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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

Zhilei Liang 1,

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].

Keywords: critical exponents, blow up, localized reaction., Porous media.
Mathematics Subject Classification: Primary: 35K15, 35K55; Secondary: 35K6.
Citation: Zhilei Liang. On the critical exponents for porous medium equation with a localized reaction in high dimensions. Communications on Pure & 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.   Google Scholar

[2]

D. Andreucci and E. Dibenedetti, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources,, Ann, 3 (1991), 393.   Google Scholar

[3]

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

[4]

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

[5]

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

[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.   Google Scholar

[7]

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

[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.  doi: 10.1006/jdeq.1996.3196.  Google Scholar

[9]

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

[10]

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

[11]

V. Samarskii, V. Galaktionov and V. Kurdyumov, et al, "Blow-up in Quasilinear Parabolic Equations,", Nauka, (1987).   Google Scholar

[12]

J. Vazquez, "The Porous Medium Equation: Mathematical Theory,", Clarendon Press, (2007).   Google Scholar

[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.  doi: 10.1007/s00028-007-0324-9.  Google Scholar

[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.  doi: 10.1016/j.aml.2006.03.008.  Google Scholar

[15]

Z. Wu, J. Zhao and J. Yin, et al, "Nonlinear Diffusion Equations,", World Scientific Publishing Co., (2001).  doi: 10.1142/9789812799791.  Google Scholar

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.   Google Scholar

[2]

D. Andreucci and E. Dibenedetti, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources,, Ann, 3 (1991), 393.   Google Scholar

[3]

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

[4]

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

[5]

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

[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.   Google Scholar

[7]

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

[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.  doi: 10.1006/jdeq.1996.3196.  Google Scholar

[9]

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

[10]

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

[11]

V. Samarskii, V. Galaktionov and V. Kurdyumov, et al, "Blow-up in Quasilinear Parabolic Equations,", Nauka, (1987).   Google Scholar

[12]

J. Vazquez, "The Porous Medium Equation: Mathematical Theory,", Clarendon Press, (2007).   Google Scholar

[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.  doi: 10.1007/s00028-007-0324-9.  Google Scholar

[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.  doi: 10.1016/j.aml.2006.03.008.  Google Scholar

[15]

Z. Wu, J. Zhao and J. Yin, et al, "Nonlinear Diffusion Equations,", World Scientific Publishing Co., (2001).  doi: 10.1142/9789812799791.  Google Scholar

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