This thesis is concerned with the semilinear parabolic cauchy problem
$$u_t-u_{xx}=f(u),qquad u(cdot,0)=u_0,quad u(0)=u(1)=0.$$
This problem is the simplest model of an heat or reaction-diffusion equation. From a mathematical point of view this problem induces a semiflow $phi$ on the state space $H^1_0([0,1])$. We are interested in the dynamical properties of $phi$.
In the case of superlinear growth of $f$ there are infinitely many equilibrium solutions, and for that case there seem to be no general results on connecting orbits of flow equivalence. The numerous results on this and similar superlinear problems, e.g. by Marek Fila, Hiroshi Matano, Peter Polacik, Pavol Quittner and others, are mainly concerned with blow-up solutions. Not much is known about global, bounded solutions. In this work we are able to describe exactly which equilibria are connected by heteroclinic orbits, and which are not. We are able to prove structural stability of certain finite dimensional invariant sets $A_{n,infty}$ for a subclass of superlinear functions (containing the model case $f(u)=u|u|^p-lambda u$). These sets also contain blow-up trajectories.
We obtain our results the following way: A superlinear function $f$ is modified outside a compact interval in such a way, that we obtain a dissipative semiflow. As the state space $H^1_0([0,1])$ is compactly embedded into $cont^0([0,1])$, the modified flow coincides with the original flow $phi$ on a neighborhood of 0. Increasing the interval on which $f$ remains unchanged, this neighborhood of 0 increases accordingly. In this way we can transfer results from the dissipative case to $phi$.
Capter 3 is the technical core of this thesis. We make sure that the technical problems in modifying $phi$ are solved. The modified function will be made constant outside a constant interval. Thus all growth conditions for the existence of a global attractor are fulfilled. The equilibrium solutions are second-order boundary value problems, so we can work with phase plane analysis and shooting curve techniques.
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