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One-dimensional viscous diffusion equation of higher order with gradient dependent potentials and sources

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Abstract

In this paper we consider the initial boundary value problem of a higher order viscous diffusion equation with gradient dependent potentials Φ(s) and sources A(s). We first show the general existence and uniqueness of global classical solutions provided that the first order derivatives of both Φ(s) and A(s) are bounded below. Such a restriction is almost necessary, namely, if one of the derivatives is unbounded from below, then the solution might blow up in a finite time. A more interesting phenomenon is also revealed for potentials or sources being unbounded from below. In fact, if either the source or the potential is dominant, then the solution will blow up definitely in a finite time. Moreover, the viscous coefficient might postpone the blow-up time. Exactly speaking, for any T > 0, the solution will never blow up during the period 0 < t < T, so long as the viscous coefficient is large enough.

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References

  1. Bai, F., Elliott, C. M., Gardiner, A., et al.: The viscous Cahn–Hilliard equation. I. Computations. Nonlinearity, 8, 131–160 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barenblatt, G., Bertsch, M., Passo, R. D., et al.: A degenerate pseudo-parabolic regularization of a nonlinear forward backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow. SIAM J. Math. Anal., 24, 1414–1439 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barenblatt, G., Zheltov, I. P., Kochina, I. N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [Strata]. J. Appl. Math. Mech., 24, 1286–1303 (1960)

    Article  MATH  Google Scholar 

  4. Benjamin, T. B., Bona, J. L., Mahony, J. J.: Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 272, 47–78 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cao, Y., Yin, J. X., Wang, C. P.: Cauchy problems of semilinear pseudo-parabolic equations. J. Differential Equations, 246, 4568–4590 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Carvalho, A. N., Dlotko, T.: Dynamics of the viscous Cahn–Hilliard equation. J. Math. Anal. Appl., 344, 703–725 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, P. J., Gurtin, M. E.: On a theory of heat conduction involving two temperatures. Z. Angew. Math. Phys., 19, 614–627 (1968)

    Article  MATH  Google Scholar 

  8. Choo, S. M., Chung, S. K.: Asymtotic behaviour of the viscous Cahn–Hilliard equation. J. Appl. Math. Comput., 11, 143–154 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. [9] Düll, W.: Some qualitative properties of solutions to a pseudoparabolic equation modeling solvent uptake in polymeric solids. Comm. Partial Differential Equations, 31, 1117–1138 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Efendiev, M., Miranville, A., Zelik, S.: Exponential attractors for a singularly perturbed Cahn–Hilliard system. Math. Nachr., 272, 11–31 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Elliott, C. M., Kostin, I. M.: Lower semicontinuity of a non-hyperbolic attractor for the viscous Cahn–Hilliard equation. Nonlinearity, 9, 687–702 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Elliott, C. M., Stuart, A. M.: Viscous Cahn–Hilliard equation. II. Analysis. J. Differential Equations, 128, 387–414 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Elliott, C. M., Zheng, S. M.: On the Cahn–Hilliard equation. Arch. Ration. Mech. Anal., 96, 339–357 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  14. Friedman, A.: Partial Differential Equations, Holt, Rinehart and Winston, 1973

    MATH  Google Scholar 

  15. Gal, C., Miranville, A.: Uniform global attractors for non-isothermal viscous and non-viscous Cahn–Hilliard equations with dynamic boundary conditions. Nonlinear Anal. Real World Appl., 10, 1738–1766 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grasselli, M., Petzeltova, H., Schimperna, G.: Asymptotic behavior of a nonisothermal viscous Cahn–Hilliard equation with inertial term. J. Differential Equations, 239, 38–60 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gurtin, M. E.: Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Phys. D, 92, 178–192 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jin, C. H., Yin, J. X., Yin, L.: Existence and blow-up of solutions of a fourth-order nonlinear diffusion equation. Nonlinear Anal. Real World Appl., 9, 2313–2325 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ke, Y. Y., Yin, J. X.: A note on the viscous Cahn–Hilliard equation. Northeast. Math. J., 20, 101–108 (2004)

    MathSciNet  MATH  Google Scholar 

  20. King, B. B., Stein, O., Winkler, M.: A fourth-order parabolic equation modeling epitaxial thin film growth. J. Math. Anal. Appl., 286, 459–490 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ladyženskaja, O., Solonnikov, V., Ural’tseva, N.: Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, 1968

  22. Liu, C. C., Yin, J. X.: Some properties of solutions for viscous Cahn–Hilliard equation. Northeast. Math. J., 14, 455–466 (1998)

    MathSciNet  MATH  Google Scholar 

  23. Liu, Y. C., Wang, F.: A class of multidimensional nonlinear Sobolev–Galpern equations. Acta Math. Appl. Sin., 17, 569–577 (1994)

    MathSciNet  Google Scholar 

  24. Miranville, A., Piétrus, A., Rakotoson, J. M.: Dynamical aspect of a generalized Cahn–Hilliard equation based on a microforce balance. Asymptot. Anal., 16, 315–345 (1998)

    MathSciNet  MATH  Google Scholar 

  25. Murray, J. D.: Mathematical Biology, Second Edition, Springer-Verlag, Berlin, 1993

    Book  MATH  Google Scholar 

  26. Nicolaenko, B., Scheurer, B., Temam, R.: Some global dynamical properties of a class of pattern formation equations. Comm. Partial Differential Equations, 14, 245–297 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  27. Novick-Cohen, A., Peletier, L.: Steady states of the one-dimensional Cahn–Hilliard equation. Proc. Roy. Soc. Edinburgh Sect. A, 123(6), 1071–1098 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  28. Padron, V.: Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation. Trans. Amer. Math. Soc., 356, 2739–2756 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Rybka, P., Hoffmann, L. K. H.: Convergence of solutions to Cahn–Hilliard equation. Comm. Partial Differential Equations, 24(5–6), 1055–1077 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sell, G., You, Y.: Dynamics of Evolutionary Equations, Springer, New York, 2001

    MATH  Google Scholar 

  31. Shang, Y. D.: Blow-up of solutions for the nonlinear Sobolev–Galpern equations. Math. Appl. (Wuhan), 13, 35–39 (2000)

    MathSciNet  MATH  Google Scholar 

  32. Ting, T. W.: Certain non-steady flows of second order fluids. Arch. Ration. Mech. Anal., 14, 1–26 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zheng, S., Milani, A.: Global attractors for singular perturbations of the Cahn–Hilliard equations. J. Differential Equations, 209, 101–139 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, P. R. China

    Yang Cao

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P. R. China

    Jing Xue Yin & Ying Hua Li

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  1. Yang Cao
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  2. Jing Xue Yin
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  3. Ying Hua Li
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Correspondence to Ying Hua Li.

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Supported by the NSF of China (Grant Nos. 11371153, 11471127, 11571062, 11671155 and 11771156) NSF of Guangdong (Grant No. 2016A030313418), NSF of Guangzhou (Grant Nos. 201607010207 and 201707010136) and the Fundamental Research Funds for the Central Universities (Grant No. DUT16LK01)

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Cao, Y., Yin, J.X. & Li, Y.H. One-dimensional viscous diffusion equation of higher order with gradient dependent potentials and sources. Acta. Math. Sin.-English Ser. 34, 959–974 (2018). https://doi.org/10.1007/s10114-017-7245-5

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  • DOI: https://doi.org/10.1007/s10114-017-7245-5

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