exterior normal derivative to $\partial\Omega$, and $\fint_{\Omega}=\frac{1}{|\Omega|}\int_{\Omega}$ means the average over $\Omega$. Here $f$ is the derivative of a double well potential $W$. We assume the following conditions for $f$:\begin{equation}f(\pm1)=0, \;f'(\pm 1)>0, \;\int_{-1}^{s}f=\int_{1}^{s}f>0 \;\text{for all} \;s\in (-1,1).\end{equation}We establish the existence of a global invariant manifold of bubble states for this equation and give the dynamics for the center of the bubble.In the second project, we consider the existence, in forward and backward time, of dynamical interior multi-spike states driven by the nonlinear Cahn-Hilliard equation:\begin{equation}\begin{cases}u_{t} = -\Delta(\epi^{2} \Delta u - f(u)) \hspace{20mm} &\text{in}\;\;\Omega\times (0, \infty),\\\partial_{n} \Delta u = \partial_{n} u= 0 &\text{on}\;\; \partial\Omega \times (0, \infty), \end{cases}\end{equation}where $\Omega\subset\mathbb{R}^{n}$ is a fixed bounded domain with smooth boundary $\partial\Omega$ and $f$ is the derivative of a double well potential $W$, that is , $xf(x)>0$ for $|x|$ large enough and $f$ has two zeros $a$ and $b$ such that $f'(a), f'(b)>0$. We construct invariant manifolds of interior multi-spike states for the nonlinear Cahn-Hilliard equation and then investigate the dynamics on it. An equation for the motion of the spikes is also derived. It turns out that the dynamics of interior spikes has a global character and each spike interacts with all the others and with the boundary. Moreover, we show that the speed of the interior spikes is super slow, which indicates the long time existence of dynamical multi-spike solutions in both positive and negative time.
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