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Abstract <p> For the hyperbolic biwave equation with nonlinear lower terms given in the first quadrantof Euclidean space, we consider a mixed problem in which the Cauchy conditions are specified onthe spatial half-line, and the Dirichlet and Wentzel conditions are specified on the temporalhalf-line. The solution is constructed by the method of characteristics in implicit form as asolution to some integro-differential equations. The solvability of these equations, as well as thedependence on the initial data and the smoothness of their solutions, is studied using theparameter continuation method and a priori estimates. For the problem under consideration, theuniqueness of the solution is proved and conditions under which there exists a classical solution areestablished. If the matching conditions are not met, then a problem with conjugation conditions isconstructed, and if the data is not smooth enough, then a mild solution is constructed.</p>

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Classical Solution to a Mixed Problem with the Dirichlet and Wentzel Conditions for the Biwave Equation with Nonlinear Lower Terms

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • V. I. Korzyuk,
  • J. V. Rudzko

摘要

Abstract

For the hyperbolic biwave equation with nonlinear lower terms given in the first quadrantof Euclidean space, we consider a mixed problem in which the Cauchy conditions are specified onthe spatial half-line, and the Dirichlet and Wentzel conditions are specified on the temporalhalf-line. The solution is constructed by the method of characteristics in implicit form as asolution to some integro-differential equations. The solvability of these equations, as well as thedependence on the initial data and the smoothness of their solutions, is studied using theparameter continuation method and a priori estimates. For the problem under consideration, theuniqueness of the solution is proved and conditions under which there exists a classical solution areestablished. If the matching conditions are not met, then a problem with conjugation conditions isconstructed, and if the data is not smooth enough, then a mild solution is constructed.