Conditions for the Existence of Global Solutions to Doubly Nonlinear Advection-diffusion Equations

Jocemar Q. Chagas, Patrícia L. Guidolin, Paulo R. Zingano

Abstract


 In this work, we consider a initial-value problem for an doubly non linear advection-diffusion equation, and we present a critical value of κ up to wich the initial-value problem has global solution independent of the initial data u0, and from which global solutions may still exists, but from initial data u0 satisfying certain conditions. For this, we suppose that the function f(x,t,u) in the advection term, writted in the divergent form, satisfies certain conditions about your variation in Rn, and we also use the decrease of the norm L1(Rn) and an control for the norm L∞(Rn) of solution u(·,t). 


Keywords


Doubly nonlinear parabolic equation; global solutions; conditions for global solutions

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References


References

O. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type. Providence: American Mathematical Society, 1968.

D. Serre, Systems of Conservation Laws, vol. 1. Cambridge: Cambridge University Press, 1999.

J. Q. Chagas, P. L. Guidolin, and P. R. Zingano, Norma do sup para equações de advecção-difusão duplamente não lineares: um caso de decrescimento, in Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, no. 5, v.1, SBMAC, 2017. (DOI: 10.5540/03.2017.005.01.0034).

J. Q. Chagas, P. L. Guidolin, and P. R. Zingano, Norma do sup para equações de advecção-difusão duplamente não lineares: o caso geral, in Proceeding Series of the Brazilian Society of Applied and Computational Mathematics, no. 7, SBMAC, 2018. (Pré-print).

E. DiBenedetto, Degenerate Parabolic Equations. New York: Springer-Verlag, 1993.

Z. Wu, J. Zhao, J. Yin, and H. Li, Nonlinear Diffusion Equations. Singapore: World Scientific, 2001.

P. B. e Silva, L. Schütz, and P. Zingano, On some energy inequalities

and supnorm estimates for advection-diffusion equations in Rn, Nonlinear

Analysis: Theory, Methods & Applications, vol. 93, pp. 9096, 2013. (DOI:

1016/j.na.2013.07.028).

J. Q. Chagas, Contribuições para a teoria de equações parabólicas duplamente não lineares com termos advectivos. PhD thesis, UFRGS, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, 2015.

A. Friedman, Partial Differential Equations. New York: Holt, Rinehart and Winston, 1969.




DOI: https://doi.org/10.5540/tema.2020.021.01.83

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