Blow-up solutions of nonlinear Volterra integro-differential equations. (English) Zbl 1235.45007
Summary: We study the finite-time blow-up theory for a class of nonlinear Volterra integro-differential equations. The conditions for the occurrence of finite-time blow-up for nonlinear Volterra integro-differential equations are provided. Moreover, the finite-time blow-up theory for nonlinear partial Volterra integro-differential equations with general kernels is also established using the blow-up results for the nonlinear Volterra integro-differential equations.
References:
| [1] | Małolepszy, T.; Okrasiński, W., Conditions for blow-up of solutions of some nonlinear Volterra integral equations, J. Comput. Appl. Math., 205, 744-750 (2007) · Zbl 1123.45003 |
| [2] | Mydlarczyk, W., The blowup solutions of integral equations, Colloq. Math., 79, 147-156 (1999) · Zbl 0919.45003 |
| [3] | Roberts, C. A.; Lasseigne, D. G.; Olmstead, W. E., Volterra equations which model explosion in a diffuse media, J. Integral Equations Appl., 5, 531-546 (1993) · Zbl 0804.45002 |
| [4] | Roberts, C. A.; Olmstead, W. E., Growth rates for blow-up solutions of nonlinear Volterra equations, Quart. Appl. Math., 54, 153-159 (1996) · Zbl 0916.45007 |
| [5] | Roberts, C. A., Recent results on blow-up and quenching for nonlinear Volterra equations, J. Comput. Appl. Math., 205, 736-743 (2007) · Zbl 1114.35096 |
| [6] | H. Brunner, Z. Yang, Blow-up behavior of Hammerstein-type Volterra integral equations, J. Integral Equations Appl. (submitted for publication).; H. Brunner, Z. Yang, Blow-up behavior of Hammerstein-type Volterra integral equations, J. Integral Equations Appl. (submitted for publication). · Zbl 1261.45003 |
| [7] | Miller, R. K., Nonlinear Volterra Integral Equations (1971), Benjamin: Benjamin Menlo Park, CA · Zbl 0209.14202 |
| [8] | Bellout, H., Blow-up of solutions of parabolic equations with nonlinear memory, J. Differential Equations, 70, 42-68 (1987) · Zbl 0648.45006 |
| [9] | Fujita, Y., Integral equation which interpolates the heat equation and the wave equation, Osaka J. Math., 27, 309-321 (1990), 28 (1990), pp. 797-804 · Zbl 0790.45009 |
| [10] | Bandle, C.; Brunner, H., Blowup in diffusion equations: a survey, J. Comput. Appl. Math., 97, 3-22 (1998) · Zbl 0932.65098 |
| [11] | Souplet, P., Blow-up in nonlocal reaction-diffusion equations, SIAM J. Math. Anal., 29, 1301-1334 (1998) · Zbl 0909.35073 |
| [12] | Souplet, P., Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source, J. Differential Equations, 153, 374-406 (1999) · Zbl 0923.35077 |
| [13] | Souplet, P., Monotonicity of solutions and blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys., 55, 28-31 (2004) · Zbl 1099.35049 |
| [14] | Li, Y.; Xie, C., Blow-up for semilinear parabolic equations with nonlinear memory, Z. Angew. Math. Phys., 55, 15-27 (2004) · Zbl 1099.35043 |
| [15] | Polyanin, A. D.; Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations (2003), Chapman & Hall, CRC: Chapman & Hall, CRC New York · Zbl 1024.35001 |
| [16] | Brunner, H., Collocation Methods for Volterra Integral and Related Functional Equations (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1059.65122 |
| [17] | Mydlarczyk, W.; Okrasiński, W., Nonlinear Volterra integral equations with convolution kernels, Bull. Lond. Math. Soc., 35, 484-490 (2003) · Zbl 1026.45007 |
| [18] | Kaplan, S., On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16, 305-330 (1963) · Zbl 0156.33503 |
| [19] | Lighthill, J. M., Contributions to the theory of the heat transfer through a laminar boundary layer, Proc. R. Soc. Lond., 202, 359-377 (1950) · Zbl 0038.11504 |
| [20] | Rebelo, M.; Diogo, T., A hybrid collocation method for a nonlinear Volterra integral equation with weakly singular kernel, J. Comput. Appl. Math., 234, 2859-2869 (2010) · Zbl 1196.65202 |
| [21] | Ma, J.; Jiang, Y.; Xiang, K., Numerical simulation of blowup in nonlocal reaction-diffusion equations using a moving mesh method, J. Comput. Appl. Math., 230, 8-21 (2009) · Zbl 1166.65067 |
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