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Nonlinear parabolic problem associated with the penetration of a magnetic field into a substance. (English) Zbl 0797.35099
Summary: We study the following initial and boundary value problem: \[ u_ t- \nabla \cdot \left( a \left( \int^ t_ 0 | \nabla u |^ 2ds \right) \nabla u \right) + f(u)=0, \] \[ u=0 \quad \text{ on } \;\partial \Omega;\qquad u(x,0) = u_ 0(x). \] In Section 1, with \(u_ 0\) in \(L^ 2(\Omega)\), \(f\) continuous such that \(f(u)+\varepsilon u\) nondecreasing for \(\varepsilon\) positive, we prove the existence of a unique solution on \((0,T)\), for each \(T>0\). In Section 2 it is proved that the unique solution \(u\) belongs to \(L^ 2(0,T;H^ 1_ 0 \cap H^ 2) \cap L^ \infty (0,T;H^ 1_ 0)\) if we assume \(u_ 0\) in \(H^ 1_ 0\) and \(f\) in \(C^ 1(\mathbb{R},\mathbb{R})\). Numerical results are given for these two cases.

MSC:
35K65 Degenerate parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35Q60 PDEs in connection with optics and electromagnetic theory
35D10 Regularity of generalized solutions of PDE (MSC2000)
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
65N99 Numerical methods for partial differential equations, boundary value problems
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References:
[1] Gordcziani, Diff. Uravan. 19 pp 1197– (1983)
[2] Laptev, Zh. Vychisl. Mat. i Mat. Fiz. 28 pp 1332– (1988)
[3] ’Quelques méthodes de résolution des problémes aux limites non-linéaires’, Dunod, Gauthier- Villars, Paris, 1969.
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