Ngoc, Le Thi PhuongYen, Dao Thi HaiLong, Nguyen Thanh2021-12-172021-12-172018-01-04Ngoc, L. T. P., Yen, D. T. H., & Long, N. T. (2018). Existence and asymptotic behavior of solutions of the Dirichlet problem for a nonlinear pseudoparabolic equation. <i>Electronic Journal of Differential Equations, 2018</i>(04), pp. 1-20.1072-6691https://hdl.handle.net/10877/15059This 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.Text20 pages1 file (.pdf)enAttribution 4.0 InternationalNonlinear pseudoparabolic equationAsymptotic behaviorExponential decayPeriodic weak solutionFaedo-Galerkin methodArticle