Study on evolutionary partial differential equations: cases of norm inflation and of the construction of a single black hole

In this thesis, we study several arguments related to the issue of well-posedness of dynamical partial differential equations. In the first part, we prove the occurrence of norm inflation, and thus ill-posedness, for the generalized improved Boussinesq equation and the viscous nonlinear wave equation in negative Sobolev spaces. This esIn this thesis, we study several arguments related to the issue of well-posedness of dynamical partial differential equations. In the second part, we do the first step of a construction of a semi-global spherically symmetric spacetime solving the Einstein-scalar field system and connecting a cosmological region to a single black hole. To do so, we prove global well-posedness of a quasilinear evolutionary equation in two variables which represents a small perturbation in the cosmological region of a Schwarzschild-de Sitter spacetime, and then determine the needed initial data on a hypersurface to transform the existing metric into an induced de Sitter metric.tablishes new lower bounds for the well-posedness regularity thresholds of these equations in Sobolev spaces. To prove norm inflation, we use a Fourier analysis argument. In the case of the viscous nonlinear wave equation, we also study failure of Ck-continuity of the solution map, and well-posedness in some cases.