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On mappings of higher order and their applications to nonlinear equations
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 |
$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].
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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