August  2018, 38(8): 3993-4017. doi: 10.3934/dcds.2018174

Blow-up and superexponential growth in superlinear Volterra equations

School of Mathematical Sciences, Dublin City University, Dublin, Ireland

* Corresponding author

Received  October 2017 Revised  March 2018 Published  May 2018

This paper concerns the finite-time blow-up and asymptotic behaviour of solutions to nonlinear Volterra integro-differential equations. Our main contribution is to determine sharp estimates on the growth rates of both explosive and nonexplosive solutions for a class of equations with nonsingular kernels under weak hypotheses on the nonlinearity. In this superlinear setting we must be content with estimates of the form $\lim_{t\toτ}A(x(t), t) = 1$, where $τ$ is the blow-up time if solutions are explosive or $τ = ∞$ if solutions are global. Our estimates improve on the sharpness of results in the literature and we also recover well-known blow-up criteria via new methods.

Citation: John A. D. Appleby, Denis D. Patterson. Blow-up and superexponential growth in superlinear Volterra equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3993-4017. doi: 10.3934/dcds.2018174
References:
[1]

J. A. D. Appleby and D. D. Patterson, Growth rates of sublinear functional and Volterra differential equations, SIAM. J. Math. Anal., 50 (2018), 2086-2110.  doi: 10.1137/16M1076885.  Google Scholar

[2]

J. A. D. Appleby and D. D. Patterson, Growth rates of solutions of superlinear ordinary differential equations, Appl. Math. Lett., 71 (2017), 30-37.  doi: 10.1016/j.aml.2017.03.012.  Google Scholar

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H. Brunner and Z. Yang, Blow-up behavior of Hammerstein-type Volterra integral equations, J. Integral Equations Appl., 24 (2012), 487-512.  doi: 10.1216/JIE-2012-24-4-487.  Google Scholar

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V. Evtukhov and A. Samoilenko, Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities, Differ. Equ., 47 (2011), 627-649.  doi: 10.1134/S001226611105003X.  Google Scholar

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L. Fu-CaiH. Shu-Xiang and X. Chun-Hong, Global existence and blow-up of solutions to a nonlocal reaction-diffusion system, Discrete Contin. Dyn. Syst., 9 (2003), 1519-1532.  doi: 10.3934/dcds.2003.9.1519.  Google Scholar

[6]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar

[7]

C. Kirk and W. Olmstead, Blow-up in a reactive-diffusive medium with a moving heat source, Z. Angew. Math. Phys., 53 (2002), 147-159.  doi: 10.1007/s00033-002-8147-6.  Google Scholar

[8]

C. KirkW. Olmstead and C. Roberts, A system of nonlinear Volterra equations with blow-up solutions, J. Integral Equations Appl., 25 (2013), 377-393.  doi: 10.1216/JIE-2013-25-3-377.  Google Scholar

[9]

J. Ma, Blow-up solutions of nonlinear Volterra integro–differential equations, Math. Comput. Model., 54 (2011), 2551-2559.  doi: 10.1016/j.mcm.2011.06.020.  Google Scholar

[10]

N. Mahmoudi, Single-point blow-up for a multi-component reaction-diffusion system, Discrete Contin. Dyn. Syst., 38 (2018), 209-230.  doi: 10.3934/dcds.2018010.  Google Scholar

[11]

T. Malolepszy, Blow-up solutions in one-dimensional diffusion models, Nonlinear Anal., 95 (2014), 632-638.  doi: 10.1016/j.na.2013.10.005.  Google Scholar

[12]

T. Malolepszy and W. Okrasiński, Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity, Appl. Math. Lett., 21 (2008), 307-312.  doi: 10.1016/j.aml.2007.03.020.  Google Scholar

[13]

T. Malolepszy and W. Okrasiñski, Blow-up time for solutions to some nonlinear Volterra integral equations, J. Math. Anal. Appl., 366 (2010), 372-384.  doi: 10.1016/j.jmaa.2010.01.030.  Google Scholar

[14]

W. Mydlarczyk, A condition for finite blow-up time for a Volterra integral equation, J. Math. Anal. Appl., 181 (1994), 248-253.  doi: 10.1006/jmaa.1994.1018.  Google Scholar

[15]

W. Mydlarczyk, The blow-up solutions of integral equations, Colloq. Math., 79 (1999), 147-156.  doi: 10.4064/cm-79-1-147-156.  Google Scholar

[16]

W. MydlarczykW. Okrasiński and C. Roberts, Blow-up solutions to a system of nonlinear Volterra equations, J. Math. Anal. Appl., 301 (2005), 208-218.  doi: 10.1016/j.jmaa.2004.07.014.  Google Scholar

[17]

W. Olmstead and C. A. Roberts, Explosion in a diffusive strip due to a source with local and nonlocal features, Methods Appl. Anal., 3 (1996), 345-357.  doi: 10.4310/MAA.1996.v3.n3.a4.  Google Scholar

[18]

C. A. Roberts, Characterizing the blow-up solutions for nonlinear Volterra integral equations, Nonlinear Anal., 30 (1997), 923-933.  doi: 10.1016/S0362-546X(97)00355-6.  Google Scholar

[19]

C. A. Roberts, Analysis of explosion for nonlinear Volterra equations, J. Comput. Appl. Math., 97 (1998), 153-166.  doi: 10.1016/S0377-0427(98)00108-3.  Google Scholar

[20]

C. A. Roberts, Recent results on blow-up and quenching for nonlinear Volterra equations, J. Comput. Appl. Math., 205 (2007), 736-743.  doi: 10.1016/j.cam.2006.01.049.  Google Scholar

[21]

C. A. Roberts and W. Olmstead, Growth rates for blow-up solutions of nonlinear Volterra equations, Quart. Appl. Math., 54 (1996), 153-159.  doi: 10.1090/qam/1373844.  Google Scholar

show all references

References:
[1]

J. A. D. Appleby and D. D. Patterson, Growth rates of sublinear functional and Volterra differential equations, SIAM. J. Math. Anal., 50 (2018), 2086-2110.  doi: 10.1137/16M1076885.  Google Scholar

[2]

J. A. D. Appleby and D. D. Patterson, Growth rates of solutions of superlinear ordinary differential equations, Appl. Math. Lett., 71 (2017), 30-37.  doi: 10.1016/j.aml.2017.03.012.  Google Scholar

[3]

H. Brunner and Z. Yang, Blow-up behavior of Hammerstein-type Volterra integral equations, J. Integral Equations Appl., 24 (2012), 487-512.  doi: 10.1216/JIE-2012-24-4-487.  Google Scholar

[4]

V. Evtukhov and A. Samoilenko, Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities, Differ. Equ., 47 (2011), 627-649.  doi: 10.1134/S001226611105003X.  Google Scholar

[5]

L. Fu-CaiH. Shu-Xiang and X. Chun-Hong, Global existence and blow-up of solutions to a nonlocal reaction-diffusion system, Discrete Contin. Dyn. Syst., 9 (2003), 1519-1532.  doi: 10.3934/dcds.2003.9.1519.  Google Scholar

[6]

G. Gripenberg, S.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34. Cambridge University Press, 1990. doi: 10.1017/CBO9780511662805.  Google Scholar

[7]

C. Kirk and W. Olmstead, Blow-up in a reactive-diffusive medium with a moving heat source, Z. Angew. Math. Phys., 53 (2002), 147-159.  doi: 10.1007/s00033-002-8147-6.  Google Scholar

[8]

C. KirkW. Olmstead and C. Roberts, A system of nonlinear Volterra equations with blow-up solutions, J. Integral Equations Appl., 25 (2013), 377-393.  doi: 10.1216/JIE-2013-25-3-377.  Google Scholar

[9]

J. Ma, Blow-up solutions of nonlinear Volterra integro–differential equations, Math. Comput. Model., 54 (2011), 2551-2559.  doi: 10.1016/j.mcm.2011.06.020.  Google Scholar

[10]

N. Mahmoudi, Single-point blow-up for a multi-component reaction-diffusion system, Discrete Contin. Dyn. Syst., 38 (2018), 209-230.  doi: 10.3934/dcds.2018010.  Google Scholar

[11]

T. Malolepszy, Blow-up solutions in one-dimensional diffusion models, Nonlinear Anal., 95 (2014), 632-638.  doi: 10.1016/j.na.2013.10.005.  Google Scholar

[12]

T. Malolepszy and W. Okrasiński, Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity, Appl. Math. Lett., 21 (2008), 307-312.  doi: 10.1016/j.aml.2007.03.020.  Google Scholar

[13]

T. Malolepszy and W. Okrasiñski, Blow-up time for solutions to some nonlinear Volterra integral equations, J. Math. Anal. Appl., 366 (2010), 372-384.  doi: 10.1016/j.jmaa.2010.01.030.  Google Scholar

[14]

W. Mydlarczyk, A condition for finite blow-up time for a Volterra integral equation, J. Math. Anal. Appl., 181 (1994), 248-253.  doi: 10.1006/jmaa.1994.1018.  Google Scholar

[15]

W. Mydlarczyk, The blow-up solutions of integral equations, Colloq. Math., 79 (1999), 147-156.  doi: 10.4064/cm-79-1-147-156.  Google Scholar

[16]

W. MydlarczykW. Okrasiński and C. Roberts, Blow-up solutions to a system of nonlinear Volterra equations, J. Math. Anal. Appl., 301 (2005), 208-218.  doi: 10.1016/j.jmaa.2004.07.014.  Google Scholar

[17]

W. Olmstead and C. A. Roberts, Explosion in a diffusive strip due to a source with local and nonlocal features, Methods Appl. Anal., 3 (1996), 345-357.  doi: 10.4310/MAA.1996.v3.n3.a4.  Google Scholar

[18]

C. A. Roberts, Characterizing the blow-up solutions for nonlinear Volterra integral equations, Nonlinear Anal., 30 (1997), 923-933.  doi: 10.1016/S0362-546X(97)00355-6.  Google Scholar

[19]

C. A. Roberts, Analysis of explosion for nonlinear Volterra equations, J. Comput. Appl. Math., 97 (1998), 153-166.  doi: 10.1016/S0377-0427(98)00108-3.  Google Scholar

[20]

C. A. Roberts, Recent results on blow-up and quenching for nonlinear Volterra equations, J. Comput. Appl. Math., 205 (2007), 736-743.  doi: 10.1016/j.cam.2006.01.049.  Google Scholar

[21]

C. A. Roberts and W. Olmstead, Growth rates for blow-up solutions of nonlinear Volterra equations, Quart. Appl. Math., 54 (1996), 153-159.  doi: 10.1090/qam/1373844.  Google Scholar

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