Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T11:36:31.715Z Has data issue: false hasContentIssue false
  • English
  • Français

Blowup of Volterra Integro-Differential Equations and Applications to Semi-Linear Volterra Diffusion Equations

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  12 September 2017

Zhanwen Yang ,
Tao Tang  and
Jiwei Zhang
Zhanwen Yang*
Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Tao Tang*
Affiliation:
Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
Jiwei Zhang*
Affiliation:
Beijing Computational Science Research Center, Zhongguancun Software Park II, Haidian District, Beijing 100094, China
*
*Corresponding author. Email addresses:[email protected] (Z. Yang), [email protected] (T. Tang), [email protected] (J. Zhang)
*Corresponding author. Email addresses:[email protected] (Z. Yang), [email protected] (T. Tang), [email protected] (J. Zhang)
*Corresponding author. Email addresses:[email protected] (Z. Yang), [email protected] (T. Tang), [email protected] (J. Zhang)
Get access

Abstract

In this paper, we discuss the blowup of Volterra integro-differential equations (VIDEs) with a dissipative linear term. To overcome the fluctuation of solutions, we establish a Razumikhin-type theorem to verify the unboundedness of solutions. We also introduce leaving-times and arriving-times for the estimation of the spending-times of solutions to ∞. Based on these two typical techniques, the blowup and global existence of solutions to VIDEs with local and global integrable kernels are presented. As applications, the critical exponents of semi-linear Volterra diffusion equations (SLVDEs) on bounded domains with constant kernel are generalized to SLVDEs on bounded domains and ℝN with some local integrable kernels. Moreover, the critical exponents of SLVDEs on both bounded domains and the unbounded domain ℝN are investigated for global integrable kernels.

Keywords

Volterra integro-differential equationsvolterra diffusion equationsblowupglobal existencerazumikhin theorem

MSC classification

Secondary: 35K55: Nonlinear parabolic equations 45K05: Integro-partial differential equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 737 - 759
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bandle, C. and Brunner, H., Blow-up in diffusion equation: a survey, J. Comput. Appl. Math., 97 (1998), pp. 2–22.Google Scholar
[2] Bellout, H., Blow-up of solutions of parabolic equations with nonlinear memory, J. Differential Equations, 70 (1987), pp. 42–68.CrossRefGoogle Scholar
[3] Blanchard, D. and Ghidouche, H., A nonlinear system for irreversible phase changes, European J. Appl. Math., 1 (1990), pp. 91–100.CrossRefGoogle Scholar
[4] Brunner, H., Li, H. and Wu, X., Numerical solution of blow-up problems for nonlinear wave equations on unbounded domains, Commun. Comput. Phys., 14 (2013), pp. 574–598.CrossRefGoogle Scholar
[5] Brunner, H., Wu, X., Zhang, J., Computational solution of blow-up problems for semi-linear parabolic PDEs on unbounded domains, SIAM J. Sci. Comput., 31 (2010), pp. 4478–4496.Google Scholar
[6] Brunner, H. and Yang, Z., Blow-up behavior of Hammerstein-type Volterra integral equations, J. Integral Equations Appl., 24 (2012), pp. 487–512.CrossRefGoogle Scholar
[7] Cho, C., A finite difference scheme for blow-up solutions of nonlinear wave equations, Numer. Math. Theor. Meth. Appl., 3 (2010), pp. 475–498.CrossRefGoogle Scholar
[8] Du, L., Mu, C. and Xiang, Z., Global existence and blow-up to a reaction-diffusion system with nonlinear memory, Commun. Pure Appl. Anal., 4 (2005), pp. 721–733.Google Scholar
[9] Engler, H., On some parabolic integro-differential equations: existence and asymptotics of solutions, In: Proceedings of the international conference on Equadiff 1982, Würzburg, Lecture notes in mathematics, 1017 (1983), pp. 161–167.Google Scholar
[10] Fujita, H., On the blowing up of solutions of the Cauchy problem for ut = Δu+ul+α , J. Fac. Sci. Univ. Tokyo Sect. IA Math., 13 (1966), pp. 109–124.Google Scholar
[11] Habetler, G. T. and Schiffman, R. L., A finite difference method for analyzing the compression of poro-viscoelastic media, Computing, 6 (1970), pp. 342–348.Google Scholar
[12] Hale, J. K., Theory of Functional Differential Equations, Springer, New York, 1977.Google Scholar
[13] Hirata, D., Blow-up for a class of semilinear integro-differential equations of parabolic type, Math. Meth. Appl. Sci., 22 (1999), pp. 1087–1100.Google Scholar
[14] Hu, B., Blow-up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, Heidelberg, 2011.CrossRefGoogle Scholar
[15] Kastenberg, W. E., Space dependent reactor kinetics with positive feed-back, Nukleonik, 11 (1968), pp. 126–130.Google Scholar
[16] Khozanov, A., Parabolic equations with nonlocal nonlinear source, Siberian Math. J., 35 (1994), pp. 545–556.Google Scholar
[17] Kirk, C. M. and Roberts, C. A., A review of quenching results in the context of nonlinear Volterra integral equations, Dyn. Contin. Discrete Impuls. Syst, Ser. A Math. Anal., 10 (2003), pp. 343–356.Google Scholar
[18] Levine, H. A., The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), pp. 262–288.Google Scholar
[19] Li, Y. X. and Xie, C. H., Blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys., 55 (2004), pp. 15–27.CrossRefGoogle Scholar
[20] Liu, M. Z., Yang, Z. W. and Hu, G. D., Asymptotic stability of numerical methods with constant stepsize for pantograph equations, BIT, 45 (2005), pp. 743–759.CrossRefGoogle Scholar
[21] Ma, J. T., Blow-up solutions of nonlinear Volterra integro-differential equations, Math. Comput. Modelling, 54 (2011), pp. 2551–2559.Google Scholar
[22] Małolepszy, T. and Okrasiński, W., Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity, Appl. Math. Lett., 21 (2008), pp. 307–312.Google Scholar
[23] Meier, P., Blow up of solutions of semilinear parabolic differential equations, Z. Angew. Math. Phys., 39 (1988), pp. 135–149.CrossRefGoogle Scholar
[24] Meier, P., On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal., 109 (1990), pp. 63–71.Google Scholar
[25] Miller, R. K., Nonlinear Volterra integral equations, J. London Math. Soc., 217(3) (1971), pp. 503–510.Google Scholar
[26] Mydlarczyk, W., The blow-up solutions of integral equations, Colloq. Math., 79 (1999), pp. 147–156.CrossRefGoogle Scholar
[27] Olmstead, W., Ignition of a combustible half space, SIAM J. Appl. Math., 43 (1983), pp. 1–15.Google Scholar
[28] Pachpatte, B. G., On a nonlinear diffusion system arising in reactor dynamics, J. Math. Anal. Appl., 94 (1983), pp. 501–508.Google Scholar
[29] Pao, C. V., Solution of a nonlinear integrodifferential system arising in nuclear reactor dynamics, J. Math. Anal. Appl., 48 (1974), pp. 470–561.CrossRefGoogle Scholar
[30] Pao, C. V., Bifurcation analysis of a nonlinear diffusion system in reactor dynamics, Appl. Anal., 9 (1979), pp. 107–125.Google Scholar
[31] Ronald, H., Huang, W. and Zegeling, P., A numerical study of blowup in the harmonic map heat flow using the MMPDE moving mesh method, Numer. Math. Theor. Meth. Appl., 6 (2013), pp. 364–383.Google Scholar
[32] Roberts, C. A., Lasseigne, D. G. and Olmstead, W. E., Volterra equations which model explosion in a diffusive medium, J. Integral Equations Appl., 5 (1993), pp. 531–546.CrossRefGoogle Scholar
[33] Roberts, C. A., Recent results on blow-up and quenching for nonlinear Volterra equations, J. Comput. Appl. Math., 205 (2007), pp. 736–743.Google Scholar
[34] Stimming, H., Numerical calculation of monotonicity properties of the blow-up time of NLS, Commun. Comput. Phys., 5 (2009), pp. 745–759.Google Scholar
[35] Souplet, P., Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), pp. 1301–1334.Google Scholar
[36] Souplet, P., Monotonicity of solutions and blow-up in semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys., 55 (2004), pp. 28–31.Google Scholar
[37] Souplet, P., Uniform blow-up profilr and boundary behaviour for a non-local reaction-diffusion equation with critical damping, Math. Methods Appl. Sci., 27 (2004), pp. 1819–1829.Google Scholar
[38] Yamada, Y., On a certain class of semilinear Volterra diffusion equations, J. Math. Anal. Appl., 88 (1982), pp. 433–457.Google Scholar
[39] Yamada, Y., Asymptotic stability for some systems of semilinear Volterra diffusion equations, J. Differ. Equations, 52 (1984), pp. 295–326.Google Scholar
[40] Yang, Z., Zhang, J. and Zhao, C., Numerical blow-up analysis of linearly implicit Euler method for nonlinear parabolic integro-differential equations, submitted, 2017.Google Scholar
[41] Zhang, J., Han, H., Brunner, H., Numerical blow-up of semi-linear parabolic PDEs on unbounded domains in ℝ2 , J. Sci. Comput., 49 (2011), pp. 367–382.Google Scholar
[42] Zhou, J., Mu, C. and Fan, M., Global existence and blow-up to a degenerate reaction–diffusion system with nonlinear memory, Nonlinear Anal.-Real., 9 (2008), pp. 1518–1534.CrossRefGoogle Scholar