Abstract |
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In qualitative theory of differential equations, an important role is played by special classes of solutions, like periodic solutions or solutions to some boundary value problems. When a system of ordinary differential equations has equilibria, i.e. constant solutions, whose stability properties are known, it is significant to search for connections between them by trajectories of solutions of the given system. These are called
homoclinic or heteroclinic, according to whether they describe a loop based at one single equilibrium or they "start" and "end" at two distinct equilibria. This thesis is devoted to the study of heteroclinic solutions for a specific class of ordinary differential
equations related to the Extended Fisher-Kolmogorov equation and the Swift-Hohenberg equation. These are semilinear fourth order
bi-stable evolution equations which appear as mathematical models for problems arising in Mechanics, Chemistry and Biology. For such
equations, the set of bounded stationary solutions is of great interest. These solve an autonomous fourth order equation. In this
thesis, we focus on such equations having a variational structure. In that case, the solutions are critical points of an associated
action functional defined in convenient functional spaces. We then look for heteroclinic solutions as minimizers of the action
functional. Our main contributions concern existence and multiplicity results of such global and local minimizers in the case where the functional is defined from sign changing Lagrangians. The underlying idea is to impose conditions which imply a lower bound on the action over all admissible functions. We then combine classical arguments of the Calculus of Variations with careful estimates on minimizing sequences to prove the existence of a minimum. |