Mixed problem with nonlocal boundary conditions for a third‐order partial differential equation of mixed type

Abstract

Abstract. We study a mixed problem with integral boundary conditions for a third-order partial differential equation of mixed type. We prove the existence and uniqueness of the solution. The proof is based on two-sided a priori estimates and on the density of the range of the operator generated by the considered problem.

Boundary value problem for second-order differential operators with integral conditions

“…Boundary value problems for parabolic equations with integral boundary conditions are studied by [27][28][29][30][31][32][33][34][35] using various methods. For instance, the potential method in [28] and [34], Fourier method in [27,[31][32][33] and the energy inequalities method has been used in [29,30,35]. In our case, we apply the method of operator differential equation.…”

Nonlocal problem for a third order partial differential equation of mixed type with an integral two-space-variables condition

“…The existence and uniqueness of the classical solution of mixed problem combining a Dirichlet and integral condition for the equation of heat. Always using the potential method, established in Kamynin [6] the existence and uniqueness of the solution of a similar problem with a more general representation, specially a third order partial differential equation of mixed type with integral condition has been investigated in Denche and Marhoune [12].…”

A third-order non-local problem with boundary integral condition for a parabolic equation

“…In recent years, boundary value problems for parabolic and hyperbolic equations with nonlocal conditions have been the subject of extensive study, see (Bouziani and Benouar[3], Denche [8], Yurchuk [13] and Al-kadhi [11]) and references therein. Our method is as follows.First we establish two-sided a priori estimates.…”