Galerkin method applied to a parabolic evolution problem with nonlocal boundary conditions. (English) Zbl 1155.35053
A model parabolic mixed problem with boundary integral condition arising in the context of thermoelasticity is considered. The Galerkin method is used to prove existence and uniqueness of a weak solution as well as the continuous dependence upon data. The proofs make specific use of the so-called Bouziani space.
Reviewer: Gudula Rünger (Chemnitz)
MSC:
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35K05 | Heat equation |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35D05 | Existence of generalized solutions of PDE (MSC2000) |
References:
| [1] | A. Bouziani, Mixed problem with integral condition for certain partial differential equations, Ph.D. Thesis, Constantine University, 1996 (in French); A. Bouziani, Mixed problem with integral condition for certain partial differential equations, Ph.D. Thesis, Constantine University, 1996 (in French) · Zbl 0864.35049 |
| [2] | Bouziani, A., Mixed problem with boundary integral conditions for a certain parabolic equation, J. Appl. Math. Stoch. Anal., 9, 3, 323-330 (1996) · Zbl 0864.35049 |
| [3] | Bouziani, A.; Benouar, N.-E., Sur un problème mixte avec uniquement des conditions aux limites intégrales pour une classe d’équations paraboliques, Maghreb Math. Rev., 9, 1-2, 55-70 (2000), (in French) |
| [4] | Bouziani, A., On the quasi static flexure of a thermoelastic rod, Commun. Appl. Anal., 6, 4, 549-568 (2002) · Zbl 1084.74526 |
| [5] | Dai, D.-Q.; Huang, Y., Remarks on a semilinear heat equation with integral boundary conditions, Nonlinear Anal. TMA, 67, 468-475 (2007) · Zbl 1113.35102 |
| [6] | Dehghan, M., Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math., 52, 39-62 (2005) · Zbl 1063.65079 |
| [7] | Dehghan, M., A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer. Methods Partial Differential Equations, 22, 220-257 (2006) · Zbl 1084.65099 |
| [8] | Dehghan, M., The one-dimensional heat equation subject to a boundary integral specification, Chaos Solitons Fractals, 32, 661-675 (2007) · Zbl 1139.35352 |
| [9] | Merazga, N.; Bouziani, A., Rothe time discretization method for a non local problem arising in thermoelasticity, J. Appl. Math. Stoch. Anal., 2005, 1, 13-28 (2005) · Zbl 1077.74019 |
| [10] | Merazga, N.; Bouziani, A., On a time-discretization method for a semilinear heat equation with purely integral conditions in a nonclassical function space, Nonlinear Anal., 66, 604-623 (2007) · Zbl 1105.35044 |
| [11] | Kufner, A.; John, O.; Fučik, S., Function Spaces (1977), Noordhoff: Noordhoff Leiden · Zbl 0364.46022 |
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