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.
\[ u_r (0, t)= u_r (1, t)+ h(t) u(1, t)=0, \] 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 |
References:
| [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: 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: Dunod, Gauthier-Villars Paris · Zbl 0189.40603 |
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