Eternal solutions for a reaction-diffusion equation with weighted reaction

Razvan Gabriel Iagar; Ariel Sánchez

doi:10.3934/dcds.2021160

Article Contents

2022, Volume 42, Issue 3: 1465-1491. Doi: 10.3934/dcds.2021160

This issue Previous Article On the structure of α-limit sets of backward trajectories for graph maps Next Article An Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515)

Eternal solutions for a reaction-diffusion equation with weighted reaction

Razvan Gabriel Iagar^, and
Ariel Sánchez

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
^* Corresponding author: Razvan Iagar

Received: December 31, 2020

Revised: April 30, 2021

Early access: October 2021

Published: March 2022

Abstract Full Text(HTML) Figure(3) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35B33, 35B36, 35C06; Secondary: 35K10, 35K57.

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

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 $

Download: Full-size image PowerPoint slide

Figure 3. The regions in the phase plane associated to the system (3.5)

Download: Full-size image PowerPoint slide

Related Papers

Cited by

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.
[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.