In this thesis, we will study two problems in parabolic Partial Differential Equations.
In the Chapter 2 we study the problem of backward uniqueness of the linear heat equation. We briefly recall some results about backward uniqueness under the assumption of Dirichlet boundary condition and results of boundary controllability in control theory. It was shown in the papers of Escauriaza, Seregin and Sverak (Russian math. Surveys 58: 211-250, 2003), Seregin and Sverak (Int. Math. Ser. 2:353-366, 2002) that a bounded solution of the heat equation in a half-space which becomes zero at some time must be identically zero, even though no assumptions are made on the boundary values of the solutions. In a recent example, Luis Escauriaza showed that this statement fails if the half-space is replaced by cones with opening angle smaller than 90 degree. The main result of the chapter is that the result remains true for cones with opening angle larger than 110 degree.
In Chapter 3 we study the profiles of the singularity of the 1-D complex Burgers equation. The Cauchy problem for the 1-D real-valued viscous Burgers equation $u_t+uu_x=u_{xx}$ is globally well-posed (Hopf in Commun Pure Appl Math 3:201-230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Polacik and Sverak (J Reine Angew math 616:205-217, 2008). They also proved that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. We study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.