Asymptotic Behavior of Approximated Solutions to Parabolic Equations with Irregular Data

Abstract

Let $\mathrm{\Omega }$ be a smooth bounded domain in ${ℝ}^N,(N\ge 3)$. We consider the asymptotic behavior of solutions to the following problem $u_t-\text{d}\text{i}\text{v}(a(x)\nabla u)+\lambda f(u)=\mu \text{in}\mathrm{\Omega }\times {ℝ}^{+},u=0\text{on}\partial \mathrm{\Omega }\times {ℝ}^{+},u(x,0)=u_0(x)\text{in}\mathrm{\Omega }$, where $u_0\in L^1(\mathrm{\Omega })$, $\mu$ is 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.