We consider the Cauchy problem for a class of strictly pseudohyperbolic fourth-order equations with variable coefficients. In particular, this class includes the Vlasov, Galpern, and Rayleigh–Bishop equations. Unique solvability of the Cauсhy problem under some conditions on right-hand sides is proved in weighted Sobolev spaces.
The work is investigating the solvability of nonlocal problems with the generalized Samarskii–Ionkin boundary condition for a special class of Sobolev-type equations, namely, for the so-called pseudoparabolic equations. We prove the existence and uniqueness theorems for regular solutions to the problems, i.e., for the solutions that have all generalized by Sobolev derivatives included in the corresponding equations. Moreover, some improvements and generalizations of the obtained results are described.
We analyze an equilibrium state of a three-dimensional elastic body with a thin two-dimensional inclusion. At the first step we assume the inclusion to be delaminated from the surrounding elastic body, which provides an interfacial crack. To describe a mutual nonpenetration between the crack faces, we impose inequality type boundary conditions at the crack faces. Additional difficulties appear in view of the Neumann type conditions considered at the external boundary of the elastic body. We analyze a passage to the limit provided that the stiffness parameter of the inclusion tends to infinity. The limit model is analyzed, which describes an equilibrium state of the elastic body with a thin rigid inclusion. Various equivalent problem formulations of the limit problem are presented.
The paper is devoted to the study of the problem of identifying a coefficient in the right-hand side of a one-dimensional third-order hyperbolic equation known as the Moore–Gibson–Thompson (MGT) equation with Dirichlet boundary conditions. An integral overdetermination condition serves as additional information required to recover the coefficient. The main goal of the work is to prove the existence of a solution in Sobolev spaces. The proof method is based on reducing the original inverse problem to an auxiliary direct problem. The solvability of the auxiliary problem is established using the parameter continuation method and the regularization method. A key step is obtaining a priori estimates for solutions in specially introduced function spaces. The paper formulates sufficient conditions on the problem’s input data that guarantee the existence of solution to the direct problem. The solution to the inverse problem is then explicitly expressed through the solution of the direct problem.
We consider the existence and uniqueness of the classical solution to the Showalter–Sidorov-type problem for some classes of quasilinear equations in Banach spaces with fractional Riemann derivatives. Equations with a degenerate operator at the highest order Riemann–Liouville derivative are studied in the case of sectoriality of the pair of operators in the linear part of the equation, i.e., when this pair generates a degenerate resolving family of operators analytic in a sector, and under the assumption that the nonlinear operator does not depend on elements of the degeneration subspace. We proved theorems on local and global unique solvability of the Showalter–Sidorovtype problem for the equation under study. The obtained abstract results are applied to the study of initial boundary value problems for a class of systems of partial differential equations of fractional order in time.
