Towards Nečas Well-Posedness for Heat and Wave Equations

In this thesis, we analyze the behavior of two important time-dependent PDEs: heat and wave equations. Our aim is to establish the well-posedness of these problems according to the Nečas Theorem, that is, to determine whether an isomorphism exists between the data space and the solution space. Our starting point are the classical results obtained by the Faedo-Galerkin technique: we wonder if those results are sharp in the above sense. In the variational formulation of the initial value problem for the heat equation, the classical result indeed establishes such an isomorphism by choosing the source term in the dual space of the test function space. For the wave equation, on the other hand, the classical result does not establish an isomorphism because it requires the source term to be overly regular, and a naive attempt to relax this regularity condition fails due to the phenomenon of resonance. We propose an analysis based first on the Laplacian eigenfunction expansion in space, which will allow us to reduce the PDE to a system of one-dimensional ODEs, and subsequently through the Fourier series in time, allowing us to expand the source term of each ODE into its fundamental harmonics. With this approach, we develop resonance-aware norms for the source term in each ODE assigning different weights to each frequency to control the resonant and non-resonant terms, constituting a first step towards the construction of the isomorphism between the solution space and the source space of the d’Alembert operator (the differential operator of the wave equation). The results obtained could potentially serve as a basis for the subsequent development and analysis of finite element methods and quasi-optimality results.