Mixed problems for a nonlinear coupled evolution system in unbounded domains: a corrected local-energy theory

We consider a coupled hyperbolic-parabolic system in an unbounded domain, with nonlinear damping |u_t|^(p-2) u_t in the hyperbolic equation and nonlinear absorption |theta|^(q-2) theta in the parabolic equation, where p, q > 2. Under globally bounded and uniformly elliptic coefficients, we prove existence and uniqueness of a local-energy weak solution for locally finite-energy data, with no growth or integrability condition imposed as |x| -> infinity. The proof has two independent parts: a bounded-domain energy solution constructed by Faedo-Galerkin approximation, and a compactly supported weighted energy inequality whose spatial cut-off error is absorbed by the two superlinear monotone terms. A correct choice of auxiliary exponents yields convergence of expanding-domain solutions and uniqueness in the full local-energy class.