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Improved Hardy-Rellich inequalities
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 |
$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $ |
$ x\in \mathbb{R}^N $ |
$ t\geq0 $ |
$ m>1 $ |
$ p\in(0, 1) $ |
$ \sigma>2(1-p)/(m-1) $ |
$ \sigma $ |
$ N>1 $ |
$ N = 1 $ |
$ \sigma $ |
$ N = 1 $ |
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. 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. |
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. 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. |


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