Convergence rate to the singular solution for the Gelfand equation and its stability

Choi, Sun-Ho

doi:10.1090/S0033-569X-2014-01370-4

Author: Sun-Ho Choi
Journal: Quart. Appl. Math. 72 (2014), 773-797
MSC (2010): Primary 34D23
DOI: https://doi.org/10.1090/S0033-569X-2014-01370-4
Published electronically: November 7, 2014
MathSciNet review: 3291828
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Abstract: We study the asymptotic behaviors of the solution to the Gelfand equation. The Gelfand equation appears in the kinetic theory of gravitational steady state and the theory of nonlinear diffusion. We present a convergence rate of the solutions of the Gelfand equation to the unique singular solution as $r$ goes to infinity and prove asymptotic stability of the solution by considering the initial value problem for the Gelfand equation. To obtain the convergence rate and the point-wise stability estimate, we construct a uniform lower bound function and use the solution for the linearized Gelfand equation.

References

S. Bae, S.-H. Choi, and S.-Y. Ha, Nonlinear instability of the Vlasov-Poisson-Boltzmann system in three dimensions, Hyperbolic Problems: Theory, Numerics, Applications, Proceedings of the Fourteenth International Conference on Hyperbolic Problems, AIMS on Applied Mathematics, Vol. 8 (2014).

Sun-Ho Choi and Seung-Yeal Ha, Dynamic instability of stationary solutions to the nonlinear Vlasov equations, Int. J. Numer. Anal. Model. Ser. B 2 (2011), no. 4, 415–421. MR 2869590

Renjun Duan, Tong Yang, and Changjiang Zhu, Existence of stationary solutions to the Vlasov-Poisson-Boltzmann system, J. Math. Anal. Appl. 327 (2007), no. 1, 425–434. MR 2277423, DOI https://doi.org/10.1016/j.jmaa.2006.04.047

I. M. Gel′fand, Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl. (2) 29 (1963), 295–381. MR 0153960

D. D. Joseph, Non-linear heat generation and the stability of the temperature distribution in conducting solids, Int. J. Heat Mass Transfer 8 (1965), 281–288.

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241–269. MR 340701, DOI https://doi.org/10.1007/BF00250508

D. D. Joseph and E. M. Sparrow, Nonlinear diffusion induced by nonlinear sources, Quart. Appl. Math. 28 (1970), 327–342. MR 272272, DOI https://doi.org/10.1090/S0033-569X-1970-0272272-0

Herbert B. Keller and Donald S. Cohen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361–1376. MR 0213694

I. Kuzin and S. Pohozaev, Entire solutions of semilinear elliptic equations, Progress in Nonlinear Differential Equations and their Applications, vol. 33, Birkhäuser Verlag, Basel, 1997. MR 1479168