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

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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

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

Received September 2010 Revised June 2011 Published October 2011

Abstract
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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.
[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.
[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.
[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.
[5]	V. Galaktionov, Blow up for quasilinear heat equations with critical Fujita exponents,, Proc. Roy. Soc. Edinburgh. Sect., 124 (1994), 517.
[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.
[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.
[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.
[9]	Y. Qi, Critical exponents of the degenerate parabolic equations,, Since in China, 38 (1995), 1153.
[10]	Y. Qi, The critical exponents of parabolic equations and blow-up in $R^N$,, Proc. Roy. Soc. Edinburgh Sect, 128 (1998), 123.
[11]	V. Samarskii, V. Galaktionov and V. Kurdyumov, et al, "Blow-up in Quasilinear Parabolic Equations,", Nauka, (1987).
[12]	J. Vazquez, "The Porous Medium Equation: Mathematical Theory,", Clarendon Press, (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. 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. 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., (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.
[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.
[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.
[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.
[5]	V. Galaktionov, Blow up for quasilinear heat equations with critical Fujita exponents,, Proc. Roy. Soc. Edinburgh. Sect., 124 (1994), 517.
[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.
[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.
[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.
[9]	Y. Qi, Critical exponents of the degenerate parabolic equations,, Since in China, 38 (1995), 1153.
[10]	Y. Qi, The critical exponents of parabolic equations and blow-up in $R^N$,, Proc. Roy. Soc. Edinburgh Sect, 128 (1998), 123.
[11]	V. Samarskii, V. Galaktionov and V. Kurdyumov, et al, "Blow-up in Quasilinear Parabolic Equations,", Nauka, (1987).
[12]	J. Vazquez, "The Porous Medium Equation: Mathematical Theory,", Clarendon Press, (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. 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. 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., (2001). doi: 10.1142/9789812799791.

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