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

• Autorzy: Baraket Sami , Mahdaoui Safia , Ouni Taieb

Identyfikatory

DOI

10.7494/OpMath.2023.43.1.5

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

singular limits; Green’s function; nonlinearity; gradient; nonlinear domain decomposition method

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2023
• Tom: Vol. 43, no. 1
• Strony: 5--18
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Baraket Sami

• Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

autor

Mahdaoui Safia

• Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia

autor

Ouni Taieb

• Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia

Bibliografia

• [1] S. Baraket, M. Dammak, T. Ouni, F. Pacard, Singular limits for a 4-dimensional semilinear elliptic problem with exponential nonlinearity, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 6, 875–895.
• [2] S. Baraket, M. Khtaifi, T. Ouni, Singular limits for 4-dimensional general stationary Q-Kuramoto–Sivashinsky equation (Q-KSE) with exponential nonlinearity, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 24 (2016), no. 3, 295–337.
• [3] M. Clapp, C. Munoz, M. Musso, Singular limits for the bi-Laplacian operator with exponential nonlinearity in R4, Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008), no. 5, 1015–1041.
• [4] M. Dammak, T. Ouni, Singular limits for 4-dimensional semilinear elliptic problem with exponential nonlinearity adding a singular source term given by Dirac masses, Differential Integral Equations 21 (2008), no. 11–12, 1019–1036.
• [5] R. Mazzeo, Elliptic theory of edge operators I, Comm. Partial Differential Equations 16 (1991), no. 10, 1615–1664.
• [6] R. Melrose, The Atiyah-Patodi-Singer Index Theorem, Research Notes in Mathematics, vol. 4, A K Peters, Ltd., Wellesley, MA, 1993.
• [7] F. Pacard, T. Rivière, Linear and Nonlinear Aspects of Vortices: The Ginzburg–Landau Model, Progress in Nonlinear Differential Equations, vol. 39, Birkäuser, 2000.
• [8] Y. Rébai, Weak solutions of nonlinear elliptic with prescribed singular set, J. Differential Equations 127 (1996), no. 2, 439–453.
• [9] J. Wei, Asymptotic behavior of a nonlinear fourth order eigenvalue problem, Comm. Partial Differential Equations 21 (1996), no. 9–10, 1451–1467.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)