The third boundary value problem for parabolic differential-difference equation in one-dimensional case

This work is concerned with the differential-difference equation u_t-(R_{2Q}u_x)_x+R_{1Q}u_x+R_{0Q}u =f(x,t),quad(x,t)in Q_T, with the boundary conditions left.left(-R_{2Q}u_x+alpha uright)right|_{x=0}=0,quad left.left(R_{2Q}u_x+beta uright)right|_{x=d}=0, quad 0<t<T, and the initial condition u|_{t=0}=phi(x),quad xin Q, where R_0,R_1, and R_2 are bounded difference operators, Q=(0,d), Q_T=Qtimes(0,T), 0<T<infty, fin L_2(Q_T), phiin L_2(0,d), and alpha,betain{Bbb R}^+. par The author studies solvability and smoothness of strong solutions of the above problem. Using semigroup theory and the theory of elliptic and parabolic functional-differential equations as well as interpolation theory, the author obtains results on existence and uniqueness of weak solutions and strong solutions as well as the smoothness of strong solutions.