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Continuous dependence of the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation in $H^{s} (\mathbb R^{n})$
Blow-up results of the positive solution for a weakly coupled quasilinear parabolic system
School of Mathematical Sciences, Shanxi University, Taiyuan 030006, China |
$ \left\{ \begin{array}{ll} u_{t} = \nabla\cdot\left(r(u)\nabla u\right)+f(u,v,x,t), & \\ v_{t} = \nabla\cdot\left(s(v)\nabla v\right)+g(u,v,x,t) &{\rm in} \ \Omega\times(0,t^{*}), \\ \frac{\partial u}{\partial\nu} = h(u), \ \frac{\partial v}{\partial\nu} = k(v) &{\rm on} \ \partial\Omega\times(0,t^{*}), \\ u(x,0) = u_{0}(x), \ v(x,0) = v_{0}(x) &{\rm in} \ \overline{\Omega}. \end{array} \right. $ |
$ \Omega $ |
$ \mathbb{R}^{n} \ (n\geq2) $ |
$ \partial\Omega $ |
$ \Omega $ |
$ (u,v) $ |
$ t^* $ |
$ t^* $ |
$ (u,v) $ |
References:
[1] |
X. L. Bai,
Finite time blow-up for a reaction-diffusion system in bounded domain, Z. Angew. Math. Phys., 65 (2014), 135-138.
doi: 10.1007/s00033-013-0330-4. |
[2] |
J. T. Ding,
Blow-up analysis for parabolic p-Laplacian equations with a gradient source term, J. Inequal. Appl., 2020 (2020), 1-11.
doi: 10.1186/s13660-020-02481-y. |
[3] |
J. T. Ding and W. Kou,
Blow-up solutions for reaction diffusion equations with nonlocal boundary conditions, J. Math. Anal. Appl., 470 (2019), 1-15.
|
[4] |
J. T. Ding and X. H. Shen,
Blow-up time estimates in porous medium equations with nonlinear boundary conditions, Z. Angew. Math. Phys., 69 (2018), 1-13.
doi: 10.1007/s00033-018-0993-y. |
[5] |
L. L. Du,
Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math., 202 (2007), 237-247.
doi: 10.1016/j.cam.2006.02.028. |
[6] |
Y. L. Du and B. C. Liu,
Time-weighted blow-up profiles in a nonlinear parabolic system with Fujita exponent, Comput. Math. Appl., 76 (2018), 1034-1055.
doi: 10.1016/j.camwa.2018.05.039. |
[7] |
C. Enache,
Blow-up, global existence and exponential decay estimates for a class of quasilinear parabolic problems, Nonlinear Anal. TMA, 69 (2008), 2864-2874.
doi: 10.1016/j.na.2007.08.063. |
[8] |
A. Friedman, Partial Differential Equation of Parabolic Type, , Prentice-Hall, Englewood Cliffs, N. J., 1964. |
[9] |
S. C. Fu and J. S. Guo,
Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions, J. Math. Anal. Anal., 276 (2002), 458-475.
doi: 10.1016/S0022-247X(02)00506-1. |
[10] |
W. Guo, W. J. Gao and B. Guo,
Global existence and blowing-up of solutions to a class of coupled reaction-convection-diffusion systems, Appl. Math. Lett., 28 (2014), 72-76.
doi: 10.1016/j.aml.2013.10.003. |
[11] |
W. Kou and J. T. Ding, Blow-up phenomena for p-Laplacian parabolic equations under nonlocal boundary conditions, Appl. Anal., 2020.
doi: 10.1080/00036811.2020.1716972. |
[12] |
F. J. Li and B. C. Liu,
Critical exponents for non-simultaneous blow-up in a localized parabolic system, Nonlinear Anal. TMA, 70 (2009), 3452-3460.
doi: 10.1016/j.na.2008.07.002. |
[13] |
G. Li, P. Fan and J. Zhu,
Blow-up estimates for a semilinear coupled parabolic system, Appl. Math. Lett., 22 (2009), 1297-1302.
doi: 10.1016/j.aml.2009.01.046. |
[14] |
F. Liang,
Global existence and blow-up for a degenerate reaction-diffusion system with nonlinear localized sources and nonlocal boundary conditions, J. Korean Math. Soc., 53 (2016), 27-43.
doi: 10.4134/JKMS.2016.53.1.027. |
[15] |
H. H. Lu,
Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems, Electron. J. Qual. Theory Differ. Equ., 49 (2009), 1-14.
doi: 10.14232/ejqtde.2009.1.49. |
[16] |
N. Mahmoudi, P. Souplet and S. Tayachi,
Improved conditions for single-point blow-up in reaction-diffusion systems, J. Differential Equations, 259 (2015), 1898-1932.
doi: 10.1016/j.jde.2015.03.024. |
[17] |
M. Marras,
Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Numer. Funct. Anal. Optim., 32 (2011), 453-468.
doi: 10.1080/01630563.2011.554949. |
[18] |
M. Marras and S. Vernier-Piro,
Finite time collapse in chemotaxis systems with logistic-type superlinear source, Math. Methods Appl. Sci., 43 (2020), 10027-10040.
doi: 10.1002/mma.6676. |
[19] |
L. E. Payne and G. A. Philippin,
Blow-up phenomena for a class of parabolic systems with time dependent coefficients, Appl. Math., 3 (2012), 325-330.
doi: 10.4236/am.2012.34049. |
[20] |
L. E. Payne and P. W. Schaefer,
Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202.
|
[21] |
J. D. Rossi and P. Souplet,
Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system, Differ. Integral Equ., 18 (2005), 405-418.
|
[22] |
X. H. Shen and J. T. Ding,
Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions, Comput. Math. Appl., 77 (2019), 3250-3263.
doi: 10.1016/j.camwa.2019.02.007. |
[23] |
P. Souplet and S. Tayachi,
Optimal condition for non-simultaneous blow-up in a reaction-diffusion system, J. Math. Soc. Japan, 56 (2004), 571-584.
doi: 10.2969/jmsj/1191418646. |
[24] |
R. P. Sperb, Maximum Principles and Their Applications,, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.
![]() ![]() |
[25] |
N. Umeda,
Large time behavior and uniqueness of solutions of a weakly coupled system of reaction-diffusion equations, Tokyo J. Math., 26 (2003), 347-372.
doi: 10.3836/tjm/1244208595. |
[26] |
J. Z. Zhang and F. S. Li,
Global existence and blow-up phenomena for divergence form parabolic equation with time-dependent coefficient in multidimensional space, Z. Angew. Math. Phys., 70 (2019), 1-16.
doi: 10.1007/s00033-019-1195-y. |
[27] |
L. L. Zhang, H. Wang and X. Q. Wang,
Global and blow-up analysis for a class of nonlinear reaction diffusion model with Dirichlet boundary conditions, Math. Methods Appl. Sci., 41 (2018), 7789-7803.
doi: 10.1002/mma.5241. |
[28] |
H. H. Zou,
Blow-up rates for semi-linear reaction-diffusion systems, J. Differential Equations, 257 (2014), 843-867.
doi: 10.1016/j.jde.2014.04.019. |
show all references
References:
[1] |
X. L. Bai,
Finite time blow-up for a reaction-diffusion system in bounded domain, Z. Angew. Math. Phys., 65 (2014), 135-138.
doi: 10.1007/s00033-013-0330-4. |
[2] |
J. T. Ding,
Blow-up analysis for parabolic p-Laplacian equations with a gradient source term, J. Inequal. Appl., 2020 (2020), 1-11.
doi: 10.1186/s13660-020-02481-y. |
[3] |
J. T. Ding and W. Kou,
Blow-up solutions for reaction diffusion equations with nonlocal boundary conditions, J. Math. Anal. Appl., 470 (2019), 1-15.
|
[4] |
J. T. Ding and X. H. Shen,
Blow-up time estimates in porous medium equations with nonlinear boundary conditions, Z. Angew. Math. Phys., 69 (2018), 1-13.
doi: 10.1007/s00033-018-0993-y. |
[5] |
L. L. Du,
Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources, J. Comput. Appl. Math., 202 (2007), 237-247.
doi: 10.1016/j.cam.2006.02.028. |
[6] |
Y. L. Du and B. C. Liu,
Time-weighted blow-up profiles in a nonlinear parabolic system with Fujita exponent, Comput. Math. Appl., 76 (2018), 1034-1055.
doi: 10.1016/j.camwa.2018.05.039. |
[7] |
C. Enache,
Blow-up, global existence and exponential decay estimates for a class of quasilinear parabolic problems, Nonlinear Anal. TMA, 69 (2008), 2864-2874.
doi: 10.1016/j.na.2007.08.063. |
[8] |
A. Friedman, Partial Differential Equation of Parabolic Type, , Prentice-Hall, Englewood Cliffs, N. J., 1964. |
[9] |
S. C. Fu and J. S. Guo,
Blow-up for a semilinear reaction-diffusion system coupled in both equations and boundary conditions, J. Math. Anal. Anal., 276 (2002), 458-475.
doi: 10.1016/S0022-247X(02)00506-1. |
[10] |
W. Guo, W. J. Gao and B. Guo,
Global existence and blowing-up of solutions to a class of coupled reaction-convection-diffusion systems, Appl. Math. Lett., 28 (2014), 72-76.
doi: 10.1016/j.aml.2013.10.003. |
[11] |
W. Kou and J. T. Ding, Blow-up phenomena for p-Laplacian parabolic equations under nonlocal boundary conditions, Appl. Anal., 2020.
doi: 10.1080/00036811.2020.1716972. |
[12] |
F. J. Li and B. C. Liu,
Critical exponents for non-simultaneous blow-up in a localized parabolic system, Nonlinear Anal. TMA, 70 (2009), 3452-3460.
doi: 10.1016/j.na.2008.07.002. |
[13] |
G. Li, P. Fan and J. Zhu,
Blow-up estimates for a semilinear coupled parabolic system, Appl. Math. Lett., 22 (2009), 1297-1302.
doi: 10.1016/j.aml.2009.01.046. |
[14] |
F. Liang,
Global existence and blow-up for a degenerate reaction-diffusion system with nonlinear localized sources and nonlocal boundary conditions, J. Korean Math. Soc., 53 (2016), 27-43.
doi: 10.4134/JKMS.2016.53.1.027. |
[15] |
H. H. Lu,
Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems, Electron. J. Qual. Theory Differ. Equ., 49 (2009), 1-14.
doi: 10.14232/ejqtde.2009.1.49. |
[16] |
N. Mahmoudi, P. Souplet and S. Tayachi,
Improved conditions for single-point blow-up in reaction-diffusion systems, J. Differential Equations, 259 (2015), 1898-1932.
doi: 10.1016/j.jde.2015.03.024. |
[17] |
M. Marras,
Bounds for blow-up time in nonlinear parabolic systems under various boundary conditions, Numer. Funct. Anal. Optim., 32 (2011), 453-468.
doi: 10.1080/01630563.2011.554949. |
[18] |
M. Marras and S. Vernier-Piro,
Finite time collapse in chemotaxis systems with logistic-type superlinear source, Math. Methods Appl. Sci., 43 (2020), 10027-10040.
doi: 10.1002/mma.6676. |
[19] |
L. E. Payne and G. A. Philippin,
Blow-up phenomena for a class of parabolic systems with time dependent coefficients, Appl. Math., 3 (2012), 325-330.
doi: 10.4236/am.2012.34049. |
[20] |
L. E. Payne and P. W. Schaefer,
Blow-up phenomena for some nonlinear parabolic systems, Int. J. Pure Appl. Math., 48 (2008), 193-202.
|
[21] |
J. D. Rossi and P. Souplet,
Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system, Differ. Integral Equ., 18 (2005), 405-418.
|
[22] |
X. H. Shen and J. T. Ding,
Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions, Comput. Math. Appl., 77 (2019), 3250-3263.
doi: 10.1016/j.camwa.2019.02.007. |
[23] |
P. Souplet and S. Tayachi,
Optimal condition for non-simultaneous blow-up in a reaction-diffusion system, J. Math. Soc. Japan, 56 (2004), 571-584.
doi: 10.2969/jmsj/1191418646. |
[24] |
R. P. Sperb, Maximum Principles and Their Applications,, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981.
![]() ![]() |
[25] |
N. Umeda,
Large time behavior and uniqueness of solutions of a weakly coupled system of reaction-diffusion equations, Tokyo J. Math., 26 (2003), 347-372.
doi: 10.3836/tjm/1244208595. |
[26] |
J. Z. Zhang and F. S. Li,
Global existence and blow-up phenomena for divergence form parabolic equation with time-dependent coefficient in multidimensional space, Z. Angew. Math. Phys., 70 (2019), 1-16.
doi: 10.1007/s00033-019-1195-y. |
[27] |
L. L. Zhang, H. Wang and X. Q. Wang,
Global and blow-up analysis for a class of nonlinear reaction diffusion model with Dirichlet boundary conditions, Math. Methods Appl. Sci., 41 (2018), 7789-7803.
doi: 10.1002/mma.5241. |
[28] |
H. H. Zou,
Blow-up rates for semi-linear reaction-diffusion systems, J. Differential Equations, 257 (2014), 843-867.
doi: 10.1016/j.jde.2014.04.019. |
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