Weak Periodic Solution for Semilinear Parabolic Problem with Singular Nonlinearities and $$L^{1}$$ Data

On nonlinear parabolic equations with singular lower order term

“…Dropping the first and second non-negative terms in the left hand side of ( 13), since u 0 ∈ L ∞ ( ) and using (3), 𝜖 < 1 n we have and passing to the limit on , we get By working in {u n ≥ 1}, we have then it follows from (15) that we can deduce that (7), by (3) and dropping the nonnegative terms, we get Therefore By the fact that u 0 ∈ L ∞ ( ) and letting goes to zero, implies that Combining ( 16) and ( 17) we obtain Hence by last inequality we deduce that u n is bounded in L 2 (0, T;H 1 0 ( )) with respect to n. ◻ Lemma 5 Let u n be a solution of problem (7), with 𝛾 < 1 and q ≤ 1 − . Suppose that f belongs to…”

“…The maximum principle implies that u n ≥ 0, and this concludes the proof. ◻ Lemma 3 Let u n be a solution of (7). Then for every 𝜔 ⊂⊂ 𝛺 there exists C 𝜔 > 0 independent on n such that u n ≥ C in × (0, T), ∀n ∈ ℕ.…”

“…Proof Define for s ≥ 0 the function We choose (u n ) as test function in (7) with ∈ L 2 (0, T;H 1 0 ( )) ∩ L ∞ ( ), ≥ 0 then we have thus, dropping the non-negative term 1 ∫ {1≤u n ≤ +1} (a(x, t) + u q n )|∇u n | 2 , and letting goes to zero, we obtain Then for the last inequality we can write as follows…”

“…Proof For n fixed, we choose 𝜖 < 1 n and using (u n ) = (u n + ) − as test function in (7), we obtain ( 16)…”

“…In the same concept the authors in [23] proved the existence of solution to problem with 𝛾 > 0, f is a nonnegative function on , and is a nonnegative bounded Radon measures on . Hence Charkaoui and Alaa [7] established the existence of weak periodic solution to singular parabolic problems with 𝛾 > 0 and f is a nonnegative integrable function periodic in time with period T. Let us observe that we refer to [8,9,11,17,24] for more details on singular parabolic problems.…”