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The four regions in the phase plane separated by the isoclines
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Eternal solutions for a reaction-diffusion equation with weighted reaction
Departamento de Matemática Aplicada, Ciencia e Ingenieria de los Materiales y Tecnologia Electrónica, Universidad Rey Juan Carlos, Móstoles, 28933, Madrid, Spain |
We prove existence and uniqueness of eternal solutions in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation
| $ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $ |
posed in
| $ \mathbb{R}^N $ |
| $ m>1 $ |
| $ 0<p<1 $ |
| $ \sigma = \frac{2(1-p)}{m-1}. $ |
Existence and uniqueness of some specific solution holds true when
| $ m+p\geq2 $ |
| $ m+p<2 $ |
| $ m+p>2 $ |
Keywords:
Eternal solutions,
Reaction-diffusion equations,
Weighted reaction,
Exponential self-similar solutions,
Phase plane analysis,
Strong reaction.
Mathematics Subject Classification:
Primary: 35B33, 35B36, 35C06; Secondary: 35K10, 35K57.
Citation:
Razvan Gabriel Iagar, Ariel Sánchez.
Eternal solutions for a reaction-diffusion equation with weighted reaction. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021160
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References:
| [1] |
P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on $ \mathbb{R}^2$,, Int. Math. Res. Not., 2006 (2006), Art. ID 83610, 20 pp.
doi: 10.1155/IMRN/2006/83610. |
| [2] |
V. Galaktionov, L. A. Peletier and J. L. Vázquez,
Asymptotics of the fast-diffusion equation with critical exponent, SIAM J. Math. Anal., 31 (2000), 1157-1174.
doi: 10.1137/S0036141097328452. |
| [3] |
B. H. Gilding and R. Kersner, Traveling Waves in Nonlinear Diffusion-Convection Reaction, , in Progress in Nonlinear Differential Equations and Their Applications, Birkhauser, 2004.
doi: 10.1007/978-3-0348-7964-4. |
| [4] |
J. Guckenheimer and Ph. Holmes, Nonlinear Oscillation, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. |
| [5] |
R. G. Iagar and Ph. Laurençot,
Eternal solutions to a singular diffusion equation with critical gradient absorption, Nonlinearity, 26 (2013), 3169-3195.
doi: 10.1088/0951-7715/26/12/3169. |
| [6] |
R. G. Iagar and A. Sánchez,
Self-similar blow-up profiles for a reaction-diffusion equation with strong weighted reaction, Adv. Nonl. Studies, 20 (2020), 867-894.
doi: 10.1515/ans-2020-2104. |
| [7] |
R. G. Iagar and A. Sánchez,
Self-similar blow-up profiles for a reaction-diffusion equation with critically strong weighted reaction, J. Dynam. Differential Equations, 31 (2019), 2061-2094.
doi: 10.1007/s10884-018-09727-w. |
| [8] |
R. Iagar, A. Sánchez and J. L. Vázquez,
Radial equivalence for the two basic nonlinear degenerate diffusion equations, J. Math. Pures Appl., 89 (2008), 1-24.
doi: 10.1016/j.matpur.2007.09.002. |
| [9] |
A. de Pablo and A. Sánchez,
Global travelling waves in reaction-convection-diffusion equations, J. Differential Equations, 165 (2000), 377-413.
doi: 10.1006/jdeq.2000.3781. |
| [10] |
A. de Pablo,
Large-time behaviour of solutions of a reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 389-398.
doi: 10.1017/S0308210500028547. |
| [11] |
A. de Pablo and J. L. Vázquez,
The balance between strong reaction and slow diffusion, Comm. Partial Differential Equations, 15 (1990), 159-183.
doi: 10.1080/03605309908820682. |
| [12] |
L. Perko, Differential Equations and Dynamical Systems, Third edition, Texts in Applied Mathematics, 7, Springer Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
| [13] |
J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford Univ. Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780199202973.001.0001.
Google Scholar
|
| [14] |
J. L. Vázquez, Asymptotic behaviour of nonlinear parabolic equations. Anomalous exponents, Degenerate Diffusions (Minneapolis, MN, 1991), IMA Vol. Math. Appl., Springer, New York, 47 (1993), 215–228.
doi: 10.1007/978-1-4612-0885-3_15. |
show all references
References:
| [1] |
P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on $ \mathbb{R}^2$,, Int. Math. Res. Not., 2006 (2006), Art. ID 83610, 20 pp.
doi: 10.1155/IMRN/2006/83610. |
| [2] |
V. Galaktionov, L. A. Peletier and J. L. Vázquez,
Asymptotics of the fast-diffusion equation with critical exponent, SIAM J. Math. Anal., 31 (2000), 1157-1174.
doi: 10.1137/S0036141097328452. |
| [3] |
B. H. Gilding and R. Kersner, Traveling Waves in Nonlinear Diffusion-Convection Reaction, , in Progress in Nonlinear Differential Equations and Their Applications, Birkhauser, 2004.
doi: 10.1007/978-3-0348-7964-4. |
| [4] |
J. Guckenheimer and Ph. Holmes, Nonlinear Oscillation, Dynamical Systems and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1990. |
| [5] |
R. G. Iagar and Ph. Laurençot,
Eternal solutions to a singular diffusion equation with critical gradient absorption, Nonlinearity, 26 (2013), 3169-3195.
doi: 10.1088/0951-7715/26/12/3169. |
| [6] |
R. G. Iagar and A. Sánchez,
Self-similar blow-up profiles for a reaction-diffusion equation with strong weighted reaction, Adv. Nonl. Studies, 20 (2020), 867-894.
doi: 10.1515/ans-2020-2104. |
| [7] |
R. G. Iagar and A. Sánchez,
Self-similar blow-up profiles for a reaction-diffusion equation with critically strong weighted reaction, J. Dynam. Differential Equations, 31 (2019), 2061-2094.
doi: 10.1007/s10884-018-09727-w. |
| [8] |
R. Iagar, A. Sánchez and J. L. Vázquez,
Radial equivalence for the two basic nonlinear degenerate diffusion equations, J. Math. Pures Appl., 89 (2008), 1-24.
doi: 10.1016/j.matpur.2007.09.002. |
| [9] |
A. de Pablo and A. Sánchez,
Global travelling waves in reaction-convection-diffusion equations, J. Differential Equations, 165 (2000), 377-413.
doi: 10.1006/jdeq.2000.3781. |
| [10] |
A. de Pablo,
Large-time behaviour of solutions of a reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 389-398.
doi: 10.1017/S0308210500028547. |
| [11] |
A. de Pablo and J. L. Vázquez,
The balance between strong reaction and slow diffusion, Comm. Partial Differential Equations, 15 (1990), 159-183.
doi: 10.1080/03605309908820682. |
| [12] |
L. Perko, Differential Equations and Dynamical Systems, Third edition, Texts in Applied Mathematics, 7, Springer Verlag, New York, 2001.
doi: 10.1007/978-1-4613-0003-8. |
| [13] |
J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations. Equations of Porous Medium Type, Oxford Univ. Press, Oxford, 2006.
doi: 10.1093/acprof:oso/9780199202973.001.0001.
Google Scholar
|
| [14] |
J. L. Vázquez, Asymptotic behaviour of nonlinear parabolic equations. Anomalous exponents, Degenerate Diffusions (Minneapolis, MN, 1991), IMA Vol. Math. Appl., Springer, New York, 47 (1993), 215–228.
doi: 10.1007/978-1-4612-0885-3_15. |
Figure 2.
Trajectories in the phase plane for different values of $ K>0 $ . Numerical experiment for $ m = 2 $ , $ p = 0.5 $ $ N = 4 $ , $ \sigma = 1 $ and $ K = 0.1 $ , respectively $ K = 8 $
Figure 3.
The regions in the phase plane associated to the system (3.5)
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