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We consider a mathematical model describing magnetization dynamics with vertical spin stiffness. The model consists of a modified form of the Landau-Lifshitz-Gilbert equation for the evolution of the magnetization vector in a rigid ferromagnet. The modification lies in the presence in the effective field of a nonlinear term describing vertical spin stiffness. We prove the global existence of weak solutions to the model by using the Faedo-Galerkin method and discuss the limit of the obtained solutions as the vertical spin stiffness parameter tends to zero. MSC: 78A25; 35Q60; 35B40

We are dealing with the classical Landau–Lifshitz equation with an unusual exchange field expressed in terms of a p -Laplacian operator for p greater than 2. We obtain weak solutions to the model by using penalization, compactness arguments, and monotonicity method.

We prove via a Galerkin approximation the time local existence of regular solutions to a Landau–Lifshitz–Bloch equation with applied current in a bounded domain. The uniqueness of the solution is also established. Moreover, we show the global in time existence of a regular solution in dimension two.

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