Research of One Mathematical Model of the Distribution of Potentials in a Crystalline Semiconductor

The article is devoted to the research of the Cauchy problem for a mathematical model of the distribution of potentials in a crystalline semiconductor. By a semiconductor we mean a substance with finite electrical conductivity, which rapidly increases with increase in the temperature. The mathematical model of the distribution of potentials is based on the semi-linear Sobolev type equation supplemented by the Dirichlet and Cauchy conditions. We use the phase space method to construct sufficient conditions for the existence of the solution to the model under study. The conditions for the continuability of the solution are given.

Keywords

Sobolev type equations; mathematical model of distribution of potentials in crystalline semiconductor; phase space method; quasi-stationary trajectories.

References

1. Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-Up in Nonlinear Sobolev Type Equations. Berlin, Walter de Gruyter, 2011. DOI:10.1134/S001226610603013X
2. Banasiak J., Pichor K., Rudnicki R. Asynchronous Exponential Growth of a General Structured Population Model. Acta Applicandae Mathematicae, 2012, vol. 119, no. 1, pp. 149-166. DOI: 10.1007/s10440-011-9666-y
3. Banasiak J., Falkiewicz A., Namayanya P. Asymptotic State Lumping in Transport and Diffusion Problems on Networks with Applications to Population Problems. Mathematical Models and Methods in Applied Sciences, 2016, vol. 26, no. 2, pp. 215-247. DOI: 10.1142/S0218202516400017
4. Korpusov M.O., Sveshnikov A.G. Three-Dimensional Nonlinear Evolution Equations of Pseudoparabolic Type in Problems of Mathematical Physics. II. Computational Mathematics and Mathematical Physics, 2003, vol. 43, no. 12, pp. 1835-1869.
5. Sviridyuk G.A. The Manifold of Solutions of an Operator Singular Pseudoparabolic Equation. DAN SSSR, 1986, vol. 289, no. 6, pp. 1-31. (in Russian)
6. Manakova N.A., Sviridyuk G.A. Non-Classical Equations of Mathematical Physics. Phase Spaces of Semilinear Sobolev Equations. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 3, pp. 31-51. DOI:10.14529/mmph160304 (in Russian)
7. Kondyukov A.O., Sukacheva T.G. Phase Space of the Initial-Boundary Value Problem for the Oskolkov System of Nonzero Order. Computational Mathematics and Mathematical Physics, 2015, vol. 55, no. 5, pp. 823-828. DOI:10.1134/S0965542515050127
8. Sviridyuk G.A., Karamova A.F. On the Phase Space Fold of a Nonclassical Equation. Differential Equation, 2005, vol. 41, no. 10, pp. 1476-1481. DOI: 10.1007/s10625-005-0300-5
9. Leng S. Introduction to Differentiable Manifolds. N.Y., Springer, 2002.
10. Sviridyuk G.A. Quasistationary Trajectories of Semilinear Dynamical Equations of Sobolev Type. Izvestiya RAN. Seriya Matematicheskaya, 1994, vol. 42, no. 3, pp. 601-614.
11. Sviridyuk G.A., Klimentev M.V. Phase Spaces of Sobolev-Type Equations with s-Monotone or Strongly Coercive Operators. Russian Mathematics, 1994, vol. 38, no 11, pp. 72-79.
12. Gaevskiy H., Zaharias K. Nichtlineare operatorgleichungen und operatordifferentialgleichungen. Berlin, Akademie Verlag, 1974. (in German)