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<p>For an abstract linear parabolic equation in a separable Hilbert space with weighted integral condition of a special type in time imposed on the solution, we prove the existence and uniqueness of a weak solution. To do this, we solve the problem approximately using the semidiscrete Galerkin method. We establish a priori estimates for the sequence of approximate solutions and then we prove that the weak limit of this sequence is the exact solution of the original problem.</p>

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WEAK SOLVABILITY OF A VARIATIONAL PARABOLIC EQUATION WITH NONLOCAL-IN-TIME CONDITION ON ITS SOLUTION

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • A. S. Bondarev,
  • A. A. Petrova,
  • O. M. Pirovskikh

摘要

For an abstract linear parabolic equation in a separable Hilbert space with weighted integral condition of a special type in time imposed on the solution, we prove the existence and uniqueness of a weak solution. To do this, we solve the problem approximately using the semidiscrete Galerkin method. We establish a priori estimates for the sequence of approximate solutions and then we prove that the weak limit of this sequence is the exact solution of the original problem.