Abstract
We prove the existence and uniqueness of a strong solution for a linear third‐order equation with integral boundary conditions. The proof uses energy inequalities and the density of the range of the operator generated.
Highlights
In the rectangle Ω = (0,1) × (0,T), we consider the equation f (x, t) = ∂3u ∂t3 + ∂ ∂x a(x, t) ∂u ∂x (1.1)with the initial conditions u(x, 0) = 0, ∂u ∂t (x, 0) 0, x ∈ (0, 1), (1.2)the final condition
34 Three-point boundary value problem In addition, we assume that the function a(x,t) and its derivatives satisfy the conditions
Over the last few years, many physical phenomena were formulated into nonlocal mathematical models with integral boundary conditions [1, 9, 10, 11]
Summary
The integral condition u(x, t)dx = 0, 0 ≤ l < 1, t ∈ (0, T). 34 Three-point boundary value problem In addition, we assume that the function a(x,t) and its derivatives satisfy the conditions. Over the last few years, many physical phenomena were formulated into nonlocal mathematical models with integral boundary conditions [1, 9, 10, 11]. The reader should refer to [13, 14] and the references therein The importance of these kinds of problems has been pointed out by Samarskii [22]. This type of boundary value problems has been investigated in [2, 3, 4, 6, 7, 8, 12, 18, 19, 20, 23, 25] for parabolic equations, in [21, 24] for hyperbolic equations, and in [15, 16, 17] for mixed-type equations. We extend this method to the study of a linear third-order partial differential equation
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