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Propagation of Perturbations in Quasilinear Multidimensional Parabolic Equations with Convective Term

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We establish estimates for the initial evolution of the supports of solutions of a broad class of quasilinear parabolic equations of arbitrary order that have the structure of the equation of strong nonlinear convective diffusion.

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REFERENCES

  1. A. S. Kalashnikov, “On some problems in the qualitative theory of nonlinear second-order degenerating parabolic equations,” Usp. Mat. Nauk., 42, No. 2, 135–176 (1987).

    Google Scholar 

  2. B. H. Gilding and L. A. Peletier, “The Cauchy problem for an equation in theory of infiltration,” Arch. Ration. Mech. Anal., 61, 127–140 (1976).

    Google Scholar 

  3. B. H. Gilding, “Properties of solutions of an equation in the theory of infiltration,” Arch. Ration. Mech. Anal., 65, 203–225 (1977).

    Google Scholar 

  4. R. E. Grundy, “Asymptotic solution of a model nonlinear convective diffusion equation,” IMA J. Appl. Math., 31, 121–137 (1983).

    Google Scholar 

  5. L. Alvarez, J. I. Diaz, and R. Kersner, “On the initial growth of the interfaces in nonlinear diffusion-convection processes,” in: W.-M. Ni, L. A. Peletier, and J. Serrin (editors), Nonlinear Diffusion Equations and Their Equilibrium States. I, Springer, New York (1987), pp. 1–20.

    Google Scholar 

  6. L. Alvarez and J. I. Diaz, “Sufficient and necessary initial mass conditions for the existence of a waiting time in nonlinear-convection processes,” J. Math. Anal. Appl., 155, 378–393 (1991).

    Google Scholar 

  7. J. I. Diaz and R. Kersner, “Non existence d'une des frontieres libres dans une equation degeneree en theorie de la filtration,” C. R. Acad. Sci., 296, 505–508 (1983).

    Google Scholar 

  8. B. H. Gilding, The Occurrence of Interfaces in Nonlinear Diffusion-Advection Processes, Memorandum 595, Dep. Appl. Math., Twente University of Technology (1986).

  9. B. H. Gilding and A. E. Shishkov, The Effect of a Convection Term on the Propagation Properties of a Nonlinear Parabolic Equation, Preprint, Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk (1998).

    Google Scholar 

  10. H. Alt and S. Luckhaus, “Quasilinear elliptic-parabolic differential equations,” Math. Z., 183, 311–341 (1983).

    Google Scholar 

  11. P. Benilan and O. Wittbold, “On mild and weak solutions of elliptic-parabolic problems,” Adv. Different. Equat., 1, 1053–1073 (1996).

    Google Scholar 

  12. F. Bernis, “Existence results for doubly nonlinear higher order parabolic equations on unbounded domains,” Math., 279, 373–394 (1988).

    Google Scholar 

  13. F. Bernis, “Qualitative properties for some nonlinear higher order degenerate parabolic equations,” Math., 14, No. 3, 319–352 (1988).

    Google Scholar 

  14. A. E. Shishkov, “Propagation of perturbations in a singular Cauchy problem for quasilinear degenerating parabolic equations,” Mat. Sb., 187, No. 9, 139–160 (1996).

    Google Scholar 

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Sapronov, D.A., Shishkov, A.E. Propagation of Perturbations in Quasilinear Multidimensional Parabolic Equations with Convective Term. Ukrainian Mathematical Journal 53, 1134–1155 (2001). https://doi.org/10.1023/A:1013381316300

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