Increasing smoothness of solutions to some huperbolic problems

Smoothing Effect and Fredholm Property for First-order Hyperbolic PDEs

“…This class of boundary operators is in detail described in [18], where the necessary and sufficient conditions for smoothing solutions are given. The results of [18] generalize the smoothing results obtained in [10,13,23,25] for the system (2.1) with time-independent coefficients and (a kind of) Dirichlet boundary conditions.…”

“…Our approach is based on the fact that for a range of boundary operators, solutions improve smoothness dynamically, more precisely, they eventually become k-times continuously differentiable for each particular k. We prove such kind of results in Section 2. Note that in some interesting cases the smoothing phenomenon was shown earlier in [10,13,23,25].…”

Perturbations of superstable linear hyperbolic systems

“…3.5 Condition (2.14) is in general not stable with respect to perturbations of b jj Example 3.7 Let us consider the superstable system perturbed in the diagonal lower order part, namely 21) and supplement it with the reflection boundary conditions…”

Increasing smoothness of solutions to a hyperbolic system on the plane with delay in the boundary conditions

“…It is demonstrated in [7] that this property holds in the case of nonhomogeneous linear problems and in the case of some nonlinear hyperbolic systems. These results allow us to distinguish a class of boundary conditions for the wave equation specified on the lateral sides of Π which are necessary and sufficient for every solution to the homogeneous wave equation to belong to the class C k [0, 1] (with k a positive integer) in some time t (see [8]).…”