Abstract:
We study the second boundary value problem for a second-order partial differential equation of mixed type in a cylindrical domain. Using the modified Galerkin method with the involvement of the time regularization method under certain conditions on the coefficients of the equation, the existence of a generalized solution to the second boundary value problem is established. Furthermore, the corresponding a priori estimate for the solution of the regularized boundary value problem is derived. Conditions on the equation’s coefficients that guarantee the uniqueness of the generalized solution to the second boundary value problem are also provided. Under certain conditions on the coefficients of the equation, a theorem on the unique regular solvability of the second boundary value problem for the mixed-type equation in a weighted Sobolev space is proved for the case where the coefficient of the second time derivative can change sign on the upper base and is negative on the lower base of the cylindrical domain. To prove this theorem, an a priori estimate for the solution of the regularized boundary value problem is established in the norm of the weighted Sobolev space.
Keywords:mixed type equation, second boundary value problem, Galerkin method, inequality, solvability, estimate, convergence
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