Nonlinear perturbations of BLMP and YTSF equations

Some nonlinear evolutive equations of Mathematical Physics present infinitely many solutions described in many paper by different methods. For example Körteweg De Vries equation and Kadomtsev - Petviashvili equation are completely integrable despite the presence of quasilinear terms. In the present paper we perturb these kind of equations by positive nonlinear terms having polynomial growth. Assuming that the quasilinear term in the original equation has divergence form, we may apply test function method and establish a range of exponents for the perturbation so that a non-existence result of global weak solutions holds. Concerning initial data condition, an important difference with other equations studied by similar methods (wave, Tricomi and so on) appears. Indeed, we present a class of quasilinear equations for which the sign assumption on the initial data can be omitted and non-existence results still hold. Our basic examples are the perturbation of Boiti Leon Manna Pempinelli equation and Yu Toda Sasa Fukuyama equation. Finally we suggest open problems for other equations and other kind of perturbations.