In this paper, we consider three nonlinear equations of the theory of magnets with gradient nonlinearities \({\left|\nabla u\right|}^{q},{\partial }_{t}{\left|\nabla u\right|}^{q},\) and \({{\partial }_{t}^{2}\left|\nabla u\right|}^{q}\) 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 ≤ 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.
