Degenerate singular parabolic problems with natural growth

• Autorzy: El Ouardy Mounim , El Hadfi Youssef , Sbai Abdelaaziz

Identyfikatory

DOI

10.7494/OpMath.2024.44.4.471

• Języki publikacji: EN

Abstrakty

EN

In this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term [formula] where Ω is a bounded open subset of RN, N > 2, Q is the cylinder Ω × (0, T), T > 0, Γ the lateral surface ∂Ω×(0, T), 2 ≤ p < N, a(x, t) and b(x, t) are positive measurable bounded functions, q ≥ 0, 0 ≤ γ < 1, and ƒ non-negative function belongs to the Lebesgue space Lm(Q) with m > 1, and u0 ∈ L∞(Ω) such that ∀ω ⊂⊂ Ω∃Dω > 0 : u0 ≥ Dω in ω. More precisely, we study the interaction between the term uq (q > 0) and the singular lower order term d(x, t)|∇u|pu−γ (0 < γ < 1) in order to get a solution to the above problem. The regularizing effect of the term uq on the regularity of the solution and its gradient is also analyzed.

Słowa kluczowe

EN

degenerate parabolic equation; singular parabolic equations; natural growth term

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2024
• Tom: Vol. 44, no. 4
• Strony: 471--503
• Opis fizyczny: Bibliogr. 44 poz.

Twórcy

autor

El Ouardy Mounim

• Sultan Moulay Slimane University, National School of Applied Sciences Khouribga, BP 77 Bd Beni Amir, Khouribga 25000, Morocco

• https://orcid.org/0000-0001-9742-0754

autor

El Hadfi Youssef

• Sultan Moulay Slimane University, National School of Applied Sciences Khouribga, BP 77 Bd Beni Amir, Khouribga 25000, Morocco

• https://orcid.org/0000-0003-4483-2709

autor

Sbai Abdelaaziz

• Sultan Moulay Slimane University, National School of Applied Sciences Khouribga, BP 77 Bd Beni Amir, Khouribga 25000, Morocco

• https://orcid.org/0000-0002-6125-3373

Bibliografia

• [1] M. Abdellaoui, H. Redwane, On some regularity results of parabolic problems with nonlinear perturbed terms and general data, Partial Differ. Equ. Appl. 3 (2022), Article no. 2.
• [2] D. Arcoya, S. Barile, P.J. Martínez-Aparicio, Singular quasi-linear equations with quadratic growth in the gradient without sign condition, J. Math. Anal. Appl. 350 (2009), 401–408.
• [3] D. Arcoya, J. Carmona, T. Leonori, P.J. Martínez-Aparicio, L. Orsina, F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differential Equations 246 (2009), 4006–4042.
• [4] D.G. Aronson, J. Serrin, Local behaviour of solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal. 25 (1967), 81–122.
• [5] A. Benkirane, B. El Haji, M. El Moumni, On the existence of solution for degenerate parabolic equations with singular terms, Pure Appl. Math. Q. 14 (2018), no. 3–4, 591–606.
• [6] L. Boccardo, L. Orsina, M.M. Porzio, Existence result for quasilinear elliptic and parabolic problems with quadratic gradients terms and sources, Adv. Calc. Var. 4 (2011), 397–419.
• [7] L. Boccardo, L. Moreno-Mérida, L. Orsina, A class of quasilinear Dirichlet problems with unbounded coefficients and singular quadratic lower order terms, Milan J. Math. 83 (2015), 157–176.
• [8] M. Chipot, On some singular nonlinear problems for monotone elliptic operators, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), 295–316.
• [9] M. Chipot, L.M. De Cave, New techniques for solving some class of singular elliptic equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018), no. 3, 487–510.
• [10] A. Dall’Aglio, L. Orsina, Nonlinear parabolic equations with natural growth conditions and L1 data, Nonlinear Anal. 27 (1996), 59–73.
• [11] A. Dall’Aglio, D. Giachetti, J.P. Puel, Nonlinear parabolic equations with natural growth in general domains, Boll. Unione Mat. Ital. 3 (2005), 653–684.
• [12] A. Dall’Aglio, D. Giachetti, S. de León, Nonlinear parabolic problem wit a very general quadratic gradient term, Differential Integral Equations 20 (2007), no. 4, 361–396.
• [13] A. Dall’Aglio, L. Orsina, F. Petitta, Existence of solutions for degenerate parabolic equations with singular terms, Nonlinear Anal. 131 (2016), 273–288.
• [14] N.A. Dao, J.I. Díaz, The extinction versus the blow-up: Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption, J. Differential Equations 263 (2017), no. 10, 6764–6804.
• [15] N.A. Dao, J.I. Díaz, Existence and uniqueness of singular solutions of p-Laplacian with absorption for Dirichlet boundary condition, Proc. Amer. Math. Soc. 145 (2017), 5235–5245.
• [16] I. De Bonis, L.M. De Cave, Degenerate parabolic equations with singular lower order terms, Differential Integral Equations 27 (2014), no. 9–10, 949–976.
• [17] I. De Bonis, D. Giachetti, Singular parabolic problems with possibly changing sign data, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 7, 2047–2064.
• [18] I. De Bonis, D. Giachetti, Nonnegative solutions for a class of singular parabolic problems involving p-Laplacian, Asymptot. Anal. 91 (2015), no. 2, 147–183.
• [19] E. DiBendetto, Degenerate parabolic equations, Springer-Verlag, New York, 1993.
• [20] Y. El Hadfi, A. Benkirane, A. Youssfi, Existence and regularity results for parabolic equations with degenerate coercivity, Complex Var. Elliptic Equ. 63 (2018), no. 5, 517–529.
• [21] Y. El Hadfi, M. El Ouardy, A. Ifzarne, A. Sbai, On nonlinear parabolic equations with singular lower order term, J. Elliptic Parabol. Equ. 8 (2022), 49–75.
• [22] M. El Ouardy, Y. El Hadfi, Some nonlinear parabolic problems with singular natural growth term, Results Math. 77 (2022), Article no. 95.
• [23] M. El Ouardy, Y. El Hadfi, A. Ifzarne, Existence and regularity results for a singular parabolic equations with degenerate coercivity, Discrete Contin. Dyn. Syst. Ser. S 15 (2022), no. 1, 117–141.
• [24] M. El Ouardy, Y. El Hadfi, A. Sbai, Existence of positive solution to nonlinear singular parabolic equations with Hardy potential, J. Pseudo-Differ. Oper. Appl. 13 (2022), Article no. 28.
• [25] M. El Ouardy, Y. El Hadfi, A. Sbai, On the existence and regularity of solutions to singular parabolic p-Laplacian equations with absorption term, Rend. Circ. Mat. Palermo (2) 72 (2023), 4119–4147.
• [26] D. Giachetti, F. Murat, An elliptic problem with a lower order term having singular behavior, Boll. Unione Mat. Ital. 2 (2009), 349–370.
• [27] D. Giachetti, F. Petitta, S. Segura de Léon, Elliptic equations having a singular quadratic gradient term and a changin sign datum, Commun. Pure Appl. Anal. 11 (2013), 1875–1895.
• [28] J.L. Lions, Quelques méthodes de résolution des problémes aux limites non linéaires, Dunod, Paris, 1969.
• [29] H.B. Keller, D.S. Choen, Some positone problems suggested by nonlinear heat generation, J. Math. Mech. 16 (1967), 1361–1376.
• [30] M. Magliocca, Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term, Nonlinear Anal. 166 (2018), 102–143.
• [31] M. Magliocca, F. Oliva, On some parabolic equations involving superlinear singular gradient terms, J. Evol. Equ. 21 (2021), 2547–2590.
• [32] P.J. Martínez-Aparicio, Singular Dirichlet problems with quadratic gradient, Bol. Uni. Mat. Ital. 2(3) (2009), 559–574.
• [33] P.J. Martínez-Aparicio, F. Petitta, Parabolic equations with nonlinear singularities, Nonlinear Anal. 74 (2011), 114–131.
• [34] A. Nachman, A. Callegari, A nonlinear singular boundary value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math. 38 (1980), 275–281.
• [35] F. Oliva, F. Petitta, A nonlinear parabolic problem with singular terms and nonregular data, Nonlinear Anal. 194 (2019), 111472.
• [36] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl. (4) 177 (1999), 143–172.
• [37] A. Sbai, Y. El Hadfi, Regularizing effect of absorption terms in singular and degenerate elliptic problems, arXiv:2008.03597, (2020).
• [38] A. Sbai, Y. El Hadfi, Degenerate elliptic problem with a singular nonlinearity, Complex Var. Elliptic Equ. 68 (2023), 701–718.
• [39] A. Sbai, Y. El Hadfi, M. El Ouardy, Singular elliptic problem involving a Hardy potential and lower order term, Asymptot. Anal. 134 (2023), no. 1–2, 213–225.
• [40] A. Sbai, Y. El Hadfi, S. Zeng, Nonlinear singular elliptic equations of p-Laplace type with superlinear growth in the gradient, Mediterr. J. Math. 20 (2023), Article no. 32.
• [41] J. Simon, Compact sets in the space Lp(0, T;B), Ann. Mat. Pura Appl. (4) 146 (1987), 65–96.
• [42] R. Souilah, Existence and regularity results for some elliptic equations with degenerate coercivity and singular quadratic lower-order terms, Mediterr. J. Math. 16 (2019), Article no. 87.
• [43] A. Youssfi, A. Benkirane, Y. El Hadfi, On bounded solutions for nonlinear parabolic equations with degenerate coercivity, Mediterr. J. Math. 13 (2016), 3029–3040.
• [44] Y. Wang, M. Wang, Solution to nonlinear elliptic equations with a gradient, Acta Math. Sci. 35B (2015), no. 5, 1023–1036.

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)