Asymptotic Behavior of Approximated Solutions to Parabolic Equations with Irregular Data

and Applied Analysis 3 For the case λ 0, we consider general bounded Radon measure μ which is independent of time. We provide the uniqueness of approximated solutions for the parabolic problem and its corresponding elliptic problem. Then we prove that the approximated solution of the parabolic equations converges to the unique approximated solution of the corresponding elliptic equations in the norm topology of L Ω ∩H1 0 Ω , for any r ∈ 1,∞ , though they all lie in some less regular spaces. Our main results can be stated as follows. Theorem 1.1. Assume that u0 ∈ L1 Ω , λ > 0, μ is a bounded Radon measure, which does not charge the sets of zero parabolic 2-capacity and is independent of time, f is a C1 function satisfying assumptions 1.3 – 1.5 . Then the semigroup {S t }t≥0, generated by approximated solutions of problem 1.1 , possesses a global attractor A in L1 Ω . Moreover, A is compact and invariant in Lp−1 Ω ∩ W 0 Ω with q < max{N/ N − 1 , 2p − 2 /p}, and attracts every bounded subset of L1 Ω in the norm topology of L Ω ∩H1 0 Ω , 1 ≤ r < ∞. Theorem 1.2. Assume that u0 ∈ L1 Ω , λ 0, μ is a bounded Radon measure independent of time. Then the approximated solution u t of problem 1.1 is unique and converges to the unique approximated solution of the corresponding elliptic equations in the norm topology of L Ω ∩H1 0 Ω , for any 1 ≤ r < ∞. Remark 1.3. Though u t and v all lie in some less-regular spaces, u t converges to v in stronger norm, that is, u t − v converges to 0 in L Ω ∩H1 0 Ω , 1 ≤ r < ∞. Such a result, in some sense, sharpens the result of 13 , where the author showed that u t converges to v in L1 Ω . We organize the paper as follows: in Section 2, we provide the existence of approximated solutions, prove the uniqueness result and some useful lemmas; in Section 3, we establish some improved regularity results on the approximated solutions. At last, in Section 4, we prove the main theorems. For convenience, for any T > 0 we use QT to denote Ω × 0, T hereafter. Also, we denote by |E| the Lebesgue measure of the set E, and denote by C any positive constant which may be different from each other even in the same line. 2. Existence Results and Useful Lemmas We begin this section by providing some existence results on the approximated solutions. Definition 2.1. A function u is called an approximated solution of problem 1.1 , if u ∈ L1 0, T ;W 0 Ω , f u ∈ L1 QT for any T > 0, and − ∫