On Blow-Up of Solutions of the Cauchy Problems for a Class of Nonlinear Equations of Ferrite Theory

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 281; no. 3; pp. 418 - 470
• Main Authors: Korpusov, M. O., Shlyapugin, G. I.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 10.05.2024; Springer; Springer Nature B.V
• Subjects: Cauchy problems; Iron compounds; Magnets; Mathematics; Mathematics and Statistics; Nonlinear equations; Nonlinearity; local solvability; estimates of the blow-up time; nonlinear Sobolev-type equation; nonlinear capacity; blow-up; 35B44
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-024-07116-x

In this paper, we consider three nonlinear equations of the theory of magnets with gradient nonlinearities ∇ u q , ∂ t ∇ u q , and ∂ t 2 ∇ 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 ≤ 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.