On the solution of evolution p(.)-Bilaplace equation with variable
DOI:
https://doi.org/10.5269/bspm.62640Abstract
A high-order parabolic p(.)-Bilaplace equation with variable exponent is studied. The well-posedness at each time step of the problem in suitable Lebesgue Sobolev spaces with variable exponent with the help of nonlinear monotone operators theory is investigated. The solvability of the proposed problem as well as some regulrarity results are shown using Roth-Galerkin method.
References
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2. Samko, S., Convolution type operators in Lp(x) (Rn), Integral transform. Spec. Funct., 7,123-144, (1998).
3. Fan, X. L., Wang, S. Y. and Zhao, D., Density of C∞(Ω) in W1,p(x) (Ω) with discontinous exponent p(x), Math. Nachr., 279(1-3), 142-149, (2006).
4. Fan, X. L. and Zhao, D., On the spaces Lp(x) (Ω) and W1,p(x) (Ω), J. Math. Anal. Appl., 263, 424-446, (2001).
5. Diening, L., Harjulehto, P.i., Hasto, P. and Ruzicka, M., Lebesgue and Sobolev spaces with variable exponents, SPIN Springer’s internal project number. December 3, (2010).
6. Georgoulis, E. H. and Houston, P., Discontinuous Galerkin methods for the biharmonic problem, IMA J. Numer. Anal., 293, 573-594, (2009).
7. Pryer, T., Discontinuous Galerkin methods for the p-biharmonic equation from a discrete variational perspective, Electr. Transac. Nume. Anal., 41, 328-349, 2014.
8. Lindqvist, P.,Notes on the p-Laplace equation, NO-7491, Trondheim, Norway.
9. Becache, E., Ciarlet, P., Hazard, C., and Luneville, E., La methode des elements finis De la theorie a la pratique II. Complements, Les presse de l’ENSTA.
10. Li, H., The W1,p stability of the Ritz projection on graded meshes, Math. Comput., 86303, 49-74, (2017).
11. Sandri, D., Sur l’approximation numerique des ecoulements quasi-newtoniens dont la viscosite suit la loi puissance ou la loi de Carreau, RAIRO Model. Math. Anal. Number, 272, 131-155, (1993).
12. Gyulov, T. and Moro sanu, G.,On a class of boundary value problems involving the pbiharmonic operator, J. Math. Anal. Appl. 367(1), 43-57, (2010).
13. Lazer, A. and McKenna, P., Large-amplitude periodic oscillations in suspension bridges: some newconnections with nonlinear analysis, Siam Review, 32(4), 537-578, (1990).
14. Theljani, A., Belhachmi, Z. and Moakher, M., High-order anisotropic diffusion operators in spaces of variable exponents and application to image inpainting and restoration problems, Nonl. Anal.: Real World Appl., 47, 251-271, (2019).
15. Chaoui, A. and Hallaci, A., On the solution of a fractional diffusion integrodifferential equation with Rothe time discretization, Numerical Functional Analysis and Optimization, DOI : 10.1080/01630563.2018.1424200.
16. Chaoui, A. and Guezane-Lakoud, A., Solution to an integrodifferential equation with integral condition, Applied Mathematics and Computation, 266(2015) 903-908.
17. Chaoui, A. and Rezgui, N., Solution to fractional pseudoparabolic equation with fractional integral condition, Red. Circ. Mat. Palermo, II. Ser., DOI 10.1007/s12215-017-0306-x.
18. Crouzeix, M. and Thomee, V., The stability in Lp and W1 p of the L2-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR878688(88f:41016).
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2022-12-28
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