Abstract
The paper concerns boundary value problems for general nonautonomous first-order quasilinear hyperbolic systems in a strip. We construct small global classical solutions, assuming that the right-hand sides are small. In the case that all data of the quasilinear problem are almost periodic, we prove that the bounded solution is also almost periodic. For the nonhomogeneous version of a linearized problem, we provide stable dissipativity conditions ensuring a unique bounded continuous solution for any smooth right-hand sides. In the autonomous case, this solution is two times continuously differentiable. In the nonautonomous case, the continuous solution is differentiable under additional dissipativity conditions, which are essential. A crucial ingredient of our approach is a perturbation theorem for general linear hyperbolic systems. One of the technical complications we overcome is the “loss of smoothness” property of hyperbolic PDEs.
Highlights
We study nonautonomous quasilinear hyperbolic systems with lower-order terms and use a different approach focusing on small solutions only
Our dissipativity conditions depend both on the boundary operator and on the coefficients of the hyperbolic system
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
Summary
BC(Π ; Rn) be the Banach space of all continuous and bounded maps u : Π → Rn with the usual sup-norm u BC = sup |u j (x, t)| : (x, t) ∈ Π, j ≤ n. BCk(Π ; Rn) denotes the space of k times continuously differentiable and bounded maps u : Π → Rn, with norm u BCk =. X → Y is denoted by L(X, Y ), with the operator norm A L(X,Y ) = sup{ Au Y :. The restriction of R to BC1(R; Rn) (respectively, to BC2(R; Rn)) is a bounded linear operator on BC1(R; Rn) (respectively, on BC2(R; Rn)). We consider two sets of stable conditions on the data of the original problem. (B3) For each j ≤ n, it holds inf x ,t. Since the constants γ j are positive for all j ≤ n, the condition (B2) allows for R j ≥ 1, what is not allowed by (B1)
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