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Problems with evolution in the boundary condition

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Abstract

One considers the problem with the boundary condition\(\frac{{\partial u}}{{\partial t}} + a(x, t) \frac{{\partial u}}{{\partial n}} = g(x, t)\) for the heat-conduction equations. By the methods of evolution equations one proves the existence and the uniqueness of weak and strong solutions and one obtains a series of estimates. The results can be generalized to the nonlinear case.

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Literature cited

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Additional information

Translated from Problemy Matematicheskogo Analiza, No. 10, pp. 105–115, 1986.

The author expresses his deep gratitude to O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva for useful discussions.

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Khazan, M.I. Problems with evolution in the boundary condition. J Math Sci 45, 1191–1199 (1989). https://doi.org/10.1007/BF01096151

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Keywords

  • Boundary Condition
  • Evolution Equation
  • Strong Solution
  • Nonlinear Case