March  2012, 11(2): 649-658. doi: 10.3934/cpaa.2012.11.649

On the critical exponents for porous medium equation with a localized reaction in high dimensions

1. 

School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China

Received  September 2010 Revised  June 2011 Published  October 2011

This paper is concerned with the critical exponents for the porous medium equation

$u_{t}=\triangle u^m+a(x)u^p, (x,t)\in R^N\times (0,T), $

where $m>1, p>0,$ and the function $a(x)\geq 0$ has a compact support. Suppose the space dimension $N\geq 2$, we prove that the global exponent $p_0$ and the Fujita type exponent $p_c$ are both $m$: if $0 < p < m$ every solution is global in time, if $ p = m $ all the solutions blow up and if $p > m$ both the blowing up solutions and the global solutions exist. While for the one-dimensional case, it is proved $p_0=\frac{m+1}{2} < m+1 = p_c$ by [E. Ferreira, A. Pablo, J. Vazquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Diff. Eqns., 231(2006) 195-211].

Citation: Zhilei Liang. On the critical exponents for porous medium equation with a localized reaction in high dimensions. Communications on Pure and Applied Analysis, 2012, 11 (2) : 649-658. doi: 10.3934/cpaa.2012.11.649
References:
[1]

K. Bimpong-Bota, P. Ortoleva and J. Ross, Far-from-equilibrium phenomena at local sites of reaction, J. Chem. Phys., 60 (1974), 3124-3133.

[2]

D. Andreucci and E. Dibenedetti, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann, Scuola, Norm, Sup, Piza, Ser, IV, XIII, Fasc, 3 (1991), 393-441.

[3]

C. Gui and X. Kang, Localization for a porous medium type equation in high dimensions, Trans. Amer. Math. Soc., 356 (2004), 4273-4285. doi: 10.1090/S0002-9947-04-03613-X.

[4]

E. Ferreira, A. Pablo and J. Vazquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Diff. Eqns., 231 (2006), 195-211. doi: 10.1016/j.jde.2006.04.017.

[5]

V. Galaktionov, Blow up for quasilinear heat equations with critical Fujita exponents, Proc. Roy. Soc. Edinburgh. Sect., A 124 (1994), 517-525.

[6]

V. Galaktionov and H. Levine, On critical Fujita exponents for heat equation with a nonlinear flux conditions on the boundary, Israel J. Math., 94 (1996), 125-146.

[7]

Z. Liang and J. Zhao, Localization for the evolution p-Laplacian equation with strongly nonlinear source term, J. Diff. Eqns., 246 (2009), 391-407. doi: 10.1016/j.jde.2008.07.038.

[8]

R. Pinsky, Existence and nonexistence of global solutions for $u_t=\Delta u+a(x)u^q$ in $R^d$, J. Diff. Eqns., 133 (1997), 152-177. doi: 10.1006/jdeq.1996.3196.

[9]

Y. Qi, Critical exponents of the degenerate parabolic equations, Since in China, 38 (1995), 1153-1162.

[10]

Y. Qi, The critical exponents of parabolic equations and blow-up in $R^N$, Proc. Roy. Soc. Edinburgh Sect, A. 128 (1998), 123-136.

[11]

V. Samarskii, V. Galaktionov and V. Kurdyumov, et al, "Blow-up in Quasilinear Parabolic Equations," Nauka, Moscow, 1987; Englishi translation:. Walter de Gruyter, Berlin/New York, 1995.

[12]

J. Vazquez, "The Porous Medium Equation: Mathematical Theory," Clarendon Press, Oxford, 2007.

[13]

Z. Wang, J. Yin and C. Wang, et al., Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources, J. Evol. Equ., 7 (2007), 615-648. doi: 10.1007/s00028-007-0324-9.

[14]

Z. Wang, J. Yin and C. Wang, Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition, Appl. Math. Lett., 20 (2007), 142-147. doi: 10.1016/j.aml.2006.03.008.

[15]

Z. Wu, J. Zhao and J. Yin, et al, "Nonlinear Diffusion Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799791.

show all references

References:
[1]

K. Bimpong-Bota, P. Ortoleva and J. Ross, Far-from-equilibrium phenomena at local sites of reaction, J. Chem. Phys., 60 (1974), 3124-3133.

[2]

D. Andreucci and E. Dibenedetti, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann, Scuola, Norm, Sup, Piza, Ser, IV, XIII, Fasc, 3 (1991), 393-441.

[3]

C. Gui and X. Kang, Localization for a porous medium type equation in high dimensions, Trans. Amer. Math. Soc., 356 (2004), 4273-4285. doi: 10.1090/S0002-9947-04-03613-X.

[4]

E. Ferreira, A. Pablo and J. Vazquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Diff. Eqns., 231 (2006), 195-211. doi: 10.1016/j.jde.2006.04.017.

[5]

V. Galaktionov, Blow up for quasilinear heat equations with critical Fujita exponents, Proc. Roy. Soc. Edinburgh. Sect., A 124 (1994), 517-525.

[6]

V. Galaktionov and H. Levine, On critical Fujita exponents for heat equation with a nonlinear flux conditions on the boundary, Israel J. Math., 94 (1996), 125-146.

[7]

Z. Liang and J. Zhao, Localization for the evolution p-Laplacian equation with strongly nonlinear source term, J. Diff. Eqns., 246 (2009), 391-407. doi: 10.1016/j.jde.2008.07.038.

[8]

R. Pinsky, Existence and nonexistence of global solutions for $u_t=\Delta u+a(x)u^q$ in $R^d$, J. Diff. Eqns., 133 (1997), 152-177. doi: 10.1006/jdeq.1996.3196.

[9]

Y. Qi, Critical exponents of the degenerate parabolic equations, Since in China, 38 (1995), 1153-1162.

[10]

Y. Qi, The critical exponents of parabolic equations and blow-up in $R^N$, Proc. Roy. Soc. Edinburgh Sect, A. 128 (1998), 123-136.

[11]

V. Samarskii, V. Galaktionov and V. Kurdyumov, et al, "Blow-up in Quasilinear Parabolic Equations," Nauka, Moscow, 1987; Englishi translation:. Walter de Gruyter, Berlin/New York, 1995.

[12]

J. Vazquez, "The Porous Medium Equation: Mathematical Theory," Clarendon Press, Oxford, 2007.

[13]

Z. Wang, J. Yin and C. Wang, et al., Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources, J. Evol. Equ., 7 (2007), 615-648. doi: 10.1007/s00028-007-0324-9.

[14]

Z. Wang, J. Yin and C. Wang, Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition, Appl. Math. Lett., 20 (2007), 142-147. doi: 10.1016/j.aml.2006.03.008.

[15]

Z. Wu, J. Zhao and J. Yin, et al, "Nonlinear Diffusion Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 2001. doi: 10.1142/9789812799791.

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