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Miyasita, Tosiya

A dynamical system for a nonlocal parabolic equation with exponential nonlinearity. (English) Zbl 1338.35234

Rocky Mt. J. Math. 45, No. 6, 1897-1917 (2015).
Author’s abstract: “We consider a nonlocal parabolic and the corresponding elliptic equation with exponential nonlinearity. At first, we study the set of a stationary solution and compute and compute its Morse index. Next, we obtain the time-global solution in the use of the Lyapunov function and define the dynamical system. Finally, we construct an exponential attractor by a squeezeng property.”
The considered parabolic problem is one-dimensional of the following form \[ u_t=\Delta u +\frac{\lambda}{2}\left( \frac{e^u}{\int_I e^udx}-1\right) \; (x\in I=(0,1), t>0) \] with conditions \(u_x(0,t)=u_x(1,t)=0; u(x,0)=u_0(x); \int_Iu(x,t)dx=0\).
Reviewer: Zdzisław Dzedzej (Gdansk)

MSC:

35K55 Nonlinear parabolic equations
35J60 Nonlinear elliptic equations
37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems

Keywords:

parabolic equation; stationary elliptic equation; Morse index; Lyapunov function; exponential attractor; squeezing property
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\textit{T. Miyasita}, Rocky Mt. J. Math. 45, No. 6, 1897--1917 (2015; Zbl 1338.35234)
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