Attractors for the nonclassical reaction–diffusion equations on time-dependent spaces
Kaixuan Zhu, Yongqin Xie, Feng Zhou
Abstract
Abstract In this paper, based on the notation of time-dependent attractors introduced by Conti, Pata and Temam in (J. Differ. Equ. 255:1254–1277, 2013), we prove the existence of time-dependent global attractors in $\mathcal{H}_{t}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>H</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math> for a class of nonclassical reaction–diffusion equations with the forcing term $g(x)\in H^{-1}(\varOmega )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>g</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mo>∈</mml:mo><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>)</mml:mo></mml:math> and the nonlinearity f satisfying the polynomial growth of arbitrary $p-1$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:math> ( $p\geq 2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mn>2</mml:mn></mml:math> ) order, which generalizes the results obtained in (Appl. Anal. 94:1439–1449, 2015) and (Bound. Value Probl. 2016: 10, 2016).