A system of nonlinear Volterra equations with blow-up solutions. (English) Zbl 1295.45005

Kirk, C. M.; Olmstead, W. E.; Roberts, C. A.

J. Integral Equations Appl. 25, No. 3, 377-393 (2013).

A system of two nonlinear convolution-type Volterra equations (of the second kind) \[ u_1(t)=\int_0^tk_1(t-s)F_1(u_2(s)+h_2(s))\;ds, \]
\[ u_2(t)=\int_0^tk_2(t-s)F_2(u_1(s)+h_1(s))\;ds \] with nonnegative kernels and positive, increasing and convex nonlinearities is considered. It is shown that a unique local solution exists which is global if the \(k_i\) decay quickly enough. It is shown that the solution is strictly increasing, and sufficient conditions for a blow-up together with a blow-up rate are obtained. The resuls are applied to fractional integral kernels with power nonlinearities, and for \(k_i(t)=e^{-ct}t^{\beta-1}\).

Reviewer: Martin Väth (Berlin)

MSC:

45G15	Systems of nonlinear integral equations
45D05	Volterra integral equations
45M20	Positive solutions of integral equations
45P05	Integral operators
26A33	Fractional derivatives and integrals

Keywords:

positive solution; convex nonlinearity; blow-up rate; nonlinear fractional integral operator; system of two nonlinear convolution-type Volterra equations; nonnegative kernel

References:

[1]	H. Brunner and Z.W. Yang, Blow-up behavior of Hammerstein-type Volterra integral equations , J. Integral Equations Appl., · Zbl 1273.45001 · doi:10.1216/JIE-2012-24-4-487
[2]	H. Hochstadt, Integral equations , Chapter 2, John Wiley & Sons, New York, 1973. · Zbl 0259.45001
[3]	M. Kirane and S.A. Malik, Profile of blowing-up solutions to a nonlinear system of fractional differential equations , Nonlinear Anal. 73 (2010), 3723-3736. · Zbl 1205.26013 · doi:10.1016/j.na.2010.06.088
[4]	M. Kirane and S. Rihani, private communication, November 13, 2010.
[5]	C.M. Kirk and W.E. Olmstead, Blow-up in a reactive-diffusive medium with a moving heat source , Z. Angew. Math. Phys. 53 (2002), 147-159. · Zbl 0994.35068 · doi:10.1007/s00033-002-8147-6
[6]	—, Blow-up solutions of the two-dimensional heat equation due to a localized moving source , Anal. Appl. 3 (2005), 1-16. · Zbl 1086.35006 · doi:10.1142/S0219530505000443
[7]	T. Malolepzy and W. Okrasinski, Blowup time for solutions to some nonlinear Volterra integral equations , J. Math. Anal. Appl. 366 (2010), 372-384. · Zbl 1188.45003 · doi:10.1016/j.jmaa.2010.01.030
[8]	W.E. Olmstead, C.A. Roberts and K. Deng, Coupled Volterra equations with blow-up solutions , J. Integral Equations Appl. 7 (1995), 499-516. · Zbl 0847.45006 · doi:10.1216/jiea/1181075901
[9]	C.A. Roberts, Recent results on blow-up and quenching for nonlinear Volterra equations , J. Comput. Appl. Math. 205 (2007), 736-743. · Zbl 1114.35096 · doi:10.1016/j.cam.2006.01.049
[10]	C.A. Roberts and W.E. Olmstead, Growth rates for blow-up solutions of nonlinear Volterra equations , Quart. Appl. Math. 54 (1996), 153-159. \noindentstyle · Zbl 0916.45007