Nonlocal boundary problems for a third-order one-dimensional nonlinear pseudoparabolic equation

We consider initial boundary value problems for a third-order nonlinear pseudoparabolic equation with one space dimension. The boundary condition is given by an integral; the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet or Neumann counterparts. By means of appropriate elliptic estimates we are able to seek solutions not only in the weighted spaces but also in the usual Sobolev spaces. The procedure is carried out in a unified way. Our results characterize a regularity of the pseudoparabolic operator that is different from that of the parabolic operator.