Equivalence of "generalized" solutions for nonlinear parabolic equations with variable exponents and diffuse measure data

Mohammed Abdellaoui, Hicham Redwane

Abstract


We prove the equivalence of suitably defined weak solutions of a nonhomogeneous initial-boundary value problem for a class of nonlinear parabolic equations. We also develop the notion of both "renormalized" and "entropy" solutions with respect to the "generalized" $p(\cdot)$-capacity, initial datum, and diffuse measure data (which does not charge the set of null $p(\cdot)$-capacity). Conditions, under which "generalized weak" solutions of the nonhomogeneous problem are in fact well-defined, are also given.

Full Text:

PDF

References


R. Adams, Sobolev Spaces, Academic Press, New York 1975.

M. Abdellaoui, E. Azroul, S. Ouaro, U. Traoré, Nonlinear parabolic capacity and renormalized solutions for PDEs with diffuse measure data and variable exponent, Annals of the University of Craiova, Mathematics and Computer Science Series 46, 2019, no. 2, 269-297.

D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory, Grundlehren der mathematischen Wissenschaften, 314, Springer-Verlag, Berlin, 1996.

E. Acerbi, G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Ration. Mech. Anal. 156 (2001), 121-140.

S.N. Antontsev, S.I. Shmarev, A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 60 (2005), 515-545.

S.N. Antontsev, V. Zhikov, Higher integrability for parabolic equations of p(x; t)-Laplacian type, Adv. Di_erential Equations 10 (2005), 1053-1080.

D.G. Aronson, Removable Singularities for Linear Parabolic Equations, Arch. Rational Mech. Anal. 17 (1964), 79-84.

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, J.L. Vazquez, An L1-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1995), 241-273.

Ph. Bénilan, M.G. Crandall, Completely accretive operators, In: Ph. Clément, et al. (Eds.), Semigroup Theory and Evolution Equations, Lect. Notes Pure Appl. Math., Marcel Dekker, New York (1991).

P. Bénilan, J. Carrillo, P. Wittbold, Renormalized entropy solutions of scalar conservation laws, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 29 (2000), no. 4, 313-327.

P. Bénilan, M.G. Crandall, A. Pazy, Evolution Equations Governed by Accretive Operators, forthcoming book.

L. Boccardo, J.I. Diaz, D. Giachetti, F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation, In: Recent Advances in Nonlinear Elliptic and Parabolic Problems, Pitman Res. Notes Math. Ser. 208, Longman Sci. Tech., Harlow, 229-246 (1989).

L. Boccardo, A. Dall'Aglio, T. Gallouëet, L. Orsina, Nonlinear parabolic equations with measure data, Journ. of Functional Anal. 147 (1997), 237-258.

L. Boccardo, A. Dall'Aglio, L. Orsina, Existence and regularity results for some elliptic equations with degenerate coercivity, Atti Sem. Mat. Fis. Univ. Modena 46 (1998), 51-81.

A. Benkirane, A. Elmahi, Almost everywhere convergence of the gradients of solutions to elliptic equations in Orlicz spaces and application, Nonlinear Analysis: T.M.A. 28 (1997), 1769-1784.

A. Benkirane, A. Elmahi, An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces, Nonlinear Analysis: T.M.A. 36 (1999), 11-24.

A. Benkirane, J.-P. Gossez, An approximation theorem in higher order Orlicz-Sobolev spaces and applications, Studia Mathematica 92 (1989), no. 3, 231-255.

L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 5, 539-551.

L. Boccardo, T. Gallouët, L. Orsina, Existence and nonexistence of solutions for some nonlinear elliptic equations, Journal D'Analyse Mathématique 73 (1997), 203-223.

D. Blanchard, Truncations and monotonicity methods for parabolic equations, Nonlinear Analysis: T.M.A. 21 (1993), 725-743.

L. Boccardo, F. Murat, J.P. Puel, Existence results for some quasilinear parabolic equations, Nonlinear Analysis: T.M.A. 13 (1989), 373-392.

M.F. Betta, A. Mercaldo, F. Murat, M.M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower-order term and right-hand side a measure. Corrected reprintof J. Math. Pures Appl. (9) 81(6) (2002) 533-566, J. Math. Pures Appl. (9) 82 (2003), no. 1, 90-124.

H. Brezis, Functional analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

D. Blanchard, H. Redwane, Renormalized solutions for a class of nonlinear evolution problems, J. Math. Pure Appl. 77 (1998), 117-151.

M. Bendahmane, P. Wittbold, Renormalized solutions for nonlinear elliptic equations with variable exponents and L1 data, Nonlinear Anal. 70 (2009), 567-583.

M. Bendahmane, P. Wittbold, A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1 data, J. Diff. Equ. 249 (2010), 1483-1515.

I. Chlebicka, A. Karppinen, Removable sets in elliptic equations with Musielak-Orlicz growth, Journal of Mathematical Analysis and Applications 501 (2021), no. 1, Art. ID 124073.

I. Chlebicka, Measure data elliptic problems with generalized Orlicz growth, Proc. Roy. Soc. Edinburgh Sect. A 153 (2023), no. 2, 588-618.

I. Chlebicka, P. Nayar, Essentially fully anisotropic Orlicz functions and uniqueness to measure data problem, Math. Methods Appl. Sci. 45 (2022), no. 14, 8503-8527.

Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406.

X. Chai, W. Niu, Existence and non-existence results for nonlinear elliptic equations with nonstandard growth, Journal of Mathematical Analysis and Applications 412 (2014), no. 2, 1045-1057.

J. Carrillo, P. Wittbold, Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems, J. Di_erential Equations 156 (1999), 93-121.

L. Diening, P. Harjulehto, P. Hästö, M. Rüžička, Lebesgue and Sobolev spaces with variable exponents, Springer, 2010.

L. Diening, Riesz potential and Sobolev embeddings of generalized Lebesgue and Sobolev spaces Lp(.) and Wk;p(.), Math. Nachr. 268 (2004), no. 1, 31-43.

L. Diening, Theoretical and Numerical Results for Electrorheological Fluids, Ph.D. Thesis, University of Freiburg, Germany, 2002.

G. Dal Maso, A. Malusa, Some properties of reachable solutions of nonlinear elliptic equations with measure data, Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e serie 25 (1997), no. 1-2, 375-396.

G. Dal Maso, F. Murat, L. Orsina, A. Prignet, Renormalized solutions for elliptic equations with general measure data, C. R. Acad. Sci. Serie I 325 (1997), 481-486. [38] J.L. Doob, Classical potential theory and its probabilistic couterpart, Springer-Verlag, 1984.

J. Droniou, A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, NoDEA 14 (2007), no. 1-2, 181-205.

J. Droniou, A. Porretta, A. Prignet, Parabolic capacity and soft measures for nonlinear equations, Potential Anal. 19 (2003), 99-161.

J. Droniou, Int_egration et Espaces de Sobolev à valeurs vectorielles, Polycopié de l'Ecole Doctorale de Mathematiques-Informatique de Marseille, available at http://www-gm3.univmrs. fr/polys.

L. Diening, M. Rüžička, Calder_on-Zygmund operators on generalized Lebesgue spaces Lp(.) and problems related to fluid dynamics, Journal für die reine und angewandte Mathematik (Crelles Journal) 563 (2003), 197-220.

A. Elmahi, Strongly nonlinear parabolic initial-boundary value problems in Orlicz spaces, Electronic Journal of Differential Equations (EJDE) Conf. 09 (2002), 203-220.

B. El Hamdaoui, J. Bennouna, A. Aberqi, Renormalized solutions for nonlinear parabolic systems in the Lebesgue-Sobolev spaces with variable exponents, Zh. Mat. Fiz. Anal. Geom. 14 (2018), no. 1, 27-53.

X. Fan, J. Shen, D. Zhao, Sobolev embedding theorems for spaces Wk;p(x)(), J. Math. Anal. Appl. 262 (2001), 749-760.

X. Fan, D. Zhao, On the spaces Lp(x)() and Wm;p(x)();, J. Math. Anal. Appl. 263 (2001), 424-446.

X. Fan, D. Zhao, On the generalized Orlicz-Sobolev space Wk;p(x)(), J. Gancu Educ. College 12 (1998), no. 1, 1-6.

O. Guibé, A. Mercaldo, Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data, Potential Anal (2006) 25, 223-258.

P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values, Math. Bohem. 132 (2007), 125-136.

P. Harjulehto, P. Hästö, Lebesgue points in variable exponent spaces, Ann. Acad. Sci. Fenn. Math. 29 (2004), no. 2, 295-306.

P. Harjulehto, P. Hästö, M. Koskenoja, Properties of capacities in variable exponent Sobolev spaces, J. Anal. Appl. 5 (2007), no. 2, 71-92.

P. Harjulehto, P. Hästö, M. Koskenoja, S. Varonen, Sobolev capacity on the space W1;p(Rn), J. Funct. Spaces Appl. 1 (2003), no. 1, 17-33.

P. Harjulehto, P. Hästö, M. Koskenoja, S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal. 25 (2006), no 3, 205-222.

P. Harjulehto, P. Hästö, _U.V. Lê, M. Nuortio, Overview of differential equations with nonstandard growth, Nonlinear Anal. 72 (2010), no. 12, 4551-4574.

P. Harjulehto, P. Hästö, V. Latvala, O. Toivanen, Critical variable exponent functionals in image restoration, Appl. Math. Lett. 26 (2013), no. 1, 56-60.

J. Heinonen, T. Kilpeläinen, O. Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford University Press, Oxford, 1993.

P. Harjulehto, V. Latvala, Fine topology of variable exponent energy superminimizers, Ann. Acad. Sci. Fenn. Math. 33 (2008), 491-510.

R. Harvey and J. Polkingn, A notion of capacity which characterizes removable singularities, Trans. Amer. Math. Soc. 1669 (1972), 183-195.

J.-P. Gossez, Nonlinear elliptic boundary-value problems with rapidly (or slowly) increasing coe_cients, Transactions of the American Mathematical Society 190 (1974), 163-205.

J.-P. Gossez, V. Mustonen, Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Analysis: T.M.A. 11 (1987), 379-392.

J.-P. Gossez, R. Manásevich, On a nonlinear eigenvalue problem in Orlicz-Sobolev spaces, Proc. Roy. Soc. Edinburgh Sect. A Mathematics, 132 (2002), 891-909.

M.A. Kranosel'skii, Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, The Netherlands, 1961.

O. Koväčik, J. Rákosnik, On spaces Lp(x) and Wk;p(x), Czechoslovak Math. J. 41 (116) (1991), 592-618.

P.-L. Lions, Mathematical Topics in Fluid Mechanics, vol. 1, Oxford Lecture Ser. Math. Appl., Oxford University Press, New York, 1996.

R. Landes, On the existence of weak solutions for quasilinear parabolic boundary value problems, Proc. Roy. Soc. Edinburgh Sect. A Mathematics 89 (1981), 217-237.

E. Lanconelli, Sul problema di Dirichlet per l'equazione del calore, Ann. Mat. Pura Appl. 97 (1973), 83-114.

F. Li, Z. Li, L. Pi, Variable exponent functionals in image restoration, Appl. Math. Comput. 216 (2010), no. 3, 870-882.

W. Luxemburg, Banach function spaces, Thesis, Technische Hogeschool te Delft, The Netherlands, 1955.

A. Malusa, A. Prignet, Stability of renormalized solutions of elliptic equations with measure data, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 52 (2004), 151-168.

J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin, 1983.

I. Nyanquini, S. Ouaro, S. Soma, Entropy solution to nonlinear multivalued elliptic problem with variable exponents and measure data, Annals of the University of Craiova, Mathematics and Computer Science Series 40 (2013), no. 2, 1-25.

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 116-162.

S. Ouaro, A. Ouédraogo, Nonlinear parabolic problem with variable exponent and L1 data, Electronic Journal of Di_erential Equations 2017 (2017), no. 32, 1-32.

S. Ouaro, U. Traoré, p(.)-parabolic capacity and decomposition of measures, Annals of the University of Craiova, Mathematics and Computer Science Series 44 (2017), no. 1, 30-63.

M. Pierre, Parabolic capacity and Sobolev spaces, Siam J. Math. Anal. 14 (1983), 522-533.

F. Petitta, Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura ed Appl. 187 (2008), no. 4, 563-604.

A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura ed Appl. (IV) 177 (1999), 143-172.

A. Porretta, Elliptic and parabolic equations with natural growth terms and measure data, PhD Thesis, Universit_a di Roma, 1999.

F. Petitta, A. Porretta, On the notion of renormalized solution to nonlinear parabolic equations with general measure data, Journal of Elliptic and Parabolic Equations 1 (2015), 201-214.

F. Petitta, A. Ponce, A. Porretta, Approximation of diffuse measures for parabolic capacities, C. R. Math. Acad. Sci. Paris 346 (2008), no. 3-4, 161-166.

F. Petitta, A. C. Ponce, A. Porretta, Diffuse measures and nonlinear parabolic equations, J. Evol. Equations 11 (2011), no. 4, 861-905.

A. Prignet, Remarks on existence and uniqueness of solutions of elliptic problems with right hand side measures, Rend. Mat. 15 (1995), 321-337.

A. Prignet, Existence and uniqueness of entropy solutions of parabolic problems with L1 data, Nonlinear Analysis: T.M.A. 28 (1997), 1943-1954.

H. Redwane, Nonlinear parabolic equation with variable exponents and diffuse measure data, J. Nonl. Evol. Equ. Appl. (6) (2019), 95-114.

M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics 146, Marcel Dekker, New York, 1991.

M. Rüžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math. 1748, Springer, Berlin, 2000.

R. Schneider, O. Reichmann and C. Schwab,Wavelet solution of variable order pseudodifferential equations, Calcolo 47 (2010), no. 2, 65-101.

M. Sanch_on, J.M. Urbano, Entropy solutions for the p(x)-Laplace equation, Trans. Amer. Math. Soc. 361 (2009), 6387-6405.

N.A.Watson, Thermal Capacity, Proceedings of the London Mathematical Society s3-37 (1978), no. 2, 342-362.

V.V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), 67-81.

V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Akad. Nauk. SSSR Ser. Mat. 50 (1986), 675-710.

V.V. Zhikov, On some variational problems, Russ. J. Math. Phys. 5 (1997), 105-116.

W.P. Ziemer, Behavior at the boundary of solutions of quasilinear parabolic equations, J. Diff. Eq. 35 (1980), 291-305.

M. Bendahmane, P. Wittbold, A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1 data, J. Diff. Equ. 249 (2010), 1483-1515.

C. Zhang, S. Zhou, Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L1 data, J. Diff. Equations. 248 (2010), 1376-1400.




DOI: https://doi.org/10.52846/ami.v50i1.1619