Abstract
We obtain conditions for the existence of a global solution and the blow-up of the solution of the Cauchy problem on a finite time interval for a nonlinear third-order partial differential equation generalizing the equation of torsion vibrations of a cylindrical rod with allowance for internal and external damping modeling the propagation of longitudinal stress waves along a one-dimensional viscoelastic rod whose the material obeys Voigt–Kelvin medium deformation law.
Notes
A classical solution is understood as a sufficiently smooth function that has all continuous derivatives of the desired order and satisfies the equation at each point in the domain where it is given.
Note that \(\beta \xi -2\sqrt {\alpha \delta }|\xi |=2\sqrt {\alpha \delta }|\xi |K(\xi )<0\), where \( K(\xi )=\mathrm {sign}\thinspace (\xi ){\beta }/{(2\sqrt {\alpha \delta })}-1 \), \(\xi \in \mathbb {R}^1 \).
The elements of the Banach space \(C[\mathbb {R}^1] \) satisfy the estimate \({\|\varphi \cdot \psi \|}_C\le {\|\varphi \|}_C {\|\psi \|}_C\); this makes it possible to operate with it like with an algebra [11, Sec. 6.1].
Here, taking into account the a priori estimates (50) and (57), the following algorithm is used [12]: we take \(v(t_0,x) \) as a new initial function and note that the function \(v(t_0,x) \) has the same a set of properties that, being assumed for the function \(v(0,x)\), made it possible to prove the existence of both the classical solution \(v(t,x) \) of Eq. (4) on the interval \(0\le t\le t_0 \) and the corresponding solution \(u(t,x) \) to Eq. (1) on the same interval \([0,t_0] \), the classical solution \(v(t,x) \) extends from the interval \([0,t_0] \) to the classical solution \(v(t,x) \), \(t\in [0,t_0+t_1] \), where the value of \(t_1 \) depends on the parameters \(\alpha \), \(\beta \), and \(\gamma \), the initial functions \(\varphi ( x) \) and \(\psi (x) \), and the nonlinearity \(\sigma (\cdot ) \). Repeating this process sufficiently many times, we obtain an unbounded extension of the time interval on which the solution is guaranteed; i.e., we obtain a classical solution of the Cauchy problem for Eq. (1) on an arbitrary time interval.
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Translated by V. Potapchouck
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Umarov, K.G. Solution Blow-Up and Global Solvability of the Cauchy Problem for a Model Third-Order Partial Differential Equation. Diff Equat 59, 51–73 (2023). https://doi.org/10.1134/S0012266123010056
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DOI: https://doi.org/10.1134/S0012266123010056