| Abstract: |
| We shall discuss some of our recent findings concerning the rigorous solution and analysis of fully non-homogeneous initial-boundary-value problems for the linearized Benjamin-Bona-Mahony equation on the spatiotemporal quarter-plane. The approach is based on complex-analytic tools and a rigorous implementation of the Fokas unified transform method. Explicit solution formulae for the forced linear problem are thus derived in terms of contour integrals and analyzed for quite general initial values and boundary conditions in classical function spaces. The a posteriori pointwise verification of the closed-form representations brings to the fore a single compatibility condition that must be obeyed by smooth data for a well-defined global solution, thereby indicating a type of boundary-smoothing effect. Subsequent boundary-behaviour analysis allows for a uniqueness theorem to be established, relying also on an energy-type argument. Additional surprising observations (e.g., asymptotic instabilities) will be highlighted. For instance, both for Dirichlet and for Neumann boundary conditions, asymptotic periodicity holds. However, for Robin boundary conditions, we find not only that solutions lack the asymptotic periodicity property, but they in fact display instability, growing in amplitude exponentially in time. This is joint work with J.L. Bona, H. Chen and S. Kamvissis. |
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