Longtime behavior  of  the  semilinearwave equation with gentle dissipation

Zhijian Yang; Zhiming  Liu; Na  Feng

doi:10.3934/dcds.2016084

Article Contents

2016, Volume 36, Issue 11: 6557-6580. Doi: 10.3934/dcds.2016084

This issue Previous Article Integrability of vector fields versus inverse Jacobian multipliers and normalizers Next Article Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems

Longtime behavior of the semilinear wave equation with gentle dissipation

1.
Department of Mathematics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001
2.
School of Mathematics and Statistics, Zhengzhou University, No.100, Science Road, Zhengzhou 450001

Received: November 30, 2015

Revised: May 31, 2016

Published: May 2017

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Abstract

The paper investigates the well-posedness and longtime dynamics of the semilinear wave equation with gentle dissipation： $u_{tt}-\triangle u+\gamma(-\triangle)^{\alpha} u_{t}+f(u)=g(x)$, with $\alpha\in(0,1/2)$. The main results are concerned with the relationships among the growth exponent $p$ of nonlinearity $f(u)$ and the well-posedness and longtime behavior of solutions of the equation. We show that (i) the well-posedness and longtime dynamics of the equation are of characters of parabolic equations as $1 \leq p < p^* \equiv \frac{N + 4\alpha}{(N-2)^+}$; (ii) the subclass $\mathbb{G}$ of limit solutions has a weak global attractor as $p^* \leq p < p^{**}\equiv \frac{N+2}{N-2}\ (N \geq 3)$.

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Mathematics Subject Classification: Primary: 35B41, 35B33; Secondary: 35B40, 35B65, 37L30.

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