Abstract
We study the Cauchy problem in the space of continuous functions for a nonlinear Sobolev type differential equation generalizing the equation of torsional vibrations of an infinite rod in a viscoelastic medium. Conditions for the existence of a global solution and for the blow-up of the solution of the Cauchy problem in finite time are considered.
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Notes
Equation (1) is also a strictly pseudohyperbolic equation: the symbol of the principal part of the differential operator \(L(\partial _x,\partial _t) \) is homogeneous with respect to the vector \( ({1}/{4},{1}/{4})\), the auxiliary algebraic equation \({\eta }^2+\alpha \xi \eta -{\xi }^2=0\) has only real distinct roots with respect to \(\eta \), and the explicit form of the symbols of minor terms implies estimates for their moduli by the majorant \(c(1+{\xi }^2) \), \(c=\mathrm {const}\thinspace \), see [1, Ch. 2, Sec. 2].
In the space \(C[\mathbb {R}^1]\) [9, Ch. VIII, Sec. 1 of the Russian translation; 10, Sec. 2], the operator \(\partial _x \) with the domain \(D(\partial _x)=C^{(1)}[\mathbb {R}^1]\) generates a \(C_0 \)-group of left translations, \(U(t;\partial _x)g(x)=g(x+t) \), while the operator \(\partial ^2_x \) with the domain \(D(\partial ^2_x)=C^{(2)}[\mathbb {R}^1]\) is the generator of the \(C_0 \)-semigroup
$$ U\left (t;\partial ^2_x\right )g(x)=\left (2\sqrt {\pi t}\right )^{-1}\int ^{+\infty }_{-\infty }{e^{-{{\xi }^2}/{(4t)}}g(x+\xi )\thinspace d\xi },\quad t\ge 0. $$
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Translated by V. Potapchouck
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Umarov, K.G. Cauchy Problem for the Equation of Torsional Vibrations of a Rod in a Viscoelastic Medium. Diff Equat 56, 1345–1362 (2020). https://doi.org/10.1134/S00122661200100110
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DOI: https://doi.org/10.1134/S00122661200100110