### Nonlocal solvability of the liminal equation of the creeping dislocations density and the topological invariant of the linear thermal deformation of the shell

#### S. N. Alekseenko^{1}, S. N. Nagornykh^{2}, D. V. Khiteva^{3}

Annotation | In the description of a deformation of a shell a coefficient introduced that links the gradient of bending in two directions. That greatly simplified the task. The stress in the shell is proportional to the expected shell deformation and to the square of the gradient of the curvature. The corresponding equation for the scalar density of dislocations is referred to as liminal. A dislocation structure of the considered problem is characterized by a kind of topological invariant for edge dislocations. The value of one of coefficients in liminal equation is closely related to the characteristics of this topological invariant. Characteristics selected in this work allowed us to prove the existence of nonlocal solutions describing the process by which the dislocation density tends to zero. But because of physical grounds and mathematical features of the liminal equations the dislocation density cannot be zero, and then the time of existence of the solution is determined from the condition that the dislocation density decreases to a certain value, characterized by a small dimensionless coefficient $\delta$. Under this assumption, the new global estimates are obtained, based on which the local solution, the existence of which was proved in previous works, extended over a finite number of steps for the whole interval in which the dislocation density is not less than a certain value, characterized by the coefficient $\delta$. An explicit formula to estimate the length of the existence interval of the solution under the made assumptions and conditions is obtained. The mathematical part of the study of the considered problem is made on the basis of the method of an additional argument. |
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Keywords | dislocations density, nonlinear first-order partial differential equation, liminality, method of an additional argument, global estimates, topological invariant |

^{1}The professor of the applied mathematics chair, Nizhniy Novgorod State Technical University, Nizhniy Novgorod; sn-alekseenko@yandex.ru^{2}The senior lecture of the applied mathematics chair, Nizhniy Novgorod State Technical University, Nizhniy Novgorod; algoritm@sandy.ru^{3}The undergraduate of the applied mathematics chair, Nizhniy Novgorod State Technical University, Nizhniy Novgorod; geheimberater@yandex.ru

**Citation**: S. N. Alekseenko, S. N. Nagornykh, D. V. Khiteva, "[Nonlocal solvability of the liminal equation of the creeping dislocations density and the topological invariant of the linear thermal deformation of the shell]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,18:1 (2016) 118–125 (In Russian)