Energy estimates and uniqueness of the weak solutions of initial-boundary value problems for semilinear hyperbolic systems (Q1589878)

The authors study a semilinear hyperbolic system of the type \[ {\partial v\over \partial t}+\Lambda(x,t) {\partial v\over \partial x} + N(x,t,v)+M(x,t)v=h(x,t) \tag{1} \] where \(\Lambda=\text{diag}(\lambda_1,\dots,\lambda_n)\) is a diagonal matrix with \(\lambda_1(x,t)<\dots<\lambda_n(x,t)\). The matrices \(\Lambda\) and \(M\) and the vectors \(N\) and \(h\) are continuous in its entries. For the equation (1) the Cauchy-Dirichlet problem is considered on the domain \([0,l]\times[0,+\infty)\) and boundary conditions on the data are dissipative in the sense that they satisfy the inequality \[ (\Lambda v\cdot v)(0,t)-(\Lambda v\cdot v)(l,t)\leq 0\quad \text{ for\;all} t\geq 0. \] After having introduced the notion of weak solution, the authors prove that some energy identities for both the smooth and the weak solutions to the Cauchy-Dirichlet problem hold. These identities are deduced by the similar ones obtained by \textit{S. K. Godunov} [Equations of mathematical physics. (Ecnaciones de la fisica matematica), (Spanish. Russian original.) Moscow: Mir Publishers (1978; Zbl 0384.35001)] in the linear case. A consequence of these energy identities is the uniqueness in the \(L^2\) sense for the weak solutions to the Cauchy-Dirichlet problem (1).