Abstract
We investigate the long-time behaviour of the solutions to the Cauchy problem
with a nonnegative initial data in L 1 (ℝ),when q ∈ (0, 1)and m ⩾ 1. We prove that the long-time profile in L 1(ℝ)of these solutions is given by the unique nonnegative entropy sourcetype solution to the conservation law u t}+(uq)x =0 with the same mass. Uniqueness of such a solution is previously established. These results extend previous results obtained for the case q>1 and m⩾1.
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References
Ph. Bénilan -H. Touré,Sur l'équation générale u t=ϕ(u)xx−ψ(u)x+v, C. R. Acad. Sci. Paris Sér. I Math.,299 (1984), pp. 919–922.
Ph. Bénilan -H. Touré,Sur l'équation générale u t=a(·, u, ϕ(·, u)x)x+v dans L1, Ann. Inst. H. Poincaré Anal. Non Linéaire,12 (1995), pp. 727–761.
A. Carpio,Unicité et comportement asymptotique pour des équations de convection-diffusion scalaires, C. R. Acad. Sci. Paris Sér. I Math.,319 (1994), pp. 51–56.
J. I. Diaz -R. Kersner,Non existence d'une des frontières libres dans une équation dégénérée en théorie de la filtration, C. R. Acad. Sci. Paris Sér. I Math.,296 (1983), pp. 505–508.
J. I. Diaz -R. Kersner,On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium, J. Diff. Eq.,69 (1987), pp. 368–403.
N. Dunford -J. T. Schwartz,Linear operators. Part I: general theory, Interscience Publishers, Inc., New York (1958).
M. Escobedo -J. L. Vazquez -E. Zuazua,Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal.,124 (1993), pp. 43–65.
M. Escobedo -J. L. Vazquez -E. Zuazua,A diffusion-convection equation in several space dimensions, Indiana Univ. Math. J.,42 (1993), pp. 1413–1440.
M. Escobedo -E. Zuazua,Large time behaviour for convection-diffusion equations in ℝn, J. Funct. Anal.,100 (1991), pp. 119–161.
B. H. Gilding,A nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4),4 (1977), pp. 393–432.
B. H. Gilding,The occurrence of interfaces in nonlinear diffusion-advection processes, Arch. Rational Mech. Anal.,100 (1988), pp. 243–263.
B. H. Gilding,Improved theory for a nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4),16 (1989), pp. 165–224.
B. H. Gilding -L. A. Peletier,The Cauchy problem for an equation in the theory of infiltration, Arch. Rational Mech. Anal.,61 (1976), pp. 127–140.
S. N. Kruzhkov,Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications, Math. Notes,6 (1969), pp. 517–523.
S. N. Kruzhkov -F. Hildebrand,The Cauchy problem for first-order quasilinear equations when the domain of dependence on the initial data is infinite, Moscow Univ. Math. Bull.,29 (1974), pp. 75–81.
Ph. Laurençot -F. Simondon,Long-time behaviour for porous medium equations with convection, Proc. Roy. Soc. Edinburgh Sect. A,128 (1998), pp. 315–336.
T. P. Liu -M. Pierre,Source-solutions and asymptotic behavior in conservation laws, J. Diff. Eq.,51 (1984), pp. 419–441.
S. I. Shmarev, On a degenerate parabolic equation in filtration theory: monotonicity and C∞-regularity of interface, Adv. Math. Sci. Appl.,5 (1995), pp. 1–29.
J. Simon,Compact sets in the space L p(0, T; B), Ann. Mat. Pura Appl. (4),146 (1987), pp. 65–96.
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Laurençot, P. Long-time behaviour for diffusion equations with fast convection. Annali di Matematica pura ed applicata 175, 233–251 (1998). https://doi.org/10.1007/BF01783685
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DOI: https://doi.org/10.1007/BF01783685