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Volterra equations which model explosion in a diffusive medium. (English) Zbl 0804.45002

This paper deals with a class of nonlinear Volterra integral equations \[ u(t) = \int^ t_{t_ 0} k(t - s) G \bigl[ u(s),s \bigr] ds, \quad t \geq t_ 0 \tag{1} \] where the nonlinearity has the form \(G[u(t),t]= r(t)\) \(g[u(t) + h(t)]\). The given nontrivial functions \(r(t)\) and \(h(t)\) are allowed to enhance the explosive behavior by being nondecreasing. The properties of the solution of equation (1) are studied. For instance it is proved that any continuous and differentiable function of (1) is positive and increasing for \(t>t_ 0\).
The basic results provide criteria involving the kernel and the nonlinearity of the solution to experience blow up. Supporting examples from solid combustion and shear band formation are provided.

MSC:

45G10 Other nonlinear integral equations
80A25 Combustion
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References:

[1] N. Bleistein and R.A. Handelsman, Asymptotic expansion of integrals , Holt, Rinehardt and Winston, New York, 1975. · Zbl 0327.41027
[2] R.A. Handelsman and W.E. Olmstead, Asymptotic solution to a class of nonlinear Volterra integral equations , SIAM J. Appl. Math. 22 (1972), 373-384. JSTOR: · Zbl 0237.45019 · doi:10.1137/0122035
[3] A.K. Kapila, Evolution of deflagration in a cold combustible subjected to a uniform energy flux , Int. J. Engrg. Sci. 43 (1981), 495-509. · Zbl 0469.76050 · doi:10.1016/0020-7225(81)90084-7
[4] D. Glenn Lasseigne and W.E. Olmstead, Ignition of a combustible solid by convection heating , J. Appl. Math. and Phys. (ZAMP) 34 (1983), 886-898. · Zbl 0529.76105 · doi:10.1007/BF00949062
[5] ——–, Ignition or nonignition of a combustible solid with marginal heating , Quart. Appl. Math. 49 (1991), 309-312. · Zbl 0731.76039
[6] H.A. Levine, The role of critical exponents in blowup theorems , SIAM Rev. 32 (1990), 262-288. JSTOR: · Zbl 0706.35008 · doi:10.1137/1032046
[7] A. Linan and F.A. Williams, Theory of ignition of a reactive solid by a constant energy flux , Comb. Sci. Tech. 3 (1971), 91-98.
[8] W.E. Olmstead, Ignition of a combustible half space , SIAM J. Appl. Math. 43 (1983), 1-15. JSTOR: · Zbl 0546.76131 · doi:10.1137/0143001
[9] W.E. Olmstead and R.A. Handelsman, Asymptotic solution to a class of nonlinear Volterra integral equations II, SIAM J. Appl. Math. 30 (1976), 180-189. JSTOR: · Zbl 0323.45007 · doi:10.1137/0130020
[10] W.E. Olmstead, S. Nemat-Nasser and L. Ni, Shear bands as discontinuities , Pittsburgh, Center for Nonlinear Analysis, Carnegie Mellon University, 92-NA-013, 1992.
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