doi: 10.3934/dcdss.2022100
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Blowing-up solutions of differential equations with shifts: A survey

1. 

Department of Mathematics, Faculty of Sciences, Khalifa University, P.O. Box 127788, Abu Dhabi, UAE

2. 

NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

* Corresponding author: Mokhtar Kirane

Received  February 2022 Revised  March 2022 Early access April 2022

Fund Project: The authors are supported by NAAM group

We present a survey on delay ordinary differential equations, delay fractional differential equations and delay partial differential equations with blowing-up or growing-up solutions. Moreover, we indicate the techniques used for the obtained results.

Citation: Mokhtar Kirane, Ahmed Alsaedi, Bashir Ahmad. Blowing-up solutions of differential equations with shifts: A survey. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022100
References:
[1]

J. A. D. Appleby and C. Kelly, Prevention of explosions in solutions of functional differential equations by noise perturbation, Dynam. Systems Appl., 15 (2006), 227-240. 

[2]

J. A. D ApplebyM. J. MacCarthy and A. Rodkina, Exact growth rates of solutions of delay–dominated differential equations with regularly varying coefficients, Proceedings of Neural, Parallel, and Scientific Computations, Dynamic, Atlanta, GA, 4 (2010), 37-42. 

[3]

C. Bandle and H. Brunner, Blow-up in diffusion equations: A survey, J. Comp. Appl. Math., 97 (1998), 3-22.  doi: 10.1016/S0377-0427(98)00100-9.

[4]

C. BandleH. Levine and Q. S. Zhang, Critical exponents of Fujita type for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl., 251 (2000), 624-648. 

[5]

A. C. CasalJ. I. Diaz and J. M. Vegas, Blow-up in some ordinary and parabolic differential equations with time-delay, Dynam. Systems Appl., 18 (2009), 29-46. 

[6]

P.-L. Chow and K. Liu, Positivity and explosion in mean $L^p$-norm of stochastic functional parabolic equations of retarded type, Stochastic Process. Appl., 122 (2012), 1709-1729.  doi: 10.1016/j.spa.2012.01.012.

[7]

K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl., 243 (2000), 85-126.  doi: 10.1006/jmaa.1999.6663.

[8]

J. DiblikR. Chupáč and M. Røužičková, Unbounded solutions of the equation $ y'(t) = \sum_{i = 1}^{n}\beta_i(t)[y(t-\delta_i)-y(t-\tau_i)]$, Appl. Math. Comput., 221 (2013), 610-619.  doi: 10.1016/j.amc.2013.07.001.

[9]

A. Domoshnitsky, Unboundedness of solutions and instability of differential equations of the second order with delayed argument, Differential Integral Equations, 14 (2001), 559-576. 

[10]

A. EreminE. IshiwataT. Ishiwata and U. Nakata, Delay-induced blow-up in a planar oscillation model, Jpn. J. Ind. Appl. Math., 38 (2021), 1037-1061.  doi: 10.1007/s13160-021-00475-x.

[11]

K. Ezzinbi and M. Jazar, Blow-up results for some nonlinear delay differential equations, Positivity, 10 (2006), 329-341.  doi: 10.1007/s11117-005-0026-x.

[12]

G. Friesecke, Exponentially growing solutions for a delay-diffusion equation with negative feedback, J. Differential Equations, 98 (1992), 1-18.  doi: 10.1016/0022-0396(92)90101-R.

[13]

H. Fujita, On the blowup of solutions of the Cauchy problem for $ u_t = \Delta u+u^{1+\alpha} $, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124. 

[14]

V. A. Galaktionov and J. L. Vazquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dynam. Systems, 8 (2002), 399-433.  doi: 10.3934/dcds.2002.8.399.

[15]

N. Garofalo, Fractional thoughts, New Developments in the Analysis of Nonlocal Operators, Contemp. Math., 723. Amer. Math. Soc., RI, 723 (2019), 1-135.  doi: 10.1090/conm/723/14569.

[16]

S. Goodchild and H. Yang, Local well-posedness of a nonlocal Burgers' equation, Involve, 9 (2016), 67-82.  doi: 10.2140/involve.2016.9.67.

[17]

I. GyöriY. Nakata and G. Röst, Unbounded and blow-up solutions for a delay logistic equation with positive feedback, Commun. Pure Appl. Anal., 17 (2018), 2845-2854.  doi: 10.3934/cpaa.2018134.

[18]

A. InoueT. Miyakawa and K. Yoshida, Some properties of solutions for semi-linear heat equations with time lag, J. Differential Equations, 24 (1977), 383-396.  doi: 10.1016/0022-0396(77)90007-9.

[19]

M. Jleli, M. Kirane and B. Samet, Blow-up phenomena for second-order differential inequalities with shifted arguments, Electron. J. Differential Equations, (2016), Paper No. 91, 12 pp.

[20]

S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.

[21]

K. Kobayashi, On the semilinear heat equations with time-lag, Hiroshima Math. J., 7 (1977), 459-472. 

[22]

K. Kobayashi, On the semi-linear heat equations with time lag, Hiroshima Math. J., 10 (1980), 189-227. 

[23]

N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972.

[24]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288.  doi: 10.1137/1032046.

[25]

J. H. Lightbourne Ⅲ and S. M. Rankin Ⅲ, Global existence for a delay differential equation, J. Differential Equations, 40 (1981), 186-192. 

[26]

S. Luckhaus, Global boundedness for a delay-differential equation, Trans. Amer. Math. Soc., 294 (1986), 767-774.  doi: 10.2307/2000215.

[27]

G. LvL. Wang and X. Wang, Positive and unbounded solution of stochastic evolution equations, Stochastic Process. Appl., 34 (2016), 927-939.  doi: 10.1080/07362994.2016.1196459.

[28]

É. Mitidieri and S. I. Pokhozhaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362. 

[29]

A. D. Myshkis, Linear Differential Equations with Delayed Argument, Second edition, Izdat. "Nauka", Moscow, 1972,352 pp.

[30]

M. Nagazawa and T. Sirao, Probabilistic treatment of the blowing up of solutions of nonlinear integral equation, Trans. Amer. Math. Soc., 139 (1969), 301-310.  doi: 10.1090/S0002-9947-1969-0239379-X.

[31]

R. D. ParshadS. BhommickE. QuansahR. Agarwal and R. K. Upadhyay, Finite time blow up in population model with competitive interference and time delay, Int. J. Nonlinear Sci. Numer. Simul., 18 (2017), 435-450.  doi: 10.1515/ijnsns-2015-0179.

[32]

O. Salieva, Blow-up solutions to some differential equations and inequalities with shifted arguments, Electron. J. Differential Equations, (2016), Paper No. 57, 8 pp.

[33]

A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Translated by M. Grinfeld, Walter de Gruyter, Berlin, New York, 1995.

[34]

P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334.  doi: 10.1137/S0036141097318900.

[35]

B. Straughan, Explosive Instabilities in Mechanics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58807-5.

[36]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51. 

[37]

H. WangY. Chen and H. Lu, Blow-up results for some reaction-diffusion equations with delay, Ann. Polon. Math., 105 (2012), 21-29.  doi: 10.4064/ap105-1-3.

show all references

References:
[1]

J. A. D. Appleby and C. Kelly, Prevention of explosions in solutions of functional differential equations by noise perturbation, Dynam. Systems Appl., 15 (2006), 227-240. 

[2]

J. A. D ApplebyM. J. MacCarthy and A. Rodkina, Exact growth rates of solutions of delay–dominated differential equations with regularly varying coefficients, Proceedings of Neural, Parallel, and Scientific Computations, Dynamic, Atlanta, GA, 4 (2010), 37-42. 

[3]

C. Bandle and H. Brunner, Blow-up in diffusion equations: A survey, J. Comp. Appl. Math., 97 (1998), 3-22.  doi: 10.1016/S0377-0427(98)00100-9.

[4]

C. BandleH. Levine and Q. S. Zhang, Critical exponents of Fujita type for inhomogeneous parabolic equations and systems, J. Math. Anal. Appl., 251 (2000), 624-648. 

[5]

A. C. CasalJ. I. Diaz and J. M. Vegas, Blow-up in some ordinary and parabolic differential equations with time-delay, Dynam. Systems Appl., 18 (2009), 29-46. 

[6]

P.-L. Chow and K. Liu, Positivity and explosion in mean $L^p$-norm of stochastic functional parabolic equations of retarded type, Stochastic Process. Appl., 122 (2012), 1709-1729.  doi: 10.1016/j.spa.2012.01.012.

[7]

K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: The sequel, J. Math. Anal. Appl., 243 (2000), 85-126.  doi: 10.1006/jmaa.1999.6663.

[8]

J. DiblikR. Chupáč and M. Røužičková, Unbounded solutions of the equation $ y'(t) = \sum_{i = 1}^{n}\beta_i(t)[y(t-\delta_i)-y(t-\tau_i)]$, Appl. Math. Comput., 221 (2013), 610-619.  doi: 10.1016/j.amc.2013.07.001.

[9]

A. Domoshnitsky, Unboundedness of solutions and instability of differential equations of the second order with delayed argument, Differential Integral Equations, 14 (2001), 559-576. 

[10]

A. EreminE. IshiwataT. Ishiwata and U. Nakata, Delay-induced blow-up in a planar oscillation model, Jpn. J. Ind. Appl. Math., 38 (2021), 1037-1061.  doi: 10.1007/s13160-021-00475-x.

[11]

K. Ezzinbi and M. Jazar, Blow-up results for some nonlinear delay differential equations, Positivity, 10 (2006), 329-341.  doi: 10.1007/s11117-005-0026-x.

[12]

G. Friesecke, Exponentially growing solutions for a delay-diffusion equation with negative feedback, J. Differential Equations, 98 (1992), 1-18.  doi: 10.1016/0022-0396(92)90101-R.

[13]

H. Fujita, On the blowup of solutions of the Cauchy problem for $ u_t = \Delta u+u^{1+\alpha} $, J. Fac. Sci. Univ. Tokyo, Sect. I, 13 (1966), 109-124. 

[14]

V. A. Galaktionov and J. L. Vazquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dynam. Systems, 8 (2002), 399-433.  doi: 10.3934/dcds.2002.8.399.

[15]

N. Garofalo, Fractional thoughts, New Developments in the Analysis of Nonlocal Operators, Contemp. Math., 723. Amer. Math. Soc., RI, 723 (2019), 1-135.  doi: 10.1090/conm/723/14569.

[16]

S. Goodchild and H. Yang, Local well-posedness of a nonlocal Burgers' equation, Involve, 9 (2016), 67-82.  doi: 10.2140/involve.2016.9.67.

[17]

I. GyöriY. Nakata and G. Röst, Unbounded and blow-up solutions for a delay logistic equation with positive feedback, Commun. Pure Appl. Anal., 17 (2018), 2845-2854.  doi: 10.3934/cpaa.2018134.

[18]

A. InoueT. Miyakawa and K. Yoshida, Some properties of solutions for semi-linear heat equations with time lag, J. Differential Equations, 24 (1977), 383-396.  doi: 10.1016/0022-0396(77)90007-9.

[19]

M. Jleli, M. Kirane and B. Samet, Blow-up phenomena for second-order differential inequalities with shifted arguments, Electron. J. Differential Equations, (2016), Paper No. 91, 12 pp.

[20]

S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.

[21]

K. Kobayashi, On the semilinear heat equations with time-lag, Hiroshima Math. J., 7 (1977), 459-472. 

[22]

K. Kobayashi, On the semi-linear heat equations with time lag, Hiroshima Math. J., 10 (1980), 189-227. 

[23]

N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972.

[24]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), 262-288.  doi: 10.1137/1032046.

[25]

J. H. Lightbourne Ⅲ and S. M. Rankin Ⅲ, Global existence for a delay differential equation, J. Differential Equations, 40 (1981), 186-192. 

[26]

S. Luckhaus, Global boundedness for a delay-differential equation, Trans. Amer. Math. Soc., 294 (1986), 767-774.  doi: 10.2307/2000215.

[27]

G. LvL. Wang and X. Wang, Positive and unbounded solution of stochastic evolution equations, Stochastic Process. Appl., 34 (2016), 927-939.  doi: 10.1080/07362994.2016.1196459.

[28]

É. Mitidieri and S. I. Pokhozhaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362. 

[29]

A. D. Myshkis, Linear Differential Equations with Delayed Argument, Second edition, Izdat. "Nauka", Moscow, 1972,352 pp.

[30]

M. Nagazawa and T. Sirao, Probabilistic treatment of the blowing up of solutions of nonlinear integral equation, Trans. Amer. Math. Soc., 139 (1969), 301-310.  doi: 10.1090/S0002-9947-1969-0239379-X.

[31]

R. D. ParshadS. BhommickE. QuansahR. Agarwal and R. K. Upadhyay, Finite time blow up in population model with competitive interference and time delay, Int. J. Nonlinear Sci. Numer. Simul., 18 (2017), 435-450.  doi: 10.1515/ijnsns-2015-0179.

[32]

O. Salieva, Blow-up solutions to some differential equations and inequalities with shifted arguments, Electron. J. Differential Equations, (2016), Paper No. 57, 8 pp.

[33]

A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations, Translated by M. Grinfeld, Walter de Gruyter, Berlin, New York, 1995.

[34]

P. Souplet, Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), 1301-1334.  doi: 10.1137/S0036141097318900.

[35]

B. Straughan, Explosive Instabilities in Mechanics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58807-5.

[36]

S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51. 

[37]

H. WangY. Chen and H. Lu, Blow-up results for some reaction-diffusion equations with delay, Ann. Polon. Math., 105 (2012), 21-29.  doi: 10.4064/ap105-1-3.

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