On the existence of solutions to the equation utt = uxxt + σ(ux)x

On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model

“…Boundary value problems for the differential equations of such a type with odd order partial derivatives were also a topic of modern research [1,3,4,6,9,11,14,[17][18][19]24]. The mixed problem for a strongly nonlinear equation of beam vibrations in a bounded domain was in detail studied in [17].…”

“…The existence of a unique classical solution, stable under perturbations of the initial data, was there proved, as well as the behavior of this solution as t → ∞ was described. The conditions for existence of local and global solutions to the mixed problem in Sobolev spaces were formulated in [1]. The case, where the degree of nonlinearity in the main part is a function of space variables was studied in [3].…”

Global Existence of Strong Solutions to a Class of Fully Nonlinear Wave Equations with Strongly Damped Terms

“…We are concerned with the following mixed problem for a class of fully nonlinear wave equations with strongly damped terms in a bounded and C ∞ domain Ω ⊂ R n : 1 1 · · · ∂x α n n , α 1 · · · α n 2, x x 1 , . .…”

Hysteresis and Stick-Slip Motion of Phase Boundaries in Dynamic Models of Phase Transitions

“…Viscosity provides energy dissipation, while time-dependent displacement-controlled loading at the ends of the bar supplies energy into the system. Similar dynamical models have been investigated before [3], [4], [6], [13], [15], [19], [27]; they all consider time-independent loads. This forces the energy to decrease with time.…”