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March  2022, 42(3): 1465-1491. doi: 10.3934/dcds.2021160

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

* Corresponding author: Razvan Iagar

Received  January 2021 Revised  May 2021 Published  March 2022 Early access  October 2021

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 $
, with
$ m>1 $
,
$ 0<p<1 $
and the critical value for the weight
$ \sigma = \frac{2(1-p)}{m-1}. $
Existence and uniqueness of some specific solution holds true when
$ m+p\geq2 $
. On the contrary, no eternal solution exists if
$ m+p<2 $
. We also classify exponential self-similar solutions with a different interface behavior when
$ m+p>2 $
. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced.
Citation: Razvan Gabriel Iagar, Ariel Sánchez. Eternal solutions for a reaction-diffusion equation with weighted reaction. Discrete & Continuous Dynamical Systems, 2022, 42 (3) : 1465-1491. doi: 10.3934/dcds.2021160
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.  Google Scholar

[2]

V. GalaktionovL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. IagarA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

V. GalaktionovL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

R. IagarA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The four regions in the phase plane separated by the isoclines
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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