Global classical solutions of an initial–boundary value problem for first-order nonhomogeneous quasilinear hyperbolic systems

We study a mixed initial–boundary value problem on the half-line for a class of first-order nonhomogeneous quasilinear strictly hyperbolic systems with a non-autonomous source term \(F(t,x)\) . Under the Lax linear degeneracy assumption, we prove the global existence and uniqueness of \(C¹\) classical solutions. The explicit dependence of F on both t and x produces additional terms in the wave decomposition and in the subsequent a priori estimates. To control these terms, we adapt John’s wave decomposition method to the non-autonomous mixed initial–boundary setting and combine it with two basic \(L¹\) estimates. The result allows the sup norms of the derivatives of the initial and boundary data to be large, while smallness is imposed only on several integral quantities. This extends the global \(C¹\) existence theory for linearly degenerate quasilinear hyperbolic systems to the case of non-autonomous source terms.