March  2022, 21(3): 891-925. doi: 10.3934/cpaa.2022003

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

Departamento de Matemática Aplicada, Ciencia e Ingenieria de los Materiales y Tecnologia Electrónica, Universidad Rey Juan Carlos, Móstoles, 28933, Madrid, Spain

* Corresponding author

Received  August 2021 Published  March 2022 Early access  December 2021

Fund Project: 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 $
.
Citation: Razvan Gabriel Iagar, Ana Isabel Muñoz, Ariel Sánchez. Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension. Communications on Pure and Applied Analysis, 2022, 21 (3) : 891-925. doi: 10.3934/cpaa.2022003
References:
[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. FerreiraA. 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. GuoC. 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. GuoC. 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. KangW. 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.

show all references

References:
[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. FerreiraA. 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. GuoC. 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. GuoC. 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. KangW. 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.

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 $
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