On the solvability of parabolic and hyperbolic problems with a boundary integral condition
The paper deals with the following two linear equations: \[ (1) \quad v_t + Lv = f, \qquad \quad (2) \quad v_{tt} + Lv = f \] where \(Lv = -( a(x,t)v_x)_x + b(x,t)v\), and the unknown function \(v(x,t)\) is defined for \(\alpha < x < \beta\), \(0 < t < T\). Each of the equations (1), (2) is equipped with initial and boundary conditions of the form \[ v_x(\alpha, t) = \mu (t), \qquad \quad \int_{\alpha}^{\beta} v(x,t) dx = m(t), \qquad \quad 0<t<T. \tag{3} \] Here \(a, b, f, \mu, m\) are known functions and \(a\) is supposed to be bounded away from zero. The author proves a priori estimates for the generalized solution of (1), (3) or (2), (3) in suitably defined function spaces. The above estimates yield the theorem about the existence and uniqueness of the generalized solution.
scientific article; zbMATH DE number 1780417