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Periodic solutions of a nonlinear parabolic equation associated with the penetration of a magnetic field into a substance. (English) Zbl 0834.35070
Summary: We prove the existence and uniqueness of a $$T$$-periodic weak solution of the nonlinear parabolic equation $u_t- {1\over {r^\gamma}} {\partial \over {\partial r}} \Biggl[a\biggl( \int_0^t |u_r (r, s)|^2 ds \biggr) r^\gamma u_r \Biggr]+ f(u) =0, \qquad (r,t)\in (0, 1)\times (0, T),$ $u_r (0, t)= u_r (1, t)+ h(t) u(1, t)=0,$ in a Sobolev space with weight. In the proof, the Galerkin method is employed. Numerical results are given.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35B10 Periodic solutions to PDEs
##### Keywords:
nonlocal term; Sobolev space with weight
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##### References:
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