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A direct method of moving planes for a fully nonlinear nonlocal system
A new second critical exponent and life span for a quasilinear degenerate parabolic equation with weighted nonlocal sources
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China |
$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), $ |
$N≥1$ |
$p>2$ |
$q$ |
$r≥1$ |
$s≥0$ |
$r+s>1$ |
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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
[7] |
K. Hayakawa,
On nonexistence of global solutions of some semilinear parabolic equation, DProc. Japan Acad., 49 (1973), 503-505.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
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. |
[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. |
[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.
|
[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. |
[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.
|
[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. |
[7] |
K. Hayakawa,
On nonexistence of global solutions of some semilinear parabolic equation, DProc. Japan Acad., 49 (1973), 503-505.
|
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
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