On local and global solvability and on blow-up of the solution of a nonlinear equation of pseudo-parabolic type

We consider an equation for which the fundamental solution on the linear part has the time derivative with an integrable singularity at t=0. (For the equations that we studied earlier, the linear parts had infinitely differentiable in t fundamental solutions.) We study the properties of the linear part of the equation by the potential method, and we reduce the Cauchy problem under consideration to an integral equation. We prove the finite-time blow-up by the test function method, and we prove the global solvability (under other hypothsis on the initial data) by a priori estimation method with Gronwall-Bellman-Bihary inequality.