Exact solutions to nonlinear equations with quadratic nonlinearity

Based on the concepts of the generalized conditional symmetry and of the invariant subspace, a constructive way to derive nonlinear equations with quadratic nonlinearity that admit exact solutions as well as many new examples is presented. This type of solution, found by Galaktionov, describes the blow-up phenomena. Examples of exact solutions that do not correspond to the invariant subspace are also presented and explained by the nonlinear generalized conditional symmetries. (author)