Existence of a Renormalized Solution to an Anisotropic Parabolic Problem for an Equation with Diffuse Measure

F. Kh. Mukminov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Mathematical physics and applications, Tome 306 (2019), pp. 192-209.

Résumé

The first initial–boundary value problem is considered for a class of anisotropic parabolic equations with variable nonlinearity exponents and a diffuse measure on the right-hand side in a cylindrical domain $(0,T)\times \Omega $. The domain $\Omega $ is bounded. The existence of a renormalized solution is proved.

Détail

Mots-clés : anisotropic parabolic equation, existence of a solution.
Keywords: diffuse measure, renormalized solution, variable nonlinearity exponents

Bibliographie

[1] Yu. A. Alkhutov and V. V. Zhikov, “Existence and uniqueness theorems for solutions of parabolic equations with a variable nonlinearity exponent”, Sb. Math., 205:3 (2014), 307–318 | DOI | DOI | MR | Zbl

[2] Alt H.W., Luckhaus S., “Quasilinear elliptic–parabolic differential equations”, Math. Z., 183 (1983), 311–341 | DOI | MR

[3] Amann H., Quittner P., “Semilinear parabolic equations involving measures and low regularity data”, Trans. Amer. Math. Soc., 356:3 (2004), 1045–1119 | DOI | MR | Zbl

[4] Ammar K., Wittbold P., “Existence of renormalized solutions of degenerate elliptic–parabolic problems”, Proc. R. Soc. Edinb. Sect. A: Math., 133:3 (2003), 477–496 | DOI | MR | Zbl

[5] Azroul E., Benboubker M.B., Rhoudaf M., “Entropy solution for some $p(x)$-quasilinear problem with right-hand side measure”, Afr. Diaspora J. Math., 13:2 (2012), 23–44 | MR | Zbl

[6] Azroul E., Redwane H., Rhoudaf M., “Existence of solutions for nonlinear parabolic systems via weak convergence of truncations”, Electron. J. Diff. Eqns., 2010 (2010), 68 | MR | Zbl

[7] Benboubker M.B., Ouaro S., Traore U., “Entropy solutions for nonlinear nonhomogeneous Neumann problems involving the generalized $p(x)$-Laplace operator and measure data”, J. Nonlinear Evol. Eqns. Appl., 2014:5 (2015), 53–76 | MR

[8] Bénilan P., Boccardo L., Gallouët T., Gariepy R., Pierre M., Vazquez J.L., “An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 4, 22:2 (1995), 241–273 | MR | Zbl

[9] Bénilan P., Brezis H., “Nonlinear problems related to the Thomas–Fermi equation”, J. Evol. Eqns., 3:4 (2003), 673–770 | DOI | MR | Zbl

[10] Benilan P., Brezis H., Crandall M.G., “A semilinear equation in $L^1(\boldsymbol R^N)$”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 4, 2:4 (1975), 523–555 | MR | Zbl

[11] Blanchard D., Murat F., “Renormalised solutions of nonlinear parabolic problems with $L^1$ data: Existence and uniqueness”, Proc. R. Soc. Edinb. Sect. A: Math., 127:6 (1997), 1137–1152 | DOI | MR | Zbl

[12] Blanchard D., Porretta A., “Nonlinear parabolic equations with natural growth terms and measure initial data”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 4, 30:3–4 (2001), 583–622 | MR | Zbl

[13] Boccardo L., Gallouët T., “Nonlinear elliptic and parabolic equations involving measure data”, J. Funct. Anal., 87:1 (1989), 149–169 | DOI | MR | Zbl

[14] Boccardo L., Gallouët T., Orsina L., “Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data”, Ann. Inst. Henri Poincaré. Anal. Non Linéaire, 13:5 (1996), 539–551 | DOI | MR | Zbl

[15] Bouajaja A., Redwane H., Marah A., “Existence and uniqueness of renormalized solutions to nonlinear parabolic equations with lower order term and diffuse measure data”, Mediterr. J. Math., 15:4 (2018), 178 | DOI | MR | Zbl

[16] Brezis H., Veron L., “Removable singularities for some nonlinear elliptic equations”, Arch. Ration. Mech. Anal., 75:1 (1980), 1–6 | DOI | MR | Zbl

[17] Carrillo J., Wittbold P., “Uniqueness of renormalized solutions of degenerate elliptic–parabolic problems”, J. Diff. Eqns., 156:1 (1999), 93–121 | DOI | MR | Zbl

[18] Dal Maso G., Murat F., Orsina L., Prignet A., “Definition and existence of renormalized solutions of elliptic equations with general measure data”, C. r. Acad. sci. Paris. Sér. 1: Math., 325:5 (1997), 481–486 | DOI | MR | Zbl

[19] Dal Maso G., Murat F., Orsina L., Prignet A., “Renormalized solutions of elliptic equations with general measure data”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 4, 28:4 (1999), 741–808 | MR | Zbl

[20] DiPerna R.J., Lions P.L., “On the Cauchy problem for Boltzmann equations: Global existence and weak stability”, Ann. Math. Ser. 2, 130:2 (1989), 321–366 | DOI | MR | Zbl

[21] Dong G., Fang X., “Existence results for some nonlinear elliptic equations with measure data in Orlicz–Sobolev spaces”, Bound. Value Probl., 2015 (2015), 18 | DOI | MR

[22] Droniou J., Gallouët T., “A uniqueness result for quasilinear elliptic equations with measures as data”, Rend. Mat. Appl. Ser. 7, 21 (2001), 57–86 | MR | Zbl

[23] Droniou J., Porretta A., Prignet A., “Parabolic capacity and soft measures for nonlinear equations”, Potential Anal., 19:2 (2003), 99–161 | DOI | MR | Zbl

[24] Dupaigne L., Ponce A.C., Porretta A., “Elliptic equations with vertical asymptotes in the nonlinear term”, J. anal. math., 98:1 (2006), 349–396 | DOI | MR | Zbl

[25] Guibé O., Mercaldo A., “Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data”, Trans. Amer. Math. Soc., 360:2 (2008), 643–669 | DOI | MR | Zbl

[26] A. K. Gushchin, “The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function”, Sb. Math., 203:1 (2012), 1–27 | DOI | DOI | MR | MR | Zbl

[27] Hewitt E., Stromberg K., Real and abstract analysis: A modern treatment of the theory of functions of a real variable, Springer, Berlin, 1965 | MR | Zbl

[28] Igbida N., Ouaro S., Soma S., “Elliptic problem involving diffuse measure data”, J. Diff. Eqns., 253:12 (2012), 3159–3183 | DOI | MR | Zbl

[29] Kozhevnikova L.M., On solutions of anisotropic elliptic equations with variable exponent and measure data, E-print, 2018, arXiv: 1808.09624 [math.AP] | MR | Zbl

[30] S. N. Kružkov, “First order quasilinear equations in several independent variables”, Math. USSR, Sb., 10:2 (1970), 217–243 | DOI | MR | Zbl | Zbl

[31] G. I. Laptev, “Weak solutions of second-order quasilinear parabolic equations with double non-linearity”, Sb. Math., 188:9 (1997), 1343–1370 | DOI | DOI | MR | Zbl

[32] Leray J., Lions J.-L., “Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty–Browder”, Bull. Soc. math. France, 93 (1965), 97–107 | DOI | MR | Zbl

[33] Liang J., Rodrigues J.F., “Quasilinear elliptic problems with nonmonotone discontinuities and measure data”, Port. Math., 53:2 (1996), 239–252 | MR | Zbl

[34] Lukkari T., “The fast diffusion equation with measure data”, Nonlinear Diff. Eqns. Appl., 19:3 (2012), 329–343 | DOI | MR | Zbl

[35] Malusa A., Porzio M.M., “Renormalized solutions to elliptic equations with measure data in unbounded domains”, Nonlinear Anal. Theory Methods Appl., 67:8 (2007), 2370–2389 | DOI | MR | Zbl

[36] Marcus M., Veron L., Boundary trace of positive solutions of semilinear elliptic equations in Lipschitz domains: The subcritical case, E-print, 2010, arXiv: math.AP/1008.4222 [math.AP] | MR

[37] F. Kh. Mukminov, “Uniqueness of the renormalized solution of an elliptic–parabolic problem in anisotropic Sobolev–Orlicz spaces”, Sb. Math., 208:8 (2017), 1187–1206 | DOI | DOI | MR | MR | Zbl

[38] F. Kh. Mukminov, “Existence of a renormalized solution to an anisotropic parabolic problem with variable nonlinearity exponents”, Sb. Math., 209:5 (2018), 714–738 | DOI | DOI | MR | Zbl

[39] Murat F., Soluciones renormalizadas de EDP elipticas no lineales, Preprint 93023, Lab. anal. numer., Univ. Paris VI, Paris, 1993 http://archive.schools.cimpa.info/anciensite/NotesCours/PDF/2009/Alexandrie_Murat_2.pdf

[40] Nyanquini I., Ouaro S., Soma S., “Entropy solution to nonlinear multivalued elliptic problem with variable exponents and measure data”, Ann. Univ. Craiova. Math. Comput. Sci. Ser., 40:2 (2013), 174–198 | MR | Zbl

[41] Orsina L., Prignet A., “Non-existence of solutions for some nonlinear elliptic equations involving measures”, Proc. R. Soc. Edinb. Sect. A: Math., 130:1 (2000), 167–187 | DOI | MR | Zbl

[42] Otto F., “$L^1$-contraction and uniqueness for quasilinear elliptic-parabolic equations”, J. Diff. Eqns., 131:1 (1996), 20–38 | DOI | MR | Zbl

[43] Petitta F., “Renormalized solutions of nonlinear parabolic equations with general measure data”, Ann. Mat. Pura Appl., 187:4 (2008), 563–604 | DOI | MR | Zbl

[44] Petitta F., “A non-existence result for nonlinear parabolic equations with singular measures as data”, Proc. R. Soc. Edinb. Sect. A: Math., 139:2 (2009), 381–392 | DOI | MR | Zbl

[45] Petitta F., Ponce A.C., Porretta A., “Diffuse measures and nonlinear parabolic equations”, J. Evol. Eqns., 11:4 (2011), 861–905 | DOI | MR | Zbl

[46] Porretta A., “Nonlinear equations with natural growth terms and measure data”, Electron. J. Diff. Eqns., 2002, Conf. 09 (2002), 183–202 | MR | Zbl

[47] Prignet A., “Existence and uniqueness of “entropy” solutions of parabolic problems with $L^1$ data”, Nonlinear Anal. Theory Methods Appl., 28:12 (1997), 1943–1954 | DOI | MR | Zbl

[48] Serrin J., “Pathological solutions of elliptic differential equations”, Ann. Sc. Norm. Super. Pisa. Cl. Sci. Ser. 3, 18:3 (1964), 385–387 | MR | Zbl

[49] Smarrazzo F., “On a class of quasilinear elliptic equations with degenerate coerciveness and measure data”, Adv. Nonlinear Stud., 18:2 (2018), 361–392 | DOI | MR | Zbl

[50] Sturm S., “Existence of very weak solutions of doubly nonlinear parabolic equations with measure data”, Ann. Acad. sci. Fenn. Math., 42 (2017), 931–962 | DOI | MR | Zbl

[51] Zaki K., Redwane H., “Nonlinear parabolic equations with blowing-up coefficients with respect to the unknown and with soft measure data”, Electron. J. Diff. Eqns., 2016 (2016), 327 | DOI | MR | Zbl

[52] Zaki K., Redwane H., “Nonlinear parabolic equations with singular coefficient and diffuse data”, Nonlinear Dyn. Syst. Theory, 17:4 (2017), 421–432 | MR | Zbl

[53] Zhang C., Zhou S., “Entropy and renormalized solutions for the $p(x)$-Laplacian equation with measure data”, Bull. Aust. Math. Soc., 82:3 (2010), 459–479 | DOI | MR | Zbl