Abstract
This paper is concerned with nonlinear diffusion equations driven by the p(·)-Laplacian with variable exponents in space. The well-posedness is first checked for measurable exponents by setting up a subdifferential approach. The main purposes are to investigate the large-time behavior of solutions as well as to reveal the limiting behavior of solutions as p(·) diverges to the infinity in the whole or in a subset of the domain. To this end, the recent developments in the studies of variable exponent Lebesgue and Sobolev spaces are exploited, and moreover, the spatial inhomogeneity of variable exponents p(·) is appropriately controlled to obtain each result.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Acerbi E., Mingione G.: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001)
Acerbi E., Mingione G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164, 213–259 (2002)
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 Linéaire 21, 25–60 (2004)
Akagi G.: Convergence of functionals and its applications to parabolic equations. Abstr. Appl. Anal. 2004, 907–933 (2004)
Akagi, G., Matsuura, K.: Well-posedness and large-time behaviors of solutions for a parabolic equation involving p(x)-Laplacian. “The Eighth International Conference on Dynamical Systems and Differential Equations,” a supplement volume of Discrete and Continuous Dynamical Systems, pp.22–31 (2011)
Akagi G., Ôtani M.: Time-dependent constraint problems arising from macroscopic critical-state models for type-II superconductivity and their approximations. Adv. Math. Sci. Appl. 14, 683–712 (2004)
Antontsev, S., Shmarev, S.: Extinction of solutions of parabolic equations with variable anisotropic nonlinearities. Tr. Mat. Inst. Steklova 261. Differ. Uravn. i Din. Sist., pp. 16–25 (2008)
Antontsev S., Shmarev S.: Anisotropic parabolic equations with variable nonlinearity. Publ. Mat. 53, 355–399 (2009)
Antontsev S., Shmarev S.: Vanishing solutions of anisotropic parabolic equations with variable nonlinearity. J. Math. Anal. Appl. 361, 371–391 (2010)
Antontsev S., Shmarev S.: Blow-up of solutions to parabolic equations with nonstandard growth conditions. J. Comput. Appl. Math. 234, 2633–2645 (2010)
Aronsson G., Evans L.C., Wu Y.: Fast/slow diffusion and growing sandpiles. J. Differ. Equ. 131, 304–335 (1996)
Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston (1984)
Barrett J.W., Prigozhin L.: Bean’s critical-state model as the p → ∞ limit of an evolutionary p-Laplacian equation. Nonlinear Anal. 42, 977–993 (2000)
Berryman J.G., Holland C.J.: Stability of the separable solution for fast diffusion. Arch. Ration. Mech. Anal. 74, 379–388 (1980)
Bendahmane M., Wittbold P., Zimmermann A.: Renormalized solutions for a nonlinear parabolic equation with variable exponents and L 1-data. J. Differ. Equ. 249, 1483–1515 (2010)
Brézis, H.: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. Math Studies, vol.5. North-Holland, Amsterdam (1973)
Chen Y., Levine S., Rao M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)
DiBenedetto, E.: Degenerate Parabolic Equations. Universitext, Springer-Verlag, New York (1993)
Diening L., Ettwein F., Růžička M.: C 1,α-regularity for electrorheological fluids in two dimensions. Nonlinear Differ. Equ. Appl. 14, 207–217 (2007)
Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017., Springer, Heidelberg (2011)
Ettwein F., Růžička M.: Existence of local strong solutions for motions of electrorheological fluids in three dimensions. Comput. Math. Appl. 53, 595–604 (2007)
Edmunds D.E., Rákosník J.: Sobolev embeddings with variable exponent. Studia Math. 143, 267–293 (2000)
Edmunds D.E., Rákosník J.: Sobolev embeddings with variable exponent II. Math. Nachr. 246/247, 53–67 (2002)
Fan X., Shen J., Zhao D.: Sobolev embedding theorems for spaces W k,p(x)(Ω). J. Math. Anal. Appl. 262, 749–760 (2001)
Fan X., Zhao D.: On the spaces L p(x)(Ω) and W m,p(x)(Ω). J. Math. Anal. Appl. 263, 424–446 (2001)
Fan X., Zhao Y., Zhao D.: Compact imbedding theorems with symmetry of Strauss-Lions type for the space W 1,p(x)(Ω). J. Math. Anal. Appl. 255, 333–348 (2001)
Fu Y., Pan N.: Existence of solutions for nonlinear parabolic problem with p(x)-growth. J. Math. Anal. Appl. 362, 313–326 (2010)
Harjulehto P., Hästö P., Lê Ú.-V., Nuortio M.: Overview of differential equations with non-standard growth. Nonlinear Anal. 72, 4551–4574 (2010)
Kováčik O., Rákosník J.: On spaces L p(x) and W k,p(x). Czechoslovak Math. J. 41, 592–618 (1991)
Manfredi J.J., Rossi J.D., Urbano J.M.: p(x)-harmonic functions with unbounded exponent in a subdomain. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 2581–2595 (2009)
Manfredi J.J., Rossi J.D., Urbano J.M.: Limits as p(x)→∞ of p(x)-harmonic functions. Nonlinear Anal. 72, 309–315 (2010)
Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, vol. 1034. Springer, Berlin (1983)
Qi Y.-W.: The degeneracy of a fast-diffusion equation and stability. SIAM J. Math. Anal. 27, 476–485 (1996)
Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748 Springer, Berlin (2000)
Samko S.: On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integr. Transforms Spec. Funct. 16, 461–482 (2005)
Sanchón M., Urbano J.M.: Entropy solutions for the p(x)-Laplace equation. Trans. Am. Math. Soc. 361, 6387–6405 (2009)
Urbano J.M., Vorotnikov D.: On the well-posedness of a two-phase minimization problem. J. Math. Anal. Appl. 378, 159–168 (2011)
Zhang C., Zhou S.: Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L 1 data. J. Differ. Equ. 248, 1376–1400 (2010)
Zhikov V.V.: On the technique for passing to the limit in nonlinear elliptic equations. Funct. Anal. Appl. 43, 96–112 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Tomomi Kojo on the occasion of his 60th birthday.
This work was supported in part by the Shibaura Institute of Technology grant for Project Research, and the Grant-in-Aid for Young Scientists (B) (No. 22740093), Ministry of Education, Culture, Sports, Science and Technology.
Rights and permissions
About this article
Cite this article
Akagi, G., Matsuura, K. Nonlinear diffusion equations driven by the p(·)-Laplacian. Nonlinear Differ. Equ. Appl. 20, 37–64 (2013). https://doi.org/10.1007/s00030-012-0153-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00030-012-0153-6