Global well-posedness and self-similar solutions for a nonlinear elliptic problem with a dynamic boundary condition

In this seminar, we present a study on a semilinear elliptic equation in the half-space Rn + subject to a nonlinear dynamic boundary condition. Using an integral formulation based on Poisson and Green kernels, we discuss the global well-posedness of the problem within the framework of Morrey spaces. This functional choice allows us to treat a broader class of singular initial data, including homogeneous pro les. We will show that, for small initial data in this norm, it is possible to obtain the existence and uniqueness of global solutions, as well as the construction of self-similar solutions.