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Well‐posedness for the 2‐D damped wave equations with exponential source terms
Author(s) -
Han Xiaosen,
Wang Mingxin
Publication year - 2010
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1320
Subject(s) - mathematics , uniqueness , bounded function , mathematical analysis , dirichlet boundary condition , infinity , domain (mathematical analysis) , eigenvalues and eigenvectors , wave equation , exponential decay , nonlinear system , boundary value problem , exponential function , exponential growth , function (biology) , boundary (topology) , physics , quantum mechanics , evolutionary biology , nuclear physics , biology
In this paper we study well‐posedness of the damped nonlinear wave equation\documentclass{article}\usepackage{amssymb}\usepackage{amsbsy}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$u_{tt}-\Delta u-\omega\Delta u_t+{\mu} u_t=g(u)$\end{document}in Ω × (0, ∞) with initial and Dirichlet boundary condition, where Ω is a bounded domain in ℝ 2 ; ω⩾0, ωλ 1 +µ>0 with λ 1 being the first eigenvalue of −Δ under zero boundary condition. Under the assumptions that g (·) is a function with exponential growth at the infinity and the initial data lie in some suitable sets we establish several results concerning local existence, global existence, uniqueness and finite time blow‐up property and uniform decay estimates of the energy. Copyright © 2010 John Wiley & Sons, Ltd.