<abstract xmlns="http://www.w3.org/1999/xhtml">In this work we study the continuity of four different notions of <italic>asymptotic behavior</italic> for a family of non-autonomous non-classical parabolic equations given by<disp-formula id="j_AMNS.2017.1.00004_eq_501_w2aab3b7d741b1b6b1aab1c14b2aAa"><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mi>γ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>ϵ</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>)</mo><mo>,</mo><mtext> in </mtext><mi>Ω</mi></mtd></mtr><mtr><mtd><mi>u</mi><mo>=</mo><mn>0</mn><mo>,</mo><mtext> on </mtext><mo>∂</mo><mi>Ω</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow></math><tex-math>$$\begin{array}{}
\displaystyle
\left\{ \begin{array}{*{20}{l}}
 {{u_t} - \gamma \left( t \right)\Delta {u_t} - \Delta u = {g_\varepsilon }\left( {t,u} \right),{\;\text{in}\;}\Omega } \hfill \\
 {u = 0,{\;\text{on}\;}\partial \Omega {\rm{.}}} \hfill \\
\end{array}\right.
\end{array}$$</tex-math></alternatives></disp-formula>in a smooth bounded domain Ω ⊂ ℝn, <italic>n</italic> ⩾ 3, where the terms <italic>gε</italic> are a small perturbation, in some sense, of a function <italic>f</italic> that depends only on <italic>u</italic>.</abstract>

In this work we study the continuity of four different notions of asymptotic behavior for a family of non-autonomous non-classical parabolic equations given by{ut−γ(t)Δut−Δu=gϵ(t,u), in Ωu=0, on ∂Ω.$$\begin{array}{}
\displaystyle
\left\{ \begin{array}{*{20}{l}}
   {{u_t} - \gamma \left( t \right)\Delta {u_t} - \Delta u = {g_\varepsilon }\left( {t,u} \right),{\;\text{in}\;}\Omega } \hfill  \\
   {u = 0,{\;\text{on}\;}\partial \Omega {\rm{.}}} \hfill  \\
\end{array}\right.
\end{array}$$in a smooth bounded domain Ω ⊂ ℝn, n ⩾ 3, where the terms gε are a small perturbation, in some sense, of a function f that depends only on u.

Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In this work we study the continuity of four different notions of asymptotic behavior for a family of non-autonomous non-classical parabolic equations given...

Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

Published Online: Feb 08, 2017

Page range: 31 - 60

Received: Oct 10, 2016

Accepted: Feb 08, 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00004

Keywords
Non-autonomous perturbations, non-autonomous dynamical systems, pullback attractors, cocyle attractor, uniform attractor, non-classical parabolic equations, evolution process, skew-product semiflow

© 2017 Matheus C. Bortolan, Felipe Rivero, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

Published Online: Feb 08, 2017

Page range: 31 - 60

Received: Oct 10, 2016

Accepted: Feb 08, 2017

DOI: https://doi.org/10.21042/AMNS.2017.1.00004

KeywordsNon-autonomous perturbations, non-autonomous dynamical systems, pullback attractors, cocyle attractor, uniform attractor, non-classical parabolic equations, evolution process, skew-product semiflow

© 2017 Matheus C. Bortolan, Felipe Rivero, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Non-autonomous perturbations, non-autonomous dynamical systems, pullback attractors, cocyle attractor, uniform attractor, non-classical parabolic equations, evolution process, skew-product semiflow