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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
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[1] Laptev, G.I., Mathematical singularities of a problem on the penetration of a magnetic field into a substance, Zh. vychisl. mat. i mat. fiz, 28, 1332-1345, (1988), (in Russian)
[2] Long, N.T.; Dinh, A.P.N., Nonlinear parabolic problem associated with the penetration of a magnetic field into a substance, Math. meth. appl. sci., 16, 281-295, (1993) · Zbl 0797.35099
[3] Lauerova, D., The existence of a periodic solution of a parabolic equation with the Bessel operator, Aplikace matematiky, 29, 1, 40-44, (1984) · Zbl 0552.35042
[4] Long, N.T.; Dinh, A.P.N., Periodic solution of a nonlinear parabolic equation involving Bessel’s operator, Computers math. applic., 25, 5, 11-18, (1993) · Zbl 0796.35088
[5] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[6] Edmunds, D.E.; Evans, W.D., Spectral theory and embeddings of Sobolev spaces, Quart. J. math., 30, 2, 431-453, (1979) · Zbl 0398.46031
[7] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non-linéaires, (1969), Dunod, Gauthier-Villars Paris · Zbl 0189.40603
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