Analysis of explosion for nonlinear Volterra equations. (English) Zbl 0932.45007

The author gives a survey of results on Volterra integral equations of the form \[ u(t) = \int_0^t k(t-s)r(s)g(u(s)+h(s)) ds,\quad t \geq 0, \] where the solution becomes infinite in finite time. The questions studied in the papers covered by the survey are sufficient conditions to ensure the existence of a blow-up solution, estimates for the time to the singularity and the asymptotic behavior at the singularity. The connections to the related partial differential equations describing various combustion or explosion processes are discussed and some results on the related problem of quenching are mentioned as well.


45G05 Singular nonlinear integral equations
80A25 Combustion
45M05 Asymptotics of solutions to integral equations
Full Text: DOI


[1] Alinhac, S., Blowup for nonlinear hyperbolic equations, () · Zbl 1257.35050
[2] Bandle, C.; Brunner, H., Numerical analysis of semilinear parabolic problems with blow-up solutions, Rev. real acad. cienc. fis. natur. Madrid, 88, 203-222, (1994) · Zbl 0862.65053
[3] Bellout, H., Blow-up of solutions of parabolic equations with nonlinear memory, J. diff. eqns., 70, 42-68, (1987) · Zbl 0648.45006
[4] Bleistein, N.; Handelsman, R.A., Asymptotic expansion of integrals, (1975), Holt, Rinehardt and Winston New York · Zbl 0327.41027
[5] Brunner, H., Numerical blow-up in Volterra integral and integro-differential equations, ()
[6] Boumenir, A., Study of the blow-up set by transformation, J. math. anal. appl., 201, 697-714, (1996) · Zbl 0863.35048
[7] Bushell, P.J.; Okrasinski, W., On the maximal interval of existence for solutions to some non-linear Volterra integral equations with convolution kernel, Bull. London math. soc., 28, 59-65, (1996) · Zbl 0841.45008
[8] Chadam, J.M.; Peirce, A.; Yin, H.-M., The blowup property of solutions to some diffusion equations with localized nonlinear reactions, J. math. anal. appl., 169, 313-328, (1992) · Zbl 0792.35095
[9] Chadam, J.M.; Yin, H.-M., A diffusion equation with localized chemical reactions, (), 101-118 · Zbl 0790.35045
[10] Cui, S.; Ma, Y., Blow-up solutions and global solutions for a class of semilinear integro-differential equations, Math. appl., 6, 531-546, (1993)
[11] Deng, K.; Roberts, C.A., Quenching for a diffusive equation with a concentrated singularity, Diff. int. eqn., 10, 369-379, (1997) · Zbl 0891.35061
[12] Deng, K., Dynamical behavior of solutions of a semilinear heat equation with nonlocal singularity, SIAM J. math. anal., 26, 98-111, (1995) · Zbl 0824.35010
[13] Dimova, S.N.; Kaschiev, M.S.; Koleva, M.G.; Vasileva, D.P., Numerical analysis of the blow-up regimes of combustion of two-component nonlinear heat-conducting medium, Comput. math. phys., 35, 303-319, (1995) · Zbl 0851.65069
[14] Fujita, H., On the blowing up of solutions to the Cauchy problem for ut = δu + u1+α, J. fac. scie. univ. Tokyo, 13, 109-124, (1966) · Zbl 0163.34002
[15] K. Fuller, C.A. Roberts, Blow-up solutions for the wave equation with a concentrated singularity, in preparation.
[16] Glassey, R.T., Finite-time blow-up for solutions of nonlinear wave equations, Math. Z., 177, 323-340, (1982) · Zbl 0438.35045
[17] Gomez, J.L.; Marquez, V.; Wolanski, N., Blow-up results and localizations of blow-up points for the heat equation with a nonlinear boundary condition, J. diff. eqns., 92, 384-401, (1991) · Zbl 0735.35016
[18] Hu, B.; Yin, H.-M., The profile near blow-up time for solution of the heat equation with a nonlinear boundary condition, Inst. math. appl. univ. minnesota, 1116, (1993)
[19] Kapila, A.K., Evolution of deflagration in a cold combustive subjected to a uniform energy flux, Int. J. engng. sci., 43, 495-509, (1981) · Zbl 0469.76050
[20] Kaplan, S., On growth of solutions of quasilinear parabolic equations, Comm. pure appl. math., 16, 305-333, (1963)
[21] Glenn Lasseigne, D.; Olmstead, W.E., Ignition of a combustible solid by convection heating, J. appl. math. phys. (ZAMP), 34, 886-898, (1983) · Zbl 0529.76105
[22] Glenn Lasseigne, D.; Olmstead, W.E., Ignition of a combustible solid with reactant consumption, SIAM J. appl. math., 47, 332-342, (1987) · Zbl 0622.45005
[23] Glenn Lasseigne, D.; Olmstead, W.E., The effect of perturbed heating on the ignition of a combustible solid, Int. J. engng. sci., 27, 1581-1587, (1989) · Zbl 0716.73004
[24] Glenn Lasseigne, D.; Olmstead, W.E., Ignition or nonignition of a combustible solid with marginal heating, Quat. appl. math., 49, 303-312, (1991) · Zbl 0731.76039
[25] Levine, H.A., The role of critical exponents in blowup theorems, SIAM rev., 32, 262-288, (1990) · Zbl 0706.35008
[26] Levine, H.A., The phenomenon of quenching: A survey, (), 275-286 · Zbl 0544.35049
[27] Levine, H.A., Advances in quenching, (), 319-346 · Zbl 0792.35017
[28] Linan, A.; Williams, F.A., Theory of ignition of a reactive solid by a constant energy flux, Comb. sci. technol., 3, 262-288, (1971)
[29] Mydlarczyk, W., A condition for finite blow-up time for a Volterra integral equation, J. math. anal. appl., 181, 248-253, (1994) · Zbl 0808.45008
[30] Olmstead, W.E., Ignition of a combustible half space, SIAM J. appl. math., 43, 1-15, (1983) · Zbl 0546.76131
[31] Olmstead, W.E.; Handelsman, R.A., Asymptotic solution to a class of nonlinear Volterra integral equations, SIAM J. appl. math., 22, 373-384, (1971) · Zbl 0237.45019
[32] Olmstead, W.E.; Handelsman, R.A., Asymptotic solution to a class of nonlinear Volterra integral equations II, SIAM J. appl. math., 30, 180-189, (1976) · Zbl 0323.45007
[33] Olmstead, W.E.; Roberts, C.A., The one-dimensional heat equation with a nonlocal initial condition, Appl. math. lett., 10, 89-94, (1997) · Zbl 0888.35042
[34] Olmstead, W.E.; Roberts, C.A., Explosion in a diffusive strip due to a source with local and nonlocal features, Math. appl. anal., 3, 345-357, (1996) · Zbl 0874.35055
[35] Olmstead, W.E.; Roberts, C.A., Quenching for the heat equation with a nonlocal nonlinearity, (), 199-205 · Zbl 0886.35158
[36] Olmstead, W.E.; Roberts, C.A., Explosion in a diffusive strip due to a concentrated nonlinear source, Meth. appl. anal., 1, 434-445, (1994) · Zbl 0838.35062
[37] Olmstead, W.E.; Roberts, C.A.; Deng, K., Coupled Volterra equations with blow-up solutions, J. int. eqns. appl., 7, 499-516, (1995) · Zbl 0847.45006
[38] C.A. Roberts, A method to determine growth rates of nonlinear Volterra equations, in: Corduneanu, Sandberg (Eds.), Volterra Equations and Applications, Gordon and Breach, UK, to appear. · Zbl 0957.45009
[39] Roberts, C.A., Characterizing the blow-up solutions for nonlinear Volterra integral equations, Nonlinear anal. theory meth. appl., 30, 923-933, (1997) · Zbl 0891.45003
[40] Roberts, C.A.; Lasseigne, D.G.; Olmstead, W.E., Volterra equations which model explosion in a diffusive medium, J. int. eqns. appl., 5, 531-546, (1993) · Zbl 0804.45002
[41] 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
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