Existence of a renormalized solution of a parabolic problem in anisotropic Sobolev--Orlicz spaces

N. A. Vorobyev ; F. Kh. Mukminov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations, Tome 163 (2019), pp. 39-64.

Résumé

We consider the first mixed problem for a certain class of anisotropic parabolic equations of the form $$ (\beta(x,u))'_t-\operatorname{div} a(t,x,u,\nabla u) -b(t,x,u,\nabla u)=\mu $$ where $\mu$ is a measure and the coefficients contain noonpower nonlinearities in the cylindrical domain $D^T=(0,T)\times\Omega$, where $\Omega\subset \mathbb{R}^n$ is a bounded domain. We prove the existence of a renormalized solution of the problem for $g_t=0$ and a function $\beta(x,r)$, which increases with respect to $r$ and satisfies the Carathéodory condition.

Détail

Mots-clés : anisotropic parabolic equation, existence of solutions
Keywords: renormalized solution, nonpower nonlinearity, $N$-function.

Bibliographie

[1] Alkhutov Yu. A., Zhikov V. V., “Teoremy suschestvovaniya i edinstvennosti reshenii parabolicheskikh uravnenii s peremennym poryadkom nelineinosti”, Mat. sb., 205:3 (2014), 3–14 | DOI | MR | Zbl

[2] Danford N., Shvarts Dzh. T., Lineinye operatory. Obschaya teoriya, IL, M., 1962

[3] Kozhevnikova L. M., “Ob entropiinykh resheniyakh anizotropnykh ellipticheskikh uravnenii s peremennymi pokazatelyami nelineinostei v neogranichennykh oblastyakh”, Sovr. mat. Fundam. napravl., 63:3 (2017), 475–493 | MR

[4] Krasnoselskii M. A., Rutitskii Ya. B., Vypuklye funktsii i prostranstva Orlicha, GIFML, M., 1958 | MR

[5] Kruzhkov S. N., “Kvazilineinye uravneniya pervogo poryadka so mnogimi nezavisimymi peremennymi”, Mat. sb., 81 (123):2 (1970), 228–255 | Zbl

[6] Laptev G. I., “Slabye resheniya kvazilineinykh parabolicheskikh uravnenii vtorogo poryadka s dvoinoi nelineinostyu”, Mat. sb., 188:9 (1997), 83–112 | DOI | MR | Zbl

[7] Lions Zh. L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972

[8] Mukminov F. Kh., “Edinstvennost renormalizovannogo resheniya pervoi smeshannoi zadachi dlya anizotropnogo parabolicheskogo uravneniya s peremennymi nelineinostyami”, Mat. sb., 208:8 (2017), 83–112 | DOI

[9] Alt H. W., Luckhaus S., “Quasilinear elliptic-parabolic differential equations”, Math. Z., 183:3 (1983), 311–341 | DOI | MR | Zbl

[10] Azroul E., Redwane H., Rhoudaf M., “Existence of solutions for nonlinear parabolic systems via weak convergence of truncations”, Electron. J. Differ. Equations., 68 (2010), 1–18 | MR

[11] Bendahmane M., Wittbold P., Zimmermann A., “Renormalized solutions for a nonlinear parabolic equation with variable exponents and $L_1$ data”, J. Differ. Equations., 249:6 (2010), 1483–1515 | DOI | MR | Zbl

[12] Benilan P., Boccardo L., Galluet T., Pierre M., Vazquez J. L., “An $L_1$-theory of existence and Uniqueness of solutions of nonlinear elliptic equations”, Ann. Scu. Norm. Super. Pisa. Cl. Sci., 22:2 (1995), 241–273 | MR | Zbl

[13] Benilan P., Brezis H., Crandall M., “A semilinear elliptic equation in $L_1$ (RN)”, Ann. Sco. Norm. Super. Pisa. Cl. Sci., 1975, no. 2, 523–555 | MR | Zbl

[14] Blanchard D., Murat F., “Renormalised solutions of nonlinear parabolic problems with $L_1$ data: existence and uniqueness”, Proc. Roy. Soc. Edinburgh. Sect. A., 127:6 (1997), 1137–1152 | DOI | MR | Zbl

[15] Brezis H., “Nonlinear elliptic equations involving measures”, Contributions to Nonlinear Partial Differential Equations. Madrid, 1981, Res. Notes Math, Pitman, Boston, 1983, 82–89 | MR

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

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

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

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

[20] 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

[21] Gossez J. P., “Some approximation properties in Orlicz–Sobolev spaces”, Stud. Math., 74:1 (1982), 17–24 | DOI | MR | Zbl

[22] Gossez J. P., “Nonlinear elliptic boundary-value problems for rapidly or slowly increasing coefficients”, Trans. Am. Math. Soc., 190:4 (1974), 163–205 | DOI | MR | Zbl

[23] Gwiazda P., Wittbold P., Wroblewska-Kaminskac A., Zimmermann A., “Renormalized solutions to nonlinear parabolic problems in generalized Musielak–Orlicz spaces”, Nonlin. Anal., 129:6 (2015), 1–36 | DOI | MR | Zbl

[24] Hewitt E., Stromberg K., Real and Abstract Analysis, Springer-Verlng, Berlin–Heidelberg–New York, 1965 | MR | Zbl

[25] Igbida N., Ouaro S., Soma S., “Elliptic problem involving diffuse measures data”, J. Differ. Equations., 253:12 (2012), 3159–3183 | DOI | MR | Zbl

[26] Kačur J., “On a solution of degenerate elliptic-parabolic systems in Orlicz–Sobolev spaces, I”, Math. Z., 203:1 (1990), 153–171 | DOI | MR | Zbl

[27] Landes R., “On the existence of weak solutions for quasilinear parabolic initial-boundary value problems”, Proc. Roy. Soc. Edinburgh Sect. A., 89:3–4 (1981), 217–237 | DOI | MR | Zbl

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

[29] Petitta F., “A nonexistence result for nonlinear parabolic equations with singular measures as data”, Proc. Roy. Soc. Edinburgh. Sect. A., 139:2 (2009), 381–392 | DOI | MR | Zbl

[30] Prignet A., “Existence and uniqueness of entropy solutions of parabolic problems with $L_1$ data”, Nonlin. Anal. Theor. Math. Appl., 28:12 (1997), 1943–1954 | DOI | MR | Zbl

[31] Redwane H., “Existence results for a class of nonlinear parabolic equations in Orlicz spaces”, Electron. J. Qualit. Theory Differ. Equations., 2010, no. 2, 1–19 | MR

[32] Zhang Ch., Zhou Sh., “Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and $L_1$ data”, J. Differ. Equations., 248:6 (2010), 1376–1400 | DOI | MR | Zbl