Gradient blow-up in generalized Burgers and Boussinesq equations

We study the influence of gradient non-linearity on the global solubility of initial-boundary value problems for a generalized Burgers equation and an improved Boussinesq equation which are used for describing one-dimensional wave processes in dissipative and dispersive media. For a large class of initial data, we obtain sufficient conditions for global insolubility and a bound for blow-up times. Using the Boussinesq equation as an example, we suggest a modification of the method of non-linear capacity which is convenient from a practical point of view and enables us to estimate the blow-up rate. We use the method of contraction mappings to study the possibility of instantaneous blow-up and short-time existence of solutions.