Journal of the Korean Society for Industrial and Applied Mathematics

  • 제21권2호
  • /
  • Pages.81-87
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  • 2017
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  • 1226-9433(pISSN)
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  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
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THE CONE PROPERTY FOR A CLASS OF PARABOLIC EQUATIONS

  • 투고 : 2017.03.29
  • 심사 : 2017.05.29
  • 발행 : 2017.06.25
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초록

In this note, we show that the cone property is satisfied for a class of dissipative equations of the form $u_t={\Delta}u+f(x,u,{\nabla}u)$ in a domain ${\Omega}{\subset}{\mathbb{R}}^2$ under the so called exactness condition for the nonlinear term. From this, we see that the global attractor is represented as a Lipshitz graph over a finite dimensional eigenspace.

키워드

참고문헌

  1. Z.B. Fang and M. Kwak, Negatively bounded solutions for a parabolic partial differential equation, Bulletin of the Korean Mathematical Society, 42 (2005), 829-836. https://doi.org/10.4134/BKMS.2005.42.4.829
  2. C. Foias, G.R. Sell, and R. Temam, Inertial manifolds for nonlinear evolutionary equations, Journal of Differential Equations, 73 (1988), 309-353. https://doi.org/10.1016/0022-0396(88)90110-6
  3. I. Kukavica, Fourier parameterization of attractors for dissipative equations in one space dimension, Journal of Dynamics and Differential Equations, 15 (2003), 473-484. https://doi.org/10.1023/B:JODY.0000009744.13730.01
  4. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences 68 (1988) Springer-Verlag, New York.
  5. S. Zelik. Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144 (2014), 1245-1327. https://doi.org/10.1017/S0308210513000073

피인용 문헌

  1. REMARKS ON THE EXISTENCE OF AN INERTIAL MANIFOLD vol.58, pp.5, 2021, https://doi.org/10.4134/jkms.j200565