Strongly quasilinear parabolic systems
DOI:
https://doi.org/10.24193/subbmath.2023.2.10Keywords:
Quasilinear parabolic systems, weak solutions, Young measures.Abstract
Using the theory of Young measures, we prove the existence of solutions to a strongly quasilinear parabolic system
Mathematics Subject Classification (2010): 35K55, 35D30, 46E30
Received 29 April 2020; Accepted 29 June 2020.
References
Adams, R.A., Fournier, J.J.F., Fournier, Sobolev spaces, 2nd ed., Vol. 140, Pure and Applied Mathematics, Amsterdam, Elsevier, Academic Press, 2003.
Aharouch, L., Azroul, E., Rhoudaf, M., Existence result for variational degenerated parabolic problems via pseudo-monotonicity, EJDE, 14(2006), 9-20, ISSN: 1072-6691.
Amann, H., Ordinary Differential Equations: An Introduction to Nonlinear Analysis, Berlin, De Gruyter, 1990.
Azroul, E., Balaadich, F., Weak solutions for generalized p-Laplacian systems via Young measures, Moroccan J. of Pure and Appl. Anal., 4(2018), no. 2, 76-83.
Azroul, E., Balaadich, F., Strongly quasilinear parabolic systems in divergence form with weak monotonicity, Khayyam J. Math., 6(2020), no 1, 57-72.
Azroul, E., Balaadich, F., Quasilinear elliptic systems in perturbed form, Int. J. Nonlinear Anal. Appl., 10(2019), no. 2, 255-266.
Azroul, E., Balaadich, F., A weak solution to quasilinear elliptic problems with perturbed gradient, Rend. Circ. Mat. Palermo, 2(2020). https://doi.org/10.1007/s12215-020-00488-4
Ball, J.M., A version of the fundamental theorem for Young measures, In: PDEs and Continuum Models of Phase Transitions (Nice, 1988), Lecture Notes in Phys., 344(1989), 207-215.
Bögelein, V., Duzaar, F., Marcellini, P., Signoriello, S., Nonlocal diffusion equations, J. Math. Anal. Appl., 432(2015), no. 1, 398-428.
Bögelein, V., Duzaar, F., Sch¨atzler, L., Scheven, C., Existence for evolutionary problems with linear growth by stability methods, J. Differential Equations, (2018), https://doi.org/10.1016/j.jde.2018.12.012.
Brezis, H., Browder, E., Strongly nonlinear parabolic initial-boundary value problems, Proc. Natl. Acad. Sci. USA, 76(1979), no. 1, 38-40.
Caddick, M., Cullen, M., Su¨li, E., Numerical approximation of young measure solutions to parabolic systems of forward-backward type, Appl. Anal. Discrete Math., 13(2019), no. 3, 649-696.
Demoulini, S., Young measure solutions for a nonlinear parabolic equation of forward- backward type, SIAM J. Math. Anal., 27(1996), no. 2, 376-403.
Di Nardo, R., Feo, F., Guibé, O., Existence result for nonlinear parabolic equations with lower order terms, Anal. Appl. (Singap.), 9(2011), no. 2, 161-186.
Dolzmann, G., Hungerbühler, N., Müller, S., Non-linear elliptic systems with measure- valued right-hand side, Math. Z., 226(1997), 545-574.
Dolzmann, G., Hungerbühler, N., Müller, S., The p-harmonic system with measure- valued right-hand side, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14(1997), no. 3, 353-364.
Dolzmann, G., Hungerbühler, N., Müller, S., Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side, J. Reine Angew. Math., 520(2000), 1-35.
Dreyfuss, P., Hungerbühler, N., Results on a Navier-Stokes system with applications to electrorheological fluid flow, Int. J. Pure Appl. Math., 14(2004), no. 2, 241-271.
El-Dessouky, A.T., On weak solutions of systems of strongly nonlinear parabolic variational inequalities, American Journal of Mathematical Analysis, 5(2017), no. 1, 7-11.
Evans, L. C., Weak convergence methods for nonlinear partial differential equations, 74(1990).
Hungerbühler, N., A refinement of Ball’s theorem on Young measures, N.Y. J. Math., 3(1997), 48-53.
Hungerbühler, N., Quasilinear parabolic systems in divergence form with weak mono- tonicity, Duke Math. J., 107(2001) 497-519.
Hungerbühler, N., Young Measures and Nonlinear PDEs, Habilitationsschrift ETH Zurich, 2001.
Landes, R., On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A89, (1981), 217-237.
Landes, R., Mustonen, V., A strong nonlinear parabolic initial boundary value problem, Ark. Mat. Vol., 25(1987), 29-40.
Marcellini, P., A variational approach to parabolic equations under general and p, q- growth conditions, Nonlinear Anal., (2019), https://doi.org/10.1016/j.na.2019.02.010
Misawa, M., Partial regularity results for evolutional p-Laplacian systems with natural growth, Manuscripta Math., 109(2002), 419-454.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Studia Universitatis Babeș-Bolyai Mathematica
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
