Skip to main content

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

  • Zhanwen Yang (a1), Tao Tang (a2) and Jiwei Zhang (a3)

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 ℝ N with some local integrable kernels. Moreover, the critical exponents of SLVDEs on both bounded domains and the unbounded domain ℝ N are investigated for global integrable kernels.

Corresponding author
*Corresponding author. Email addresses: (Z. Yang), (T. Tang), (J. Zhang)
Hide All
[1] Bandle, C. and Brunner, H., Blow-up in diffusion equation: a survey, J. Comput. Appl. Math., 97 (1998), pp. 222.
[2] Bellout, H., Blow-up of solutions of parabolic equations with nonlinear memory, J. Differential Equations, 70 (1987), pp. 4268.
[3] Blanchard, D. and Ghidouche, H., A nonlinear system for irreversible phase changes, European J. Appl. Math., 1 (1990), pp. 91100.
[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. 574598.
[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. 44784496.
[6] Brunner, H. and Yang, Z., Blow-up behavior of Hammerstein-type Volterra integral equations, J. Integral Equations Appl., 24 (2012), pp. 487512.
[7] Cho, C., A finite difference scheme for blow-up solutions of nonlinear wave equations, Numer. Math. Theor. Meth. Appl., 3 (2010), pp. 475498.
[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. 721733.
[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. 161167.
[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. 109124.
[11] Habetler, G. T. and Schiffman, R. L., A finite difference method for analyzing the compression of poro-viscoelastic media, Computing, 6 (1970), pp. 342348.
[12] Hale, J. K., Theory of Functional Differential Equations, Springer, New York, 1977.
[13] Hirata, D., Blow-up for a class of semilinear integro-differential equations of parabolic type, Math. Meth. Appl. Sci., 22 (1999), pp. 10871100.
[14] Hu, B., Blow-up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, Springer, Heidelberg, 2011.
[15] Kastenberg, W. E., Space dependent reactor kinetics with positive feed-back, Nukleonik, 11 (1968), pp. 126130.
[16] Khozanov, A., Parabolic equations with nonlocal nonlinear source, Siberian Math. J., 35 (1994), pp. 545556.
[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. 343356.
[18] Levine, H. A., The role of critical exponents in blowup theorems, SIAM Rev., 32 (1990), pp. 262288.
[19] Li, Y. X. and Xie, C. H., Blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys., 55 (2004), pp. 1527.
[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. 743759.
[21] Ma, J. T., Blow-up solutions of nonlinear Volterra integro-differential equations, Math. Comput. Modelling, 54 (2011), pp. 25512559.
[22] Małolepszy, T. and Okrasiński, W., Blow-up conditions for nonlinear Volterra integral equations with power nonlinearity, Appl. Math. Lett., 21 (2008), pp. 307312.
[23] Meier, P., Blow up of solutions of semilinear parabolic differential equations, Z. Angew. Math. Phys., 39 (1988), pp. 135149.
[24] Meier, P., On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal., 109 (1990), pp. 6371.
[25] Miller, R. K., Nonlinear Volterra integral equations, J. London Math. Soc., 217(3) (1971), pp. 503510.
[26] Mydlarczyk, W., The blow-up solutions of integral equations, Colloq. Math., 79 (1999), pp. 147156.
[27] Olmstead, W., Ignition of a combustible half space, SIAM J. Appl. Math., 43 (1983), pp. 115.
[28] Pachpatte, B. G., On a nonlinear diffusion system arising in reactor dynamics, J. Math. Anal. Appl., 94 (1983), pp. 501508.
[29] Pao, C. V., Solution of a nonlinear integrodifferential system arising in nuclear reactor dynamics, J. Math. Anal. Appl., 48 (1974), pp. 470561.
[30] Pao, C. V., Bifurcation analysis of a nonlinear diffusion system in reactor dynamics, Appl. Anal., 9 (1979), pp. 107125.
[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. 364383.
[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. 531546.
[33] Roberts, C. A., Recent results on blow-up and quenching for nonlinear Volterra equations, J. Comput. Appl. Math., 205 (2007), pp. 736743.
[34] Stimming, H., Numerical calculation of monotonicity properties of the blow-up time of NLS, Commun. Comput. Phys., 5 (2009), pp. 745759.
[35] Souplet, P., Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29 (1998), pp. 13011334.
[36] Souplet, P., Monotonicity of solutions and blow-up in semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys., 55 (2004), pp. 2831.
[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. 18191829.
[38] Yamada, Y., On a certain class of semilinear Volterra diffusion equations, J. Math. Anal. Appl., 88 (1982), pp. 433457.
[39] Yamada, Y., Asymptotic stability for some systems of semilinear Volterra diffusion equations, J. Differ. Equations, 52 (1984), pp. 295326.
[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.
[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. 367382.
[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. 15181534.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 1
Total number of PDF views: 36 *
Loading metrics...

Abstract views

Total abstract views: 223 *
Loading metrics...

* Views captured on Cambridge Core between 12th September 2017 - 20th March 2018. This data will be updated every 24 hours.