• Strichartz estimates for the heat equation
• liquid crystals
• least energy solution

Blood disorders are common and dangerous deseases in our society. Someone of these patologies cause a deformation of the shape of the red blood cells. In these cases, it can be interesting to compare blood to liquid crystals, a state of matter intermediate between the solid state and the liquid state. In the master's thesis we study the Beris-Edward model for liquid crystals in the case of absence of velocity. For an appropriate choice of the non-linearity, it can be proved the existence of a least energy solution for the stationary problem. This solution satisfies also some regularity properties and it can be seen as a saddle point for the energy. Then the article treats the evolution equation, which is the heat equation with a non-linearity. At first we proved some Strichartz estimates for the Heat Operator. Then we proved the local existence and uniqueness of the solution in the Strichartz Spaces, the global existence for small initial data and the decay of the solution as time goes to infinity. In the end we also talk about the stability of stationary solutions.