# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021160
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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

* Corresponding author: Razvan Iagar

Received  January 2021 Revised  May 2021 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 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 $. Some transformations to reaction-convection-diffusion equations and traveling wave solutions are also introduced. Citation: Razvan Gabriel Iagar, Ariel Sánchez. Eternal solutions for a reaction-diffusion equation with weighted reaction. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021160 ##### References:  [1] P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on$ \mathbb{R}^2$,, Int. Math. Res. Not., 2006 (2006), Art. ID 83610, 20 pp. doi: 10.1155/IMRN/2006/83610. Google Scholar [2] V. 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. 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. 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. 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. 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. 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. 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. 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 The four regions in the phase plane separated by the isoclines 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 \$
The regions in the phase plane associated to the system (3.5)
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