Electron. J. Diff. Eqns., Vol. 2007(2007), No. 76, pp. 1-10.

Blowup and asymptotic stability of weak solutions to wave equations with nonlinear degenerate damping and source terms

Qingying Hu, Hongwei Zhang

Abstract:
This article concerns the blow-up and asymptotic stability of weak solutions to the wave equation
$$ u_{tt}-\Delta u +|u|^kj'(u_t)=|u|^{p-1}u \quad \hbox{in }\Omega \times (0,T), $$
where p›1 and j' denotes the derivative of a $C^1$ convex and real value function j. We prove that every weak solution is asymptotically stability, for every m such that 0‹m‹1, p‹k+m and the the initial energy is small; the solutions blows up in finite time, whenever p›k+m and the initial data is positive, but appropriately bounded.

Editors note: A reader informed us that that parts of the introduction were copied from reference [2], without giving the proper credit. Also that the first statement in Lemma 4.3 maybe false; so that theorem 4.5 has not been proved. The authors agreed to post a new proof, if they succeed in proving the lemma.

Submitted February 27, 2007. Published May 22, 2007.
Math Subject Classifications: 35B40.
Key Words: Wave equation; degenerate damping and source terms; asymptotic stability; blow up of solutions.

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Qingying Hu
Department of Mathematics
Henan University of Technology
Zhengzhou 450052, China
email: slxhqy@yahoo.com.cn
Hongwei Zhang
Department of Mathematics
Henan University of Technology
Zhengzhou 450052, China
email: wei661@yahoo.com.cn

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