A Problem with Nonlocal Boundary Conditions for a Quasilinear Parabolic Equation

Abstract The solvability of the nonlocal boundary value problem 𝑢 𝑡 = 𝑎(𝑡, 𝑥, 𝑢, 𝑢 𝑥 )𝑢 𝑥𝑥 + 𝑏(𝑡, 𝑥, 𝑢, 𝑢 𝑥 ), 0 ≤ 𝑡 ≤ 𝑇, |𝑥| ≤ 𝑙, 𝑢(0, 𝑥) = 0, 𝑢(𝑡, –𝑙) = 𝑢(𝑡, 𝑙), 𝑢 𝑥 (𝑡, –𝑙) = 𝑢 𝑥 (𝑡, 𝑙) in a class of functions is investigated for a quasilinear parabolic equation. The solution uniqueness follows from the maximum principle.

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