Convective stability in scalar balance laws

We fill the two main remaining gaps in the full classification of non-degenerate planar traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under smooth perturbations. On one hand, we investigate the impact on the classification of allowing to restrict perturbations by further localization constraints, the so-called \emph{convective} stability analysis. On the other hand, in the multidimensional case, we extend the existing analysis to general perturbations, relaxing the artificial restriction, assumed so far, that initial perturbations are supported away from characteristic points. A striking feature of the latter analysis is that it involves infinite-dimensional families of genuinely multidimensional traveling waves that live near the perturbed planar traveling wave.