A variety of important physical problems are modelled by pseudoparabolic equations. The authors study an initial boundary value problem for the nonlinear equation:
\(W_{z^*t}=H(z,t,W,W_{z*}\), \(W_ t,W_{zt})\), \(W=W(z,t)\) in \((\mathbb{C}/\Gamma)\times I\) subject to the initial condition: \(W(z,0)=a(z)\), \(z\in \mathbb{C}\) and Riemann type boundary conditions: \(W^ +(\zeta,t)-W^ -(\zeta,t)=g(\zeta,t)\), \((\zeta,t)\in\Gamma\times I\) and \(W(\infty,t)=M(t)\), \(t\in I\). The curve \(\Gamma\) is simple, positively oriented and of class \(C'\) with non-vanishing tangent. The complex valued function \(H\) is measurable on \(\mathbb{C}\times I\times \mathbb{C}^ 5\). It is Lipschitz continuous with respect to the last five variables and has Lipschitz constant in the last variable that is strictly less than one. The functions \(g\), \(M\), and \(a\) are given.
The authors construct solutions without requiring smallness assumptions in the argument. They first derive an a priori estimate for the elliptic equation \(W_{z^*}+\mu_ 1W_ z+\mu_ 2W^*_ z+a_ 1W+a_ 2W^*=f\) and construct a bound. A related linear problem is then solved via successive approximation to establish an estimate for the solution. Based on the estimate of the linear problem, the solution of the nonlinear problem follows by embedding and Newton’s method.