Solvability of nonlocal boundary value problems for some systems of evolution differential equations (Q1814478)

In the domain \(Q=\{(x,t): \alpha<x<\beta, 0<t<T\}\) two system of differential equations are considered: \[ L_ 1u(x,t)\equiv u_{tt}- (a(x)u_ x)_{xtt}-(b(x,t)u_ x)_ x+c(x,t)u=f(x,t), \leqno (1) \] \[ L_ 2u(x,t)\equiv u_{tt}-(a(x)u_ x)_{xt}-(b(x,t)u_ x)_ x+c(x,t)u=f(x,t), \leqno (2) \] where \(u,f\) are \(m\)-dimensional vector functions, \(a,b,c\) are symmetric \(m\times m\) matrices, \(a\in C^ 1[\alpha,\beta]\), \(b,c\in C^ 1(\bar Q)\), \(a(\alpha)=a(\beta)\), \(b(\alpha,t)=b(\beta,t)\), and, for every \(z\in R^ m\), the inequalities \(a(x)zz\geq\nu zz\), \(b(x,t)zz\geq0\), \(c(x,t)zz\geq0\), \(\nu>0\) are satisfied in \(\bar Q\). The systems (1) and (2) are associated with the boundary conditions \[ u(\alpha,t)=\lambda u(\beta,t),\quad \lambda u_ x(\alpha,t)=u_ x(\beta,t),\quad u(x,0)=u_ t(x,0),\quad \lambda=\hbox{const}\neq 1 \leqno (3) \] \[ v(\alpha,t)=\lambda v(\beta,t),\lambda v_ x(\alpha,t)=v_ x(\beta,t), v(x,T)=v_ t(x,T)=0. \leqno (4) \] In this paper, the existence and uniqueness of some strong solutions in certain normed function spaces are proved, using various inequalities between some norms.