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Asymptotic estimates of weak solutions for a parabolic p(x)-Laplace equation with variable exponents and absorption

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Abstract

This paper deals with a parabolic p(x)-Laplace equation with absorption and nonlocal source. Firstly, we study the uniqueness and existence of weak solutions in some Sobolev spaces with variable exponents. Secondly, by using some kinds of ordinary differential inequalities, we obtain the extinction and non-extinction criteria for weak solutions, where the extinction rate is given also. Thirdly, we use the auxiliary function methods and different embedding inequalities to classify blow-up and global existence of solutions under some conditions on the initial data and the diffusion coefficient. Moreover, the estimates on blow-up time and rates of weak solutions and large time behavior of global solutions are studied, respectively.

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References

  1. Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164, 213–259 (2002)

    Article  MathSciNet  Google Scholar 

  2. Antontsev, S.N., Rodrigues, J.F.: On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52, 19–36 (2006)

    Article  MathSciNet  Google Scholar 

  3. Antontsev, S.N., Shmarev, S.I.: Anisotropic parabolic equations with variable nonlinearity. Pub. Math. 53, 355–399 (2009)

    Article  MathSciNet  Google Scholar 

  4. Antontsev, S.N., Shmarev, S.I.: Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-Up, Atlantis Studies in Differential Equations, vol. 4. Atlantis Press, Amsterdam (2015)

    Book  Google Scholar 

  5. Antontsev, S.N., Chipot, M., Shmarev, S.I.: Uniqueness and comparison theorems for solutions of doubly nonlinear parabolic equations with nonstandard growth conditions. Commu. Pure Appl. Anal. 12, 1527–1546 (2013)

    Article  MathSciNet  Google Scholar 

  6. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Math. Appl. 66, 1383–1406 (2006)

    Article  MathSciNet  Google Scholar 

  7. Diening, L., Harjulehto, P., Hästö, P., Rûžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer-Verlag, Heidelberg (2011)

    Book  Google Scholar 

  8. Guo, B., Zhang, J.J., Liao, M.L.: Classification of blow-up and global existence of solutions to an initial Neumann problem. arXiv:2009.04624

  9. Guo, B., Gao, W.J.: Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the \(p(x, t)\)-Laplace operator and a non-local term. Discrete Contin. Dyn. Syst. 36, 715–730 (2016)

    Article  MathSciNet  Google Scholar 

  10. Li, Y., Zhang, Z.C., Zhu, L.P.: Classification of certain qualitative properties of solutions for the quasilinear parabolic equations. Sci. China Math. 61, 855–868 (2018)

    Article  MathSciNet  Google Scholar 

  11. Nhan, L.C., Chuong, L.X.: Existence and nonexistence of global solutions for doubly nonlinear diffusion equations with logarithmic nonlinearity. Electron. J. Qual. Theory Differ. Equ. 67, 1–25 (2018)

    Article  MathSciNet  Google Scholar 

  12. Nhan, L.C., Chuong, Q.V., Truong, L.X.: Potential well method for \(p(x)\)-Laplacian equations with variable exponent sources. Nonlinear Anal. 56, 103155 (2020)

    Article  MathSciNet  Google Scholar 

  13. Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22, 273–303 (1975)

    Article  MathSciNet  Google Scholar 

  14. Sattinger, D.H.: Stability of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal 28, 226–244 (1968)

    Article  MathSciNet  Google Scholar 

  15. Wang, Y.X., Liu, B.C., Sun, Y.R.: Asymptotic property of singular solutions in some nonstandard parabolic equation. Nonlinear Anal. 60, 103301 (2021)

    Article  MathSciNet  Google Scholar 

  16. Zhou, J., Yang, D.: Upper bound estimate for the blow-up time of an evolution \(m\)-Laplace equation involving variable source and positive initial energy. Comput. Math. Appl. 69, 1463–1469 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors would like to express their sincerely thanks to the Editors and the Reviewers for the constructive comments to improve this paper.

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  1. College of Science, China University of Petroleum, Qingdao, 266580, People’s Republic of China

    Bingchen Liu, Ke Li & Fengjie Li

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  1. Bingchen Liu
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  2. Ke Li
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  3. Fengjie Li
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Correspondence to Fengjie Li.

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This paper is supported by Shandong Provincial Natural Science Foundation (ZR2021MA003, ZR2020MA020)

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Liu, B., Li, K. & Li, F. Asymptotic estimates of weak solutions for a parabolic p(x)-Laplace equation with variable exponents and absorption. Anal.Math.Phys. 12, 47 (2022). https://doi.org/10.1007/s13324-022-00659-9

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  • DOI: https://doi.org/10.1007/s13324-022-00659-9

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