Existence and uniqueness of generalized solution of two point nonlocal boundary conditions of second mixed problem for second order linear parabolic equation

The aim of this paper is to prove existence and uniqueness of a generalized solution for certain parabolic equations with initial and nonlocal boundary conditions : begin{align*}begin{cases}&dfrac{partial v}{partial t} - a(x, t)dfrac{partial^2 u}{partial x^2} + b(x, t)dfrac{partial v}{partial x} + c(x, t) v =f(x, t)&&v(x, 0) = varphi(x), 0leq xleq ell&&v(0,t) =psi(t), 0leq tleq T&&displaystyle{int_0^Tfrac{v^2(ell, t) -v^2(xi, t)}{ell -xi} },dt = 0, 0< xi < ellend{cases}end{align*}The proofs are based on a priori estimates established in non-classical function spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.