This report presents my main results in three separate chapters. The first one is about nonlinear dispersive partial differential equations. The Cauchy problem is studied through a method of normal form and dispersion inequalities. Two unique continuation properties are established via a Carleman inequality and an estimate of Bourgain. The consideration of the viscosity and viscoelasticity is the subject of the second chapter. Described mathematically by a nonlocal operator, one shows the dissipation and the stability of the energy. The convergence to the entropy solution of the hyperbolic conversion law is also proved. In the third chapter, completely independent of the first two, we focus on the modeling of plants and their environment. The interactions between plants, pathogens and pests are drawn up.