Nonlinear Parabolic Problems in Anisotropic Sobolev Spaces

Year 2024, Volume: 14 Issue: 2, 109 - 143, 31.07.2024

Mohammed El Ansari Youssef Akdim

https://izlik.org/JA37DS27YH

Abstract

In this paper, our goal is to establish results on existence of renormalized
solutions for a class of Stefan problems of the form β(u)t − div(a(x, Du) + F(u)) ∋ f,
posed in an open bounded Ω, where data belongs to L
1 −data, β is a maximal monotone graph and div(a(x, Du)) is a Leary-Lions operator with anisotropic growth conditions.

Keywords

anisotropic Sobolev spaces , maximal monotone graph , Stefan type problems

References

• [1] Y. Akdim, M. El Ansari, S. Lalaoui Rhali, Solvability of some Stefan type problems with L1− data, Rend. Mat. Appl., 44, 2023, 281-307.
• [2] H.-W. Alt, S. Luckhaus, Quasi-linear elliptic-parabolic differential equations, Math. Z., 183, 1983, 311-341.
• [3] K. Ammar, P. Wittbold, Existence of renormalized solutions of degenerate elliptic parabolic problems, Proc. Poy. Soc. Edinburgh, 133, 2003, 477-496. [4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publisher, 1976.
• [5] M. Bendahmane, P. Wittbold, Renormalized solutions for nonlinear elliptic equations with variable exponents and L1-data, Nonlinear Anal. TMA, 70, 2009, 567-583.
• [6] Ph. B´enilan, M.G. Crandall, Completely accretive operators. In Semigroup Theory and Evolution Equations, Lecture Notes in Pure and Appl. Math., 135, 1991, 41-75.
• [7] D. Blanchard, A. Porretta, Stefan problems with nonlinear diffusion and convection, J. Diff. Equ., 210, 2005, 383-428.
• [8] D. Blanchard, F. Murat, H. Redwane, Existence and Uniqueness of a Renormalized Solution for a Fairly General Class of Nonlinear Parabolic Problems, J. Diff. Equ, 177, 2001, 331-347.
• [9] H. Br´ezis, M. Crandall, Uniqueness of solutions of the initial value problem for ut − ∆φ(u) = 0, J. Math. Pures. Appl., 58, 1979, 153-163.
• [10] J. Crank, Free and Moving Boundary Problems, North-Holland, Amsterdam, 1977.
• [11] E. Dibenedetto, Friedman, A. The ill-posed Hele-Shaw model and the Stefan problem for supercooled water, Trans. Amer. Math. Soc., 282, 1984, 183-204.
• [12] R.J. DiPerna, P-L, Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math., 130, 1989, 321-366.
• [13] I. Fragal`a, F. Gazzola, B. Kawohl, Existence and nonexistence results for anisotropic quasilinear elliptic equations, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 215, 2004, 715-734.
• [14] A. Friedman, Variational principles and free-boundary problems, Wiley, New York, 1982.