Blow-up of solutions of a class of strongly non-linear dissipative wave equations of Sobolev type with sources

We consider the abstract Cauchy problem for a first-order ordinary differential equation with non-linear operator coefficients. The results are applied to some strongly non-linear dissipative wave equations of Sobolev type. We obtain sufficient conditions for the problem to be globally soluble, as well as sufficient conditions for the solutions to blow up in finite time. These conditions are close to being necessary. Under certain supplementary assumptions on the non-linear operators, we prove that the problem is soluble in any finite cylinder. Under certain conditions on the norm of the initial functions, we prove that the solution of the problem blows up in finite time. We give examples of equations of Sobolev type satisfying these conditions.