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Статья
2017

On unique solvability of one nonlinear nonlocal with respect to the solution gradient nonstationary problem


A. S. IvanovaA. S. Ivanova, M. F. PavlovaM. F. Pavlova
Русская математика
https://doi.org/10.3103/S1066369X17030082
Abstract / Full Text

We consider a parabolic equation whose space operator is a product of a nonlinear bounded function which depends on a nonlocal characteristic with respect to a solution gradient and a strongly monotone potential operator. We prove the existence and uniqueness of the solution in the class of the vector-valued functions with values in the Sobolev space.

Author information
  • Kazan (Volga Region) Federal University, ul. Kremlyovskaya 18, Kazan, 420008, RussiaA. S. Ivanova & M. F. Pavlova
References
  1. Zheng, S., Chipot, M. “Nonlinear Nonlocal Evolution Problems”, Asymptot. Anal. 45, No. 3–4, 301–312 (2005).
  2. Chang, N.-H., Chipot, M. “Nonlinear Nonlocal Evolution Problems”, RACSAM, Rev. R. Acad. Cienc. Exactas Fis. Nat., Ser. A Mat. 97, No. 3, 423–445 (2003).
  3. Chipot, M., Molinet, L. “Asymptotic Behavior of Some Nonlocal Diffusion Problems”, Appl. Anal. 80, No. 3/4, 279–315 (2001).
  4. Chipot, M., Lovat, B. “Existence and Uniqueness Results for a Class of Nonlocal Elliptic and Parabolic Problems”, Dyn. Contin. Discrete Impuls. Syst., Ser. A:Math. Anal. 8, No. 1, 35–51 (2001).
  5. Pavlova, M. F. “On the Solvability of Nonlocal Nonstationary Problems with Double Degeneration”, Diff. Equat. 47, No. 8, 1161–1175 (2011).
  6. Glazyrina, O. V., Pavlova, M. F. “On the Unique Solvability of a Certain Nonlocal Nonlinear Problem with a Spatial Operator Strongly Monotone with Respect to the Gradient”, Russian Mathematics 56, No. 3, 83–86 (2012).
  7. Glazyrina, O. V., Pavlova, M. F. “On the Solvability of an Evolution Variational Inequality with a Nonlocal Space Operator”, Diff. Equat. 50, No. 7, 873–887 (2014).
  8. Glazyrina, O. V., Pavlova, M. F. “Study of the Convergence of the Finite-Element Method for Solving Parabolic Equations with a Nonlinear Nonlocal Space Operator”, Diff. Equat. 51 (7), 876–889 (2015).
  9. Karchevskii, M. M., and Pavlova, M. F. Equations of Mathematical Physics. Additional Chapters (Lan’, St.-Petersburg, 2016) [in Russian].