Differential and Integral Equations

Smoothness of inertial manifolds for semilinear evolution equations in complex Banach spaces

Satoru Takagi

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We study inertial manifolds for a semilinear evolution equation $du/dt+Au=F(t,u)$ in a complex Banach space. It is known that various conditions ensure existence of inertial manifolds for the equation, however, Miklavčič gave a sharp but simple condition so as to show the existence of inertial manifolds. In this paper, we show smoothness of inertial manifolds using the sharp condition with additional assumptions on $F$, and also apply to a scalar reaction diffusion equation $u_t-u_{xx}=f(t,x,u,u_x)$ with the Dirichlet boundary conditions.

Article information

Differential Integral Equations, Volume 21, Number 1-2 (2008), 63-80.

First available in Project Euclid: 20 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37L25: Inertial manifolds and other invariant attracting sets
Secondary: 34G20: Nonlinear equations [See also 47Hxx, 47Jxx] 35K20: Initial-boundary value problems for second-order parabolic equations 35K58: Semilinear parabolic equations 35K90: Abstract parabolic equations 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25]


Takagi, Satoru. Smoothness of inertial manifolds for semilinear evolution equations in complex Banach spaces. Differential Integral Equations 21 (2008), no. 1-2, 63--80. https://projecteuclid.org/euclid.die/1356039059

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