Long-time existence for a Boussinesq-like system with strong topography variations

In this paper, we investigate the long-time existence of solutions for a Boussinesq-like system modeling surface water waves in shallow water with strong-bottom topography variations (i.e., b = O(1), where b measures the bathymetry). The system is derived under the long-wavelength, small-amplitude regime, with h = 1 − b representing the mean water depth. The strong-bottom topography may exhibit slow (∇h = O(ε)) or fast (∇h = O(1)) oscillations, where the small parameter ε measures the comparable long-wave and weak-nonlinearity effects. We establish the long-time existence results in two typical regimes. Slow oscillation (∇h = O(ε)). For both 1D and 2D cases, we prove the local existence on the time scale O(1/ε) under some restrictions on the parameters ensuring the symmetry of the main dispersive terms. To overcome key challenges arising from h-dependent coefficients and the incomplete control of ∇V in 2D, we design adapted energy functionals and exploit the cancellation of lower-order linear terms in 2D. Fast oscillation (∇h = O(1)). In 1D, we extend the natural O(1) existence time to O(1/ε) for a special case, maintaining the symmetric principal dispersive structure. The proof uses a strategy that first establishes time regularity before recovering spatial derivatives, supported by carefully designed energy estimates and the cancellation of lower-order terms. These results provide the first rigorous long-time existence theory for the Boussinesq-like system with strong topography variations in both 1D and 2D, significantly extending previous studies limited to fully symmetric cases.