Geodesic
Extinction of Solutions of Parabolic Equations with Variable Anisotropic Nonlinearities
Informatics and Automation, Differential equations and dynamical systems, Tome 261 (2008), pp. 16-25.

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We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions that generalize the evolutional $p(x,t)$-Laplacian. We study the property of extinction of solutions in finite time. In particular, we show that the extinction may take place even in the borderline case when the equation becomes linear as $t\to\infty$. 
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S. N. Antontsev; S. I. Shmarev. Extinction of Solutions of Parabolic Equations with Variable Anisotropic Nonlinearities. Informatics and Automation, Differential equations and dynamical systems, Tome 261 (2008), pp. 16-25. http://geodesic.mathdoc.fr/item/TRSPY_2008_261_a1/

[1] Acerbi E., Mingione G., Seregin G. A., “Regularity results for parabolic systems related to a class of non-Newtonian fluids”, Ann. Inst. H. Poincaré Anal. Non Lin., 21 (2004), 25–60 | MR | Zbl

[2] Andreu-Vaillo F., Caselles V., Mazón J. M., Parabolic quasilinear equations minimizing linear growth functionals, Progr. Math., 223, Birkhäuser, Basel, 2004 | MR | Zbl

[3] Antontsev S. N., Shmarëv S. I., “Suschestvovanie i edinstvennost reshenii vyrozhdayuschikhsya parabolicheskikh uravnenii s peremennymi pokazatelyami nelineinosti”, Fund. i prikl. mat., 12:4 (2006), 3–19 | MR

[4] Antontsev S. N., Shmarev S. I., “A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions”, Nonlin. Anal. Theory Meth. and Appl., 60 (2005), 515–545 | MR | Zbl

[5] Antontsev S., Shmarev S., “Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions”, Nonlin. Anal. Theory Meth. and Appl., 65 (2006), 728–761 | DOI | MR | Zbl

[6] Antontsev S., Shmarev S., “Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions”, Handbook of differential equations: Stationary partial differential equations, Vol. 3, eds. M. Chipot, P. Quittner, North-Holland, Amsterdam, 2006, 1–100 | Zbl

[7] Antontsev S., Shmarev S., “Parabolic equations with anisotropic nonstandard growth conditions”, Free boundary problems. Theory and applications, Intern. Ser. Numer. Math., 154, Birkhäuser, Basel, 2007, 33–44 | MR | Zbl

[8] Antontsev S. N., Díaz J. I., Shmarev S., Energy methods for free boundary problems: Applications to nonlinear PDEs and fluid mechanics, Progr. Nonlin. Diff. Equat. and Appl., 48, Bikhäuser, Basel, 2002 | MR

[9] Antontsev S. N., Rodrigues J. F., “On stationary thermo-rheological viscous flows”, Ann. Univ. Ferrara. Sez. 7 Sci. Mat., 52 (2006), 19–36 | DOI | MR | Zbl

[10] Antontsev S. N., Shmarëv S. I., “O lokalizatsii reshenii ellipticheskikh uravnenii s neodnorodnym anizotropnym vyrozhdeniem”, Sib. mat. zhurn., 46:5 (2005), 963–984 | MR | Zbl

[11] Chen Y., Levine S., Rao M., “Variable exponent, linear growth functionals in image restoration”, SIAM J. Appl. Math., 66 (2006), 1383–1406 | DOI | MR | Zbl

[12] Diening L., “Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(\cdot)}$ and $W^{k,p(\cdot)}$”, Math. Nachr., 268 (2004), 31–43 | DOI | MR | Zbl

[13] Edmunds D. E., Rákosník J., “Sobolev embeddings with variable exponent”, Stud. math., 143:3 (2000), 267–293 | MR | Zbl

[14] Harjulehto P., Hästö P., “An overview of variable exponent Lebesgue and Sobolev spaces”, Future trends in geometric function theory, Rep. Univ. Jyväskylä Dep. Math. Stat., 92, Univ. Jyväskylä, Jyväskylä, 2003, 85–93 | MR | Zbl

[15] Kováčik O., Rákosník J., “On spaces $L^{p(x)}$ and $W^{k,p(x)}$”, Czechosl. Math. J., 41 (1991), 592–618 | MR | Zbl

[16] Musielak J., Orlicz spaces and modular spaces, Lect. Notes Math., 1034, Springer, Berlin, 1983 | MR | Zbl

[17] Růžička M., Electrorheological fluids: Modeling and mathematical theory, Lect. Notes Math., 1748, Springer, Berlin, 2000 | MR

[18] Samko S., “On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators”, Integr. Transf. and Spec. Funct., 16 (2005), 461–482 | DOI | MR | Zbl

[19] Zhikov V. V., “On some variational problems”, Russ. J. Math. Phys., 5:1 (1997), 105–116 | MR | Zbl

[20] Zhikov V. V., “O plotnosti gladkikh funktsii v prostranstve Soboleva–Orlicha”, Zap. nauch. sem. POMI, 310, 2004, 67–81 | MR | Zbl