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1 2023

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On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially “dominated” nonlinearity and singular weight

Baraket Sami, Mahdaoui Safia, Ouni Taieb

Opuscula Mathematica

2023

Vol. 43, no. 1

5--18

Let Ω be a bounded domain in R4 with smooth boundary and let x1, x2, . . . , xm be m-points in Ω. We are concerned with the problem [formula], where the principal term is the bi-Laplacian operator, H(x, u, Dku) is a functional which grows with respect to Du at most like |Du|q, 1 ≤ q ≤ 4, f : Ω → [0,+∞[ is a smooth function satisfying f(pi) > 0 for any i = 1, . . . , n, αi are positives numbers and g : R → [0,+∞[ satisfy |g(u)| ≤ ceu. In this paper, we give sufficient conditions for existence of a family of positive weak solutions (uρ) ρ>0 in Ω under Navier boundary conditions u = Δu = 0 on ∂Ω. The solutions we constructed are singular as the parameters ρ tends to 0, when the set of concentration S = {x1, . . . , xm} ⊂ Ω and the set Λ := {p1, . . . , pn} ⊂ Ω are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method.