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Traveling fronts for a class of non-local convolution differential equations. (English) Zbl 0889.45011

The authors prove the existence of solutions for a class of non-local convolution differential equations \[ u_t= r(u)+ p(u)\Gamma(\omega* q(u)), \] where \[ \omega* q(u)|_\zeta\equiv \int^{+\infty}_{-\infty} \omega(\zeta- y)q(u(y))dy, \] under the assumptions that \(r\in C^1(\mathbb{R})\), \(r(x)<0\) if \(x>0\) and \(r(x)>0\) if \(x<0\), \(p\in C^1(\mathbb{R})\), \(p'\leq 0\), \(p(0)>0\) and there is at most one root of \(p(u)= 0\), \(q\in C^1(\mathbb{R})\), \(q>0\), \(q'>0\) if \(x>0\), \(\Gamma\in C^1(\mathbb{R})\), \(\Gamma>0\), \(\Gamma\) bounded, \(\Gamma'>0\), the kernel \(\omega\) is an even and positive function with unit integral; moreover, if \(f(v)\equiv r(v)+ p(v)\Gamma(q(v))\) then there are three and only three roots of \(f(v)= 0\), say, \(u_0< u_1<u_2\) with \(f'(u_0)< 0\), \(f'(u_2)< 0\) and \(u_0>0\), \(p(u_2)\geq 0\).
The authors prove a local existence theorem and a global existence theorem of traveling fronts for the class of differential equations given above.
Reviewer: D.M.Bors (Iaşi)

MSC:

45K05 Integro-partial differential equations
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[1] Mody R S., J Neurophysiology 8 pp 309– (1945)
[2] Andersen P., Physiological Basis of the Alpha Rhythm (1968)
[3] Steriade M., Thalamic Oscillations and Signaling (1990)
[4] von Krosigk, M., Bal, T. and McCormick, D A. 1993.Cellular mechanisms of a synchronized oscillation in the thalamus, 361–364. Science Wash.DC.
[5] Kim U., J Neurophys 74 pp 1301– (1995)
[6] Golomb D., J Neurophysiol 75 pp 750– (1996)
[7] Chen Z., Wave propagation mediated by GABAB synapse and rebound excitation in an inhibition network (1996)
[8] Ermentrout B., Proceedings of the Royal Society of Edinburgh 123 pp 461– (1993)
[9] Bates P., Traveling waves in convolution model for phase transitions (1996)
[10] Fife P., Arch Rat Mech Ana 65 pp 335– (1997)
[11] Dal Passo R., The heat equation with a non-local density dependent advection term (1996)
[12] De Masi A., Proceedings of the Royal Society of Edinburgh 124 pp 1013– (1994) · Zbl 0819.45005 · doi:10.1017/S0308210500022472
[13] De Masi A., Traveling fronts in non-local evolution equations (1995) · Zbl 0847.45008
[14] Orlandi E., CARR Reports in Mathematical Physics 124 (1995)
[15] Chen X., Existence, uniqueness, and asymptotic stability of traveling waves in non-local evolution equations (1996)
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