A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources

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September 2017, 16(5): 1697-1706. doi: 10.3934/cpaa.2017081

A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources

Lingwei Ma and Zhong Bo Fang ^,

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

^* Corresponding author

Received September 2016 Revised March 2017 Published May 2017

Fund Project: This work is supported by the National Natural Science Foundation of China [Grant No. 11671188]

In this paper, we consider positive solutions of a Cauchy problem for the following quasilinear degenerate parabolic equation with weighted nonlocal sources:

$u_{t}=\Delta_{p}u+ \left(\int_{\mathbb{R}^{N}}K(x)u^{q}(x, t)dx\right)^{\frac{r-1}{q}}u^{s+1}, (x, t) \in \mathbb{R}^{N} \times(0, T), $

where

$N≥1$

$p>2$

$q$

$r≥1$

$s≥0$

, and

$r+s>1$

. We classify global and non-global solutions of the equation in the coexistence region by finding a new second critical exponent via the slow decay asymptotic behavior of an initial value at spatial infinity, and the life span of non-global solution is studied.

Keywords: Weighted nonlocal sources, quasilinear degenerate parabolic equation, second critical exponent, life span.

Mathematics Subject Classification: 35K65, 35B30, 35B40.

Citation: Lingwei Ma, Zhong Bo Fang. A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1697-1706. doi: 10.3934/cpaa.2017081

References:

[1]	N. V. Afanas'eva and A. F. Tedeev, Theorems on the existence and nonexistence of solutions of the Cauchy problem for degenerate parabolic equations with nonlocal source, Ukr. Math. J., 57 (2005), 1687-1711. doi: 10.1007/s11253-006-0024-6. Google Scholar
[2]	J. I. Diaz and J. E. Saá, Uniqueness of very singular self-similar solution of a quasilinear degenerate parabolic equation with absorption, Publ. Mat., 36 (1992), 19-38. doi: 10.5565/PUBLMAT_36192_02. Google Scholar
[3]	H. Fujita, On the blowing up of solution of the Cauchy problem for $u_{t}=Δ u+u^{1+α}$, J. Fac. Sci. Univ. Tokyo Sect. Ⅰ, 13 (1966), 109-124. Google Scholar
[4]	J. Furter and M. Grinfield, Local vs. non-local interactions in populations dynamics, J. Math. Biol., 27 (1989), 65-80. doi: 10.1007/BF00276081. Google Scholar
[5]	V. A. Galaktionov, Conditions for nonexistence in the large and localization of solutions of the Cauchy problem for a class of nonlinear parabolic equations, Zh. Vychisl. Mat. i Mat. Fiz., 23 (1983), 1341-1354. Google Scholar
[6]	V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34 (1998), 1005-1027. doi: 10.1016/S0362-546X(97)00716-5. Google Scholar
[7]	K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equation, DProc. Japan Acad., 49 (1973), 503-505. Google Scholar
[8]	T. Y. Lee and W. M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378. doi: 10.2307/2154114. Google Scholar
[9]	Y. H. Li and C. L. Mu, Life span and a new critical exponent for a degenerate parabolic equation, J. Differential Equations, 207 (2004), 392-406. doi: 10.1016/j.jde.2004.08.024. Google Scholar
[10]	C. L. Mu, Y. H. Li and Y. Wang, Life span and a new critical exponent for a quasilinear degenerate parabolic equation with slow decay initial values, Nonlinear Anal. Real World Appl., 11 (2010), 198-206. doi: 10.1016/j.nonrwa.2008.10.048. Google Scholar
[11]	C. L. Mu, R. Zeng and S. M. Zhou, Life span and a new critical exponent for a doubly degenerate parabolic equation with slow decay initial values, J. Math. Anal. Appl., 384 (2011), 181-191. doi: 10.1016/j.jmaa.2010.12.042. Google Scholar
[12]	C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. doi: 10.1007/978-1-4612-0873-0.
[13]	Y. W. Qi, Critical exponents of degenerate parabolic equations, Sci. China Ser. A, 38 (1995), 1153-1162. Google Scholar
[14]	F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. Google Scholar
[15]	C. X. Yang, F. Y. Ji and S. S. Zhou, The second critical exponent for a semilinear nonlocal parabolic equation, J. Math. Anal. Appl., 418 (2014), 231-237. doi: 10.1016/j.jmaa.2014.03.095. Google Scholar
[16]	J. N. Zhao, On the Cauchy problem and initial traces for the evolution P-Laplacian equations with strongly nonlinear sources, J. Differential Equations, 121 (1995), 329-383. doi: 10.1006/jdeq.1995.1132. Google Scholar

show all references

References:

[1]	N. V. Afanas'eva and A. F. Tedeev, Theorems on the existence and nonexistence of solutions of the Cauchy problem for degenerate parabolic equations with nonlocal source, Ukr. Math. J., 57 (2005), 1687-1711. doi: 10.1007/s11253-006-0024-6. Google Scholar
[2]	J. I. Diaz and J. E. Saá, Uniqueness of very singular self-similar solution of a quasilinear degenerate parabolic equation with absorption, Publ. Mat., 36 (1992), 19-38. doi: 10.5565/PUBLMAT_36192_02. Google Scholar
[3]	H. Fujita, On the blowing up of solution of the Cauchy problem for $u_{t}=Δ u+u^{1+α}$, J. Fac. Sci. Univ. Tokyo Sect. Ⅰ, 13 (1966), 109-124. Google Scholar
[4]	J. Furter and M. Grinfield, Local vs. non-local interactions in populations dynamics, J. Math. Biol., 27 (1989), 65-80. doi: 10.1007/BF00276081. Google Scholar
[5]	V. A. Galaktionov, Conditions for nonexistence in the large and localization of solutions of the Cauchy problem for a class of nonlinear parabolic equations, Zh. Vychisl. Mat. i Mat. Fiz., 23 (1983), 1341-1354. Google Scholar
[6]	V. A. Galaktionov and H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34 (1998), 1005-1027. doi: 10.1016/S0362-546X(97)00716-5. Google Scholar
[7]	K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic equation, DProc. Japan Acad., 49 (1973), 503-505. Google Scholar
[8]	T. Y. Lee and W. M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378. doi: 10.2307/2154114. Google Scholar
[9]	Y. H. Li and C. L. Mu, Life span and a new critical exponent for a degenerate parabolic equation, J. Differential Equations, 207 (2004), 392-406. doi: 10.1016/j.jde.2004.08.024. Google Scholar
[10]	C. L. Mu, Y. H. Li and Y. Wang, Life span and a new critical exponent for a quasilinear degenerate parabolic equation with slow decay initial values, Nonlinear Anal. Real World Appl., 11 (2010), 198-206. doi: 10.1016/j.nonrwa.2008.10.048. Google Scholar
[11]	C. L. Mu, R. Zeng and S. M. Zhou, Life span and a new critical exponent for a doubly degenerate parabolic equation with slow decay initial values, J. Math. Anal. Appl., 384 (2011), 181-191. doi: 10.1016/j.jmaa.2010.12.042. Google Scholar
[12]	C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. doi: 10.1007/978-1-4612-0873-0.
[13]	Y. W. Qi, Critical exponents of degenerate parabolic equations, Sci. China Ser. A, 38 (1995), 1153-1162. Google Scholar
[14]	F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. Google Scholar
[15]	C. X. Yang, F. Y. Ji and S. S. Zhou, The second critical exponent for a semilinear nonlocal parabolic equation, J. Math. Anal. Appl., 418 (2014), 231-237. doi: 10.1016/j.jmaa.2014.03.095. Google Scholar
[16]	J. N. Zhao, On the Cauchy problem and initial traces for the evolution P-Laplacian equations with strongly nonlinear sources, J. Differential Equations, 121 (1995), 329-383. doi: 10.1006/jdeq.1995.1132. Google Scholar

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