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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

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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

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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.  Google Scholar
[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.  Google Scholar
[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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