Classical bounded and almost periodic solutions to quasilinear first-order hyperbolic systems in a strip

Abstract

We consider boundary value problems for quasilinear first-order one-dimensional hyperbolic systems in a strip. The boundary conditions are supposed to be of a smoothing type, in the sense that the L 2 -generalized solutions to the initial-boundary value problems become eventually C 2 -smooth for any initial L 2 -data. We investigate small global classical solutions and obtain the existence and uniqueness result under the condition that the evolution family generated by the linearized problem has exponential dichotomy on R. We prove that the dichotomy survives under small perturbations in the leading coefficients of the hyperbolic system. Assuming that the coefficients of the hyperbolic system are almost periodic, we prove that the bounded solution is almost periodic also.

Bounded and almost periodic solvability of nonautonomous quasilinear hyperbolic systems

“…Moreover, we will use the fact that, if a function w(x, t) has bounded and continuous partial derivatives up to the second order in both x ∈ [0, 1] and in t ∈ R and is Bohr almost periodic in t uniformly in x (or, simply, almost periodic), the last property is true for ∂ x w(x, t) and ∂ t w(x, t) also. Specifically, the almost periodicity of ∂ t w(x, t) follows from [6, Theorem 2.5], while the almost periodicity of ∂ x w(x, t) is shown in [20,Section 5.2]. We are, therefore, reduced to showing that the approximating sequence V k , constructed in Sect.…”

“…The smoothing property allowed us in [20] to solve the problem (1.1)- (1.3) where the boundary conditions (1.2) are specified to be of the reflection type, without the requirement of the smallness of D L(BC(Π,R n )) . In [20], we used the assumption that the evolution family generated by a linearized problem has exponential dichotomy on R and proved that the dichotomy survives under small perturbations in the coefficients of the hyperbolic system. For more general boundary conditions (in particular, for (1.2)) when the operator C is not nilpotent, the issue of the robustness of exponential dichotomy for hyperbolic PDEs remains a challenging open problem.…”

Bounded and Almost Periodic Solvability of Nonautonomous Quasilinear Hyperbolic Systems

“…The smoothing property allowed us in [17] to solve the problem (1.1)-(1.2) where the boundary conditions (1.2) are specified to be of the reflection type, without the requirement of the smallness of D L(BC(Π,R n )) . In [17] we used the assumption that the evolution family generated by a linearized problem has exponential dichotomy on R and proved that the dichotomy survives under small perturbations in the coefficients of the hyperbolic system, see also [18]. For more general boundary conditions (in particular, for (1.2)) when the operator C is not nilpotent, the issue of the robustness of exponential dichotomy for hyperbolic PDEs remains a challenging open problem.…”