A Nonlocal Boundary Value Mixed Problem for Second-order Hyperbolic-parabolic Equations in Nonclassical Function Spaces

Aims/ We use an energy method to solve a following boundary-value problem for hyperbolic-parabolic equation with an integral condition :∂2v∂t2+∂v∂t−∂∂x(a(x,t)∂v∂x)+b(x,t)v=F(x,t)v(x,0) =ψ1(x),0≤x≤`,∂v∂t(x,0) =ψ2(x),0≤x≤`,u(`,t) =E(t),0≤t≤T,∫`0xv(x,t)dx=H(t),0≤t≤T.The proof is based on an energy inequality and on the fact that the range of the operator generatedis dense.