THE POINTWISE ESTIMATES OF SOLUTIONS FOR A NONLINEAR CONVECTION DIFFUSION REACTION EQUATION

doi:10.1016/S0252-9602(16)30117-5

Abstract

Abstract:

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension. First, the pointwise estimates of solutions are obtained, furthermore, we obtain the optimal L^p, 1≤p≤+∞, convergence rate of solutions for small initial data. Then we establish the local existence of solutions, the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data. Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.

Key words: convection diffusion reaction equation, pointwise estimate, Green function, energy method

Guowei LIU. THE POINTWISE ESTIMATES OF SOLUTIONS FOR A NONLINEAR CONVECTION DIFFUSION REACTION EQUATION[J].Acta mathematica scientia,Series B, 2017, 37(1): 79-96.

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References

[1] Adams R A. Sobolev Space. New York:Academic Press, 1975
[2] Deuring P, Eymard R, Mildner M. L²-stability independent of diffusion for a finite element-finite volume discretization of a linear convection-diffusion equation. SIAM J Numer Anal, 2015, 53:508-526
[3] Escobedo M, Zuazua E. Large time behaviour for convection-diffusion equations in Rn. J Funct Anal, 1991, 100:119-161
[4] Escobedo M, Vazquez J L, Zuazua E. Asymptotic behaviour and source type solutions for a diffusion convection equation. Arch Rational Mech Anal, 1993, 124:43-65
[5] Even L C. Partial Differential Equations. Graduate Studies in Mathematics. Providence, RI:American Mathematical Society, 1998
[6] Hoff D, Zumbrun K. Multi-dimensional diffusion wave for the Navier-Stokes equations of compressible flow. Indiana Univ Math J, 1995, 44:603-676
[7] Hoff D, Zumbrun K. Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves. Z Angew Math Phys, 1997, 48:603-676
[8] Hosono T, Kawashima S. Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system. Math Models Methods Appl Sci, 2006, 16:1839-1859
[9] Laurencot P H. Long time behaviour for diffusion equations with fast convection. Ann Mat Pura Appl, 1998, 175:233-251
[10] Laurencot P H, Simondon F. Large time behaviour for porous medium equations with convection. Proc Roy Soc Edinburgh, 1998, 128A:315-336
[11] Liu T P, Wang W K. The pointwise estimates of diffusion wave for the Navier-Stokes sysytems in odd multi-dimensions. Commun Math Phys, 1998, 196:145-173
[12] Li F B, Wang W K. The pointwise estimate of solutions to the parabolic conservation law in multi-dimensions. Nonlinear Diff Equ Appl, 2014, 21:87-103
[13] Liu H L. Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations. Math Comp, 2015, 84:2263-2295
[14] Lu Y G, Koley C K U, Lu X Z. Decay rate for degenerate convection diffusion equations in both one and several space dimensions. Acta Math Sci, 2015, 35B:281-302
[15] Oleinik O A, Kalashnikov A S, Chzou Y L. The Cauchy problem and boundary problems for equations of the type of unsteady filtration. Izv Akad Nauk SSSR Ser Mat, 1958, 22:667-704
[16] Riviere B, Yang X. Convergence analysis of a coupled method for time-dependent convection-diffusion equations. Numer Methods Partial Diff Equ, 2014, 30:133-157
[17] Wang W K, Yang T. The pointwise estimates of solutions for Euler equations with damping in multi-dimensions. J Diff Equ, 2001, 173:410-450
[18] Wang W K, Yang X F, The pointwise estimates of solutions to the isentropic Navier-Stokes equations in enen space dimensions. J Hyperbolic Diff Equ, 2005, 2:673-695
[19] Wang W K, Wu Z G. Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions. J Diff Equ, 2010, 248:1617-1636
[20] Wang W K, Wang W J. The pointwise estimates of solutions for semilinear disspative wave equation in multi-dimensions. J Math Anal Appl, 2010, 366:226-241
[21] Xu H M, Ma H L. Global existence and decay estimate of solution to one dimenion convection-difussion equation. Wuhan Univ J Nat Sci, 2013, 18:461-464
[22] Xu H M, Wang J. Pointwise estimation of one dimensional convection diffusion equation. Wuhan Univ J Nat Sci, 2014, 60:129-131