The Existence and Behavior of Solutions for Nonlocal Boundary Problems

Abstract

The purpose of this work is to investigate the uniqueness and existence of local solutions for the boundary value problem of a quasilinear parabolic equation. The result is obtained via the abstract theory of maximal regularity. Applications are given to some model problems in nonstationary radiative heat transfer and reaction-diffusion equation with nonlocal boundary flux conditions.

Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation

“…Combining estimates (13), (15), (16), (17) and (20) we have w t Eα ≤ M 1 Aϕ Eα + M 6 (a, σ, α, x * , q, T ) Using integral inequality, we can write, w t Eα ≤ e M 6 (a,σ,α,x * ,q,T ) M 1 Aϕ Eα + M 7 (a, σ, α, x * , q, T ) Aϕ Eα + +M 8 (a, σ, α, x * , q, T ) ρ C[0,T ] + M 12 (α, T ) f C(Eα) .…”

On source identification problem for a delay parabolic equation

“…Such conditions usually are identified as nonlocal boundary conditions and reflect situations when the data on the domain boundary cannot be measured directly, or when the data on the boundary depend on the data inside the domain. The well-posedness of various nonlocal boundary value problems for partial differential and difference equations has been studied extensively by many researchers (see [1][2][3][4][5][6][7][8][9][10][11][12] and the references given therein).…”