Asymptotic Behavior of Approximated Solutions to Parabolic Equations with Irregular Data

Abstract

LetΩbe a smooth bounded domain inℝN,(N≥3). We consider the asymptotic behavior of solutions to the following problemut-div(a(x)∇u)+λf(u)=μ  in  Ω×ℝ+,u=0  on  ∂Ω×ℝ+,  u(x,0)=u0(x)  in  Ω, whereu0∈L1(Ω),μis a finite Radon measure independent of time. We provide the existence and uniqueness results on the approximated solutions. Then we establish some regularity results on the solutions and consider the long-time behavior.