Asymptotic behavior of impulsive parabolic problem with infinite-dimensional impulsive set

Abstract In the paper we consider semilinear parabolic equation with a nonlinear term <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ε</m:mi> <m:mo>⁢</m:mo> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\varepsilon f(u)} , whose trajectories undergo impulsive perturbations when they meet the impulsive set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>M</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mo>∥</m:mo> <m:mi>u</m:mi> <m:mo>∥</m:mo> <m:mo>=</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {M=\{\|u\|=R\}} . We prove that for sufficiently small ε this problem generates an impulsive dynamical system which possesses a compact uniform attractor in the phase space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> {L^{2}} .