Existence and asymptotic behavior of solutions of the Dirichlet problem for a nonlinear pseudoparabolic equation
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Date
2018-01-04
Authors
Ngoc, Le Thi Phuong
Yen, Dao Thi Hai
Long, Nguyen Thanh
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
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.
Description
Keywords
Nonlinear pseudoparabolic equation, Asymptotic behavior, Exponential decay, Periodic weak solution, Faedo-Galerkin method
Citation
Ngoc, 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.