Abstract
We consider the first mixed problem for a certain class of anisotropic parabolic equations of the form \( {\left(\beta \left(x,u\right)\right)}_t^{\prime }-\operatorname{div}\kern0.5em a\left(t,x,u,\nabla u\right)-b\left(t,x,u,\nabla u\right)=u \) where μ is a measure and the coefficients contain nonpower nonlinearities in the cylindrical domain DT = (0, T)×Ω, where Ω ⊂ ℝn is a bounded domain. We prove the existence of a renormalized solution of the problem for gt = 0 and a function β(x, r), which increases with respect to r and satisfies the Carath´eodory condition.
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References
Yu. A. Alkhutov and V. V. Zhikov, “Existence and uniqueness theorems for solutions of parabolic equations with a variable nonlinearity exponent,” Mat. Sb., 205, No. 3, 3–14 (2014).
H. W. Alt and S. Luckhaus, “Quasilinear elliptic-parabolic differential equations,” Math. Z., 183, No. 3, 311–341 (1983).
E. Azroul, H. Redwane, and M. Rhoudaf, “Existence of solutions for nonlinear parabolic systems via weak convergence of truncations,” Electron. J. Differ. Equations, 68, 1–18 (2010).
M. Bendahmane, P. Wittbold, and A. Zimmermann, “Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1 data,” J. Differ. Equations, 249, No. 6, 1483– 1515 (2010).
P. Benilan, L. Boccardo, T. Galluet, M. Pierre, and J. L. Vazquez, “An L1-theory of existence and Uniqueness of solutions of nonlinear elliptic equations,” Ann. Scu. Norm. Super. Pisa. Cl. Sci., 22, No. 2, 241–273 (1995).
P. Benilan, H. Brezis, and M. Crandall, “A semilinear elliptic equation in L1 (RN),” Ann. Sco. Norm. Super. Pisa. Cl. Sci., No. 2, 523–555 (1975).
D. Blanchard and F. Murat, “Renormalised solutions of nonlinear parabolic problems with L1 data: existence and uniqueness,” Proc. Roy. Soc. Edinburgh. Sect. A, 127, No. 6, 1137–1152 (1997).
H. Brezis, “Nonlinear elliptic equations involving measures,” in: Contributions to Nonlinear Partial Differential Equations. Madrid, 1981, Res. Notes Math., Pitman, Boston (1983), pp. 82–89.
J. Carrillo and P. Wittbold, “Uniqueness of renormalized solutions of degenerate elliptic-parabolic problems,” J. Differ. Equations, 156, No. 1, 93–121 (1999).
G. Dal Maso, F. Murat, L. Orsina, and A. Prignet, “Renormalized solutions for elliptic equations with general measure data,” Ann. Scu. Norm. Super. Pisa. Cl. Sci., 28, No. 4, 741–808 (1999).
R. J. Di Perna and P. L. Lions, “On the Cauchy problem for Boltzmann equations: global existence and weak stability,” Ann. Math., 130, No. 2, 321–366 (1989).
J. Droniou, A. Porretta, and A. Prignet, “Parabolic capacity and soft measures for nonlinear equations,” Potential Anal., 19, No. 2, 99–161 (2003).
N. Dunford and J. T. Schwartz, Linear Operators. General Theory, Interscience, New York–London (1958).
L. Dupaigne, A. C. Ponce, and A. Porretta, “Elliptic equations with vertical asymptotes in the nonlinear term,” J. Anal. Math., 98, No. 1, 349–396 (2006).
J. P. Gossez, “Some approximation properties in Orlicz–Sobolev spaces,” Stud. Math., 74, No. 1, 17–24 (1982).
J. P. Gossez, “Nonlinear elliptic boundary-value problems for rapidly or slowly increasing coefficients,” Trans. Am. Math. Soc., 190, No. 4, 163–205 (1974).
P. Gwiazda, P. Wittbold, A. Wroblewska-Kaminskac, and A. Zimmermann, “Renormalized solutions to nonlinear parabolic problems in generalized Musielak–Orlicz spaces,” Nonlin. Anal., 129, No. 6, 1–36 (2015).
E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlng, Berlin–Heidelberg–New York (1965).
N. Igbida, S. Ouaro, and S. Soma, “Elliptic problem involving diffuse measures data,” J. Differ. Equations, 253, No. 12, 3159–3183 (2012).
J. Kačur, “On a solution of degenerate elliptic-parabolic systems in Orlicz–Sobolev spaces, I,” Math. Z., 203, No. 1, 153–171 (1990).
L. M. Kozhevnikova, “On entropy solutions of anisotropic elliptic equations with variable nonlinearity exponents in unbounded domains,” Sovr. Mat. Fundam. Napr., 63, No. 3, 475–493 (2017).
M. A. Krasnosel’sky and Ya. B. Rutitsky, Convex Functions and Orlicz Spaces [Russian], GIFML, Moscow (1958).
S. N. Kruzhkov, “First-order quasilinear equations with many independent variables,” Mat. Sb., 81 (123), No. 2, 228–255 (1970).
R. Landes, “On the existence of weak solutions for quasilinear parabolic initial-boundary value problems,” Proc. Roy. Soc. Edinburgh Sect. A, 89, No. 3–4, 217–237 (1981).
G. I. Laptev, “Weak solutions of second-order quasilinear parabolic equations with double nonlinearity,” Mat. Sb., 188, No. 9, 83–112 (1997).
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaire, Dunod–Gauthier-Villars, Paris (1969).
F. Kh. Mukminov, “Uniqueness of a renormalized solution of the first mixed problem for an anisotropic parabolic equation with variable nonlinearities,” Mat. Sb., 208, No. 8, 83–112 (2017).
L. Orsina and A. Prignet, “Non-existence of solutions for some nonlinear elliptic equations involving measures,” Proc. Roy. Soc. Edinburgh. Sect. A, 130, No. 1, 167–187 (2000).
F. Petitta, “A nonexistence result for nonlinear parabolic equations with singular measures as data,” Proc. Roy. Soc. Edinburgh. Sect. A., 139, No. 2, 381–392 (2009).
A. Prignet, “Existence and uniqueness of entropy solutions of parabolic problems with L1 data,” Nonlin. Anal. Theor. Math. Appl., 28, No. 12, 1943–1954 (1997).
H. Redwane, “Existence results for a class of nonlinear parabolic equations in Orlicz spaces,” Electron. J. Qualit. Theory Differ. Equations, No. 2, 1–19 (2010).
Ch. Zhang and Sh. Zhou, “Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L1 data,” J. Differ. Equations, 248, No. 6, 1376–1400 (2010).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 163, Differential Equations, 2019.
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Vorob’yov, N.A., Mukminov, F.K. Existence of a Renormalized Solution of a Parabolic Problem in Anisotropic Sobolev–Orlicz Spaces. J Math Sci 258, 37–64 (2021). https://doi.org/10.1007/s10958-021-05535-8
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DOI: https://doi.org/10.1007/s10958-021-05535-8
Keywords and phrases
- anisotropic parabolic equation
- renormalized solution
- nonpower nonlinearity
- existence of solutions
- N-function