On the Numerical Solution of a Semilinear Sobolev Equation Subject to Nonlocal Dirichlet Boundary Condition

A semilinear Sobolev equation ∂t(u − ∆u) − ∆u = f(∇u) with a Dirichlet-type integral boundary condition is investigated in this contribution. Using the Rothe method which is based on a semi-discretization of the problem under consideration with respect to the time variable, we prove the existence and uniqueness of a weak solution. Moreover, a suitable approach for the numerical solution based on Legendre spectral-method is presented.