On blow-up of solutions of the Cauchy problems for a class of nonlinear equations of ferrite theory

In this paper, we consider three nonlinear equations of the theory of magnets with gradient nonlinearities $|\nabla u|^q$, $\partial_t|\nabla u|^q$, and $\partial^2_t|\nabla u|^q $ are considered. For the corresponding Cauchy problems, we obtain results on local-in-time unique solvability in the weak sense and on blow-up for a finite time. These three equations have the same critical exponent $q=3/2$ since weak solutions behave differently for $1<q\leq 3/2$ and for $q>3/2$. By the method of nonlinear capacity proposed by S. I. Pokhozhaev, we obtain a priori estimates, which imply the absence of local and global weak solutions.