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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.

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
Full Text: DOI
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