## Some boundary problems of the spin systems and the systems of ferro- magnetic chain. I: Nonlinear boundary problems.(English)Zbl 0644.35060

Some nonlinear boundary problems for the system of ferro-magnetic chain $(1)\;z_ t = z\times z_{xx}+f(x,t,z)$ and the corresponding spin system $(2)\;z_ t = \epsilon z_{xx}+z\times z_{xx}+f(x,t,z)$ are considered in the rectangular $$Q_ T=\{0\leq x\leq \ell$$; $$0\leq t\leq T\}$$ with the nonlinear boundary conditions $(3)\;z_ x(0,t)=\text{grad}\psi_ 0(t,z(0,t)),\quad -z_ x(\ell,t)=\text{grad}\psi_ 1(t,z(\ell,t))$ and the initial condition $$(4)\quad z(x,0)=\phi(x),$$ where $$z=(u,v,w)$$, $$f(x,t,z)$$ and $$\phi(x)$$ are the three-dimensional vector functions; $$\psi_ 0(t,z)$$ and $$\psi_ 1(t,z)$$ are scalar functions, “$$\times$$” denotes the cross product operator of two three-dimensional vectors and “grad” denotes the gradient operator with respect to the vector variable z. The system (1) contains the so-called Landau-Lifschitz equation of isotropic Heisenberg ferro-magnetic chain as the special case. The three-dimensional generalized global vector solution z(x,t) of the nonlinear boundary problem (3) and (4) for the spin system (2) is established by the method of discretization of space variable $$x\in [0,\ell]$$. The generalized solution z(x,t) satisfies the spin system (2) in generalized sense and the nonlinear boundary conditions (3) and the initial condition (4) in classical sense. By the limiting process $$\epsilon\to 0$$, the weak global solution $$z(x,t)\in L_{\infty}(0,T;H^ 1(0,\ell))\cap C^{(1/2,1/4)}(Q_ T)$$ of the nonlinear boundary problem (3) and (4) is obtained for the strongly degenerate system of ferro-magnetic chain (1).
Reviewer: B.L.Guo

### MSC:

 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35Q99 Partial differential equations of mathematical physics and other areas of application 35L65 Hyperbolic conservation laws