Article Contents
Article Contents

# Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

• * Corresponding author

R. I. and A. S. are partially supported by the Spanish project PID2020-115273GB-I00. A. I. M. is partially supported by the Spanish project RTI2018-098743-B-100

• We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight:

$\partial_tu = \Delta u^m+|x|^{\sigma}u^p,$

posed in any space dimension $x\in \mathbb{R}^N$, $t\geq0$ and with exponents $m>1$, $p\in(0, 1)$ and $\sigma>2(1-p)/(m-1)$. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are compactly supported and might present two different types of interface behavior and three different possible good behaviors near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of $\sigma$. This paper generalizes in dimension $N>1$ previous results by the authors in dimension $N = 1$ and also includes some finer classification of the profiles for $\sigma$ large that is new even in dimension $N = 1$.

Mathematics Subject Classification: Primary: 35A24, 35B44, 35C06; Secondary: 35K10, 35K57, 35K65.

 Citation:

• Figure 1.  Orbits from $P_0$ and $P_2$ in the phase space for $\sigma$ small. Experiment for $m = 3$, $p = 0.5$, $N = 4$ and $\sigma = 3.5$

Figure 2.  A plot of the regions $D_1$, $D_2$ and $D_3$ in the phase space

Figure 3.  The planes $(\Pi_1)$ and $(\Pi_2)$ in the phase space

Figure 4.  Orbits from $P_2$ and $P_0$ for different values of $\sigma$. Experiments for $m = 3$, $p = 0.5$, $N = 4$ and $\sigma = 4.822$, respectively $\sigma = 6$

•  [1] D. Andreucci and E. DiBenedetto, On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources, Ann. Scuola Norm. Sup. Pisa, 18 (1991), 363-441. [2] D. Andreucci and A. F. Tedeev, Universal bounds at the blow-up time for nonlinear parabolic equations, Adv. Differ. Equ., 10 (2005), 89-120. [3] C. Bandle and H. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., 316 (1989), 595-622.  doi: 10.2307/2001363. [4] P. Baras and R. Kersner, Local and global solvability of a class of semilinear parabolic equations, J. Differ. Equ., 68 (1987), 238-252.  doi: 10.1016/0022-0396(87)90194-X. [5] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer Verlag, New York-Berlin, 1982. [6] T. Date, Classification and analysis of two-dimensional real homogeneous quadratic differential equation systems, J. Differ. Equ., 32 (1979), 311-334.  doi: 10.1016/0022-0396(79)90037-8. [7] R. Ferreira and A. de Pablo, Grow-up for a quasilinear heat equation with a localized reaction in higher dimensions, Rev. Mat. Complut., 31 (2018), 805-832.  doi: 10.1007/s13163-018-0267-4. [8] R. Ferreira, A. de Pablo and J. L. Vázquez, Classification of blow-up with nonlinear diffusion and localized reaction, J. Differ. Equ., 231 (2006), 195-211.  doi: 10.1016/j.jde.2006.04.017. [9] Y. Giga and N. Umeda, On blow-up at space infinity for semilinear heat equations, J. Math. Anal. Appl., 316 (2006), 538-555.  doi: 10.1016/j.jmaa.2005.05.007. [10] J. Guckenheimer and Ph. Holmes, Nonlinear oscillation, dynamical systems and bifurcations of vector fields, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1990. [11] J. S. Guo, C. S. Lin and M. Shimojo, Blow-up behavior for a parabolic equation with spatially dependent coefficient, Dynam. Systems Appl., 19 (2010), 415-433. [12] J. S. Guo, C. S. Lin and M. Shimojo, Blow-up for a reaction-diffusion equation with variable coefficient, Appl. Math. Lett., 26 (2013), 150-153.  doi: 10.1016/j.aml.2012.07.017. [13] J. S. Guo and M. Shimojo, Blowing up at zero points of potential for an initial boundary value problem, Commun. Pure Appl. Anal., 10 (2011), 161-177.  doi: 10.3934/cpaa.2011.10.161. [14] J. S. Guo and P. Souplet, Excluding blowup at zero points of the potential by means of Liouville-type theorems, J. Differ. Equ., 265 (2018), 4942-4964.  doi: 10.1016/j.jde.2018.06.025. [15] R. G. Iagar and A. Sánchez, Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction with linear growth, J. Dynam. Differ. Equ., 31 (2019), 2061-2094.  doi: 10.1007/s10884-018-09727-w. [16] R. G. Iagar and A. Sánchez, Blow up profiles for a reaction-diffusion equation with critical weighted reaction, Nonlinear Anal., 191 (2020), 111628, 24 pp. doi: 10.1016/j.na.2019.111628. [17] R. G. Iagar and A. Sánchez, Self-similar blow-up profiles for a reaction-diffusion equation with strong weighted reaction, Adv. Nonlinear Stud., 20 (2020), 867-894.  doi: 10.1515/ans-2020-2104. [18] R. G. Iagar and A. Sánchez, Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction, J. Differ. Equ., 272 (2021), 560-605.  doi: 10.1016/j.jde.2020.10.006. [19] R. G. Iagar and A. Sánchez, Self-similar blow-up profiles for a reaction-diffusion equation with critically strong weighted reaction, J. Dynam. Differ. Equ., (2021), 34 pp. doi: 10.1007/s10884-020-09920-w. [20] R. G. Iagar and A. Sánchez, Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction, arXiv: 2103.04500. [21] R. G. Iagar and A. Sánchez, Eternal solutions for a reaction-diffusion equation with weighted reaction, Discret. Cont. Dynam. Syst., (2021), 27 pp. doi: 10.3934/dcds.2021160. [22] X. Kang, W. Wang and X. Zhou, Classification of solutions of porous medium equation with localized reaction in higher space dimensions, Differ. Integral Equ., 24 (2011), 909-922. [23] A. A. Lacey, The form of blow-up for nonlinear parabolic equations, Proc. Royal Society Edinburgh Sect. A, 98 (1984), 183-202.  doi: 10.1017/S0308210500025609. [24] Z. Liang, On the critical exponents for porous medium equation with a localized reaction in high dimensions, Commun. Pure Appl. Anal., 11 (2012), 649-658.  doi: 10.3934/cpaa.2012.11.649. [25] A. de Pablo and A. Sánchez, Self-similar solutions satisfying or not the equation of the interface, J. Math. Anal. Appl., 276 (2002), 791-814.  doi: 10.1016/S0022-247X(02)00450-X. [26] A. de Pablo and J. L. Vázquez, The balance between strong reaction and slow diffusion, Commun. Partial Differ. Equ., 15 (1990), 159-183.  doi: 10.1080/03605309908820682. [27] A. de Pablo and J. L. Vázquez, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differ. Equ., 93 (1991), 19-61.  doi: 10.1016/0022-0396(91)90021-Z. [28] A. de Pablo and J. L. Vázquez, An overdetermined initial and boundary-value problem for a reaction-diffusion equation, Nonlinear Anal., 19 (1992), 259-269.  doi: 10.1016/0362-546X(92)90144-4. [29] L. Perko, Differential equations and dynamical systems. Third edition, in Texts in Applied Mathematics, Springer Verlag, New York, 2001. doi: 10.1007/978-1-4613-0003-8. [30] R. G. Pinsky, Existence and nonexistence of global solutions for $u_t = \Delta u+a(x)u^p$ in $\mathbb{R}^d$, J. Differ. Equ., 133 (1997), 152-177.  doi: 10.1006/jdeq.1996.3196. [31] R. G. Pinsky, The behavior of the life span for solutions to $u_t = \Delta u+a(x)u^p$ in $\mathbb{R}^d$, J. Differ. Equ., 147 (1998), 30-57.  doi: 10.1006/jdeq.1998.3438. [32] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-up in quasilinear parabolic problems, de Gruyter Expositions in Mathematics, 19, W. de Gruyter, Berlin, 1995. doi: 10.1515/9783110889864.535. [33] J. Sotomayor, Generic bifurcations of dynamical systems, in Proceedings of a Symposium Held at University of Bahia, Salvador, Brasil, Academic Press, New York, 1973, 561-582. [34] R. Suzuki, Existence and nonexistence of global solutions of quasilinear parabolic equations, J. Math. Soc. Japan, 54 (2002), 747-792.  doi: 10.2969/jmsj/1191591992. [35] J. L. Vázquez, The porous medium equation. Mathematical theory, in Oxford Monographs in Mathematics, Oxford University Press, 2007.

Figures(4)