Entropy solution for a nonlinear parabolic problem with homogeneous Neumann boundary condition involving variable exponents
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U. Traoré
urbain.traore@yahoo.fr
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DOI:
https://doi.org/10.4067/S0719-06462021000300385Abstract
In this paper we prove the existence and uniqueness of an entropy solution for a non-linear parabolic equation with homogeneous Neumann boundary condition and initial data in \(L^1\). By a time discretization technique we analyze the existence, uniqueness and stability questions. The functional setting involves Lebesgue and Sobolev spaces with variable exponents.
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M. Abdellaoui, E. Azroul, S. Ouaro and U. Traoré, “Nonlinear parabolic capacity and renormalized solutions for PDEs with diffuse measure data and variable exponent”, An. Univ. Craiova Ser. Mat. Inform., vol. 46, no. 2, pp. 269–297, 2019.
M. Abdellaoui and E. Azroul, “Nonlinear parabolic equations with data soft measure data”, J. Nonlinear Evol. Equ. Appl., vol. 2019, no. 7, pp. 115–133, 2020.
B. Andreianov, M. Bendahmane, K. H. Karlsenc and S. Ouaro, “Well-posedness results for triply nonlinear degenerate parabolic equations.”, vol. 247, no. 1, pp. 277–302, 2009.
F. Andreu, J. M. Mazón, S. Segura de Léon and J. Toledo, “Existence and uniqueness for a degenerate parabolic equation with (L^1) data”, Trans. Amer. Math. Soc., vol. 351, no. 1, pp. 285–306, 1999.
F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, “(L^1) existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions”, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 20, no. 1, pp. 61–89, 2017.
M. Bendahmane, P. Wittbold and A. Zimmermann, “Renormalized solutions for a nonlinear parabolic equation with variable exponents and (L^1) data”, J. Differential Equations, vol. 249, no. 6, pp. 1483–1515, 2010.
F. Benzekri and A. El Hachimi, “Doubly nonlinear parabolic equations related to the p- Laplacian operator: Semi-discretization”, Electron. J. Differential Equations, vol. 2003, no. 113, pp. 1–14, 2003.
P. Bénilan, L. Boccardo, T. Gallouët, M. Pierre and J. L. Vazquez, “An (L^1) theory of existence and uniqueness of nonlinear elliptic equations”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), vol. 22, no. 2, pp. 214–273, 1995.
B. K. Bonzy, I. Nyanquini and S. Ouaro, “Existence and uniqueness of weak and entropy solutions for homogeneous Neumann boundary-value problems involving variable exponents”, Electron. J. Differential Equations, vol. 2012, no. 12, pp. 1–19, 2012.
A. Dall‘Aglio, “Approximated solutions of equations with (L^1) data. Application to the H- convergence of quasi-linear parabolic equations”, Ann. Mat. Pura Appl.(4), vol. 170, pp. 207– 240, 1996.
L. Diening, P. Harjulehto, P. Hästö and M. RůžiÄka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, vol. 2017, New York: Springer-Verlag Heidelberg, 2011.
R. J. DiPerna and P.-L. Lions, “On the Cauchy problem for Boltzmann equations: global existence and weak stability”, Ann. of Math. (2), vol. 130, no. 2, pp. 321–366, 1989.
A. Eden, B. Michaux and J.-M. Rakotoson, “Semi-discretized nonlinear evolution equations as discrete dynamical systems and error analysis”, Indiana Univ. Math. J., vol. 39, no. 3, pp. 737–783, 1990.
X. L. Fan and D. Zhao, “On the generalized Orlicz-Sobolev space (displaystyle W^{k,p(x)}(Omega))”, Journal of Gansu Education College, vol. 12, no. 1, pp. 1–6, 1998.
G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l‘ingénierie pétrolière, Mathématiques et Applications 22, Berlin: Springer, 1996.
A. El Hachimi and M. R. Sidi Ammi, “Thermistor problem: a nonlocal parabolic problem”, in Differential Equations and Mechanics. Electron. J. Diff. Eqns., vol. 2004, no. 11, 2004, pp. 117–128.
A. El Hachimi, J. Igbida and A. Jamea, “Existence result for nonlinear parabolic problems with (L^1)-data”, Appl. Math. (Warsaw), vol. 37, no. 4, pp. 483–508, 2010.
S. Ouaro and S. Soma, “Weak and entropy solutions to nonlinear Neumann boundary value- problems with variable exponents”, Complex Var. Elliptic Equ., vol. 56, nos. 7-9, pp. 829–851, 2011.
S. Ouaro and A. Tchousso, “Well-posedness result for a nonlinear elliptic problem involving variable exponent and Robin type boundary condition”, Afr. Diaspora J. Math., vol. 11, no. 2, pp. 36–64, 2011.
S. Ouaro and A. Ouédraogo, “Nonlinear parabolic problems with variable exponent and (L^1)- data”, Electron. J. Differential Equations, vol. 2017, no. 32, pp. 1–32, 2017.
S. Ouaro and U. Traoré, “Nonlinear parabolic problem with variable exponent and measure data”, J. Nonlinear Evol. Equ. Appl., vol. 2020, no. 5, pp. 65–93, 2020.
M. Sanchón and J. M. Urbano, “Entropy solution for p(x)-Laplace equation”, Trans. Amer. Math. Soc., vol. 361, no. 2, pp. 6387–6404, 2009.
H. Redwane, “Nonlinear parabolic equation with variable exponents and diffuse measure data”, J. Nonl. Evol. Equ. Appl., vol. 2019, no. 6, pp. 95–114, 2020.
M. RůžiÄka, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics, vol. 1748, Berlin: Springer-Verlag, 2000.
L.-L. Wang, Y.-H. Fan and W.-G. Ge, “Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator”, Nonlinear Anal., vol. 71, no. 9, pp. 4259–4270, 2009.
J. Yao, “Solutions for Neumann boundary value problems involving p(x)-Laplace operator”, Nonlinear Anal., vol. 68, no. 5, pp. 1271–1283, 2008.
C. Zhang and S. Zhou, “Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and (L^1) data”, Journal of Differential Equations, vol. 248, no. 6, pp. 1376–1400, 2010.
D. Zhao, W. J. Qiang and X. L. Fan, “On generalized Orlicz spaces (L^{p(x)}(Omega))”, J. Gansu Sci., vol. 9, no. 2, pp. 1–7, 1997 [in Chinese].
A. Zimmermann, “Renormalized solutions for a nonlinear parabolic equation with variable exponents and (L^1) data”, Ph. D. Thesis, T. U. Berlin, 2010.
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