Existence and asymptotic behavior of solutions of the Dirichlet problem for a nonlinear pseudoparabolic equation

Abstract

This article concerns the initial-boundary value problem for non-linear pseudo-parabolic equation ut - uxxt - (1 + μ(ux))uxx + (1 + σ(ux))u = ƒ(x, t), 0 < x < 1, 0 < t < T, u(0, t) = u(1, t) = 0, u(x, 0) = ũ0(x), where ƒ, ũ0, μ, σ are given functions. Using the Faedo-Galerkin method and the compactness method, we prove that there exists a unique weak solution u such that u ∈ L∞ (0, T; H¹₀ ∩ H²), u′ ∈ L²(0, T; H¹₀) and ∥u∥L∞(QT) ≤ max{∥ũ0∥L∞(Ω), ∥ƒ∥L∞(QT}. Also we prove that the problem has a unique global solution with H1-norm decaying exponentially as t → +∞. Finally, we establish the existence and uniqueness of a weak solution of the problem associated with a periodic condition.