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Problems with evolution in the boundary condition

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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

  1. 1.

    Chzhou Yui-Lin, “Boundary-value problems for nonlinear parabolic equations,” Mat. Sb.,47, No. 4, 431–484 (1959).

  2. 2.

    M. I. Khazan, “Differentiability of nonlinear semigroups and classical solvability of nonlinear boundary-value problems for the equation ϕ(ut=uxx,” Dokl. Akad. Nauk SSSR,228, No. 4, 805–808 (1976).

  3. 3.

    N. Sauer, “Linear evolution equations in two Banach spaces,” Proc. R. Soc. Edinburgh,91A, No. 3–4, 287–303 (1982).

  4. 4.

    M. I. Khazan, “The existence and the approximation of the solutions of nonlinear evolution equations,” Latv. Mat Ezhegodnik, No. 26, 114–131 (1982).

  5. 5.

    M. I. Khazan, “Nonlinear and quasilinear evolution equations: existence, uniqueness, and comparison of solutions; the rate of convergence of the difference method,” J. Sov. Math.,27, No. 2 (1984).

  6. 6.

    S. G. Krein and M. I. Khazan, “Differential equatiosn in a Banach space,” J. Sov. Math.,30, No. 3 (1985).

  7. 7.

    O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations [in Russian], Nauka, Moscow (1973).

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