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Long-time asymptotic behaviors of solutions of $N$-dimensional dissipative partial differential equations
The generalized Liénard systems
| 1. | Department of Mathematics, University of Turku, FIN-20014 Turku, Finland, Finland |
$\frac{dx}{dt} = \frac{1}{a(x)}[h(y)-F(x)],$
$\frac{dy}{dt}= -a(x)g(x),\qquad\qquad\qquad\qquad\qquad$ (0.1)
where $a$ is a positive and continuous function on $R=(-\infty, \infty)$, and $F$, $g$ and $h$ are continuous functions on $R$. Under the assumption that the origin is a unique equilibrium, we obtain necessary and sufficient conditions for the origin of system (0.1) to be globally asymptotically stable by using a nonlinear integral inequality. Our results substantially extend and improve several known results in the literature.
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