Four mixed problems for the vibrating string equation with boundary and nonlocal Dirichlet and Neumann conditions

Abstract

We construct weak solutions u ( x , t ) of the wave equation for four mixed problems with zero initial conditions, the boundary condition u (0, t ) = µ ( t ) or u x (0, t ) = µ ( t ) , and the nonlocal condition u ( l , t ) – α u ( x 0 , t ) = ν ( t ) or u x ( l , t ) – α u x ( x 0 , t ) = ν ( t ) , where 0 ≤ x 0 < l and α is an arbitrary constant. On the rectangle Q T = [0 ≤ x ≤ l ] × [0 ≤ t ≤ T ] , consider the class ( Q T ) introduced by Il’in in [1], namely, the class of two-variable functions u ( x , t ) that are continuous in Q T and have generalized partial derivatives u x ( x , t ) and u t ( x , t ) that belong to L 2 ( Q T ) , as well as to L 2 [0 ≤ x ≤ l ] for all t ∈ [0, T ] and to L 2 [0 ≤ t ≤ T ] for all x ∈ [0, l ] . We search for weak solutions from ( Q T ) to mixed problems for the wave equation