Boundedness and Regularity of Solutions for Nonlinear Parabolic Problem with two Singularities

Mecheter, Rabah; Khelifi, Hichem

doi:10.2478/tmmp-2025-0026

Abstract

We examine the boundedness and regularity of weak solutions to a type of nonlinear parabolic equations with singular natural growth gradient terms. These equations also have a singular right-hand side with f ∈ L^m(Q_T ), (m ≥ 1). With suitable test functions, we can use Stampacchia’s lemma to prove that the solutions u(x, t) are bounded when $m > \frac{N}{p} + 1$ m > {N \over p} + 1 . Furthermore, we can establish a regularity result if f ∈ L¹(Q_T ).

References

ANDREU, F.—SEGURA, S.—BOCCARDO, L.—ORSINA, L.: Existence results for L¹ data of some quasi-linear parabolic problems with a quadratic gradient term and source, Mathematical Models Methods Appl. Sci. 12 (2002), no. 1, 1–16.
Search in Google Scholar Back to article
ARCOYA, D.—CARMONA, J.—LEONORI, T.—APARICIO, J. M. —ORSINA, L.— PETITTA, F.: Existence and nonexistence of solutions for singular quadratic quasilinear equations, J. Differential Equ. 246 (2009), 4006-4042.
Search in Google Scholar Back to article
BOCCARDO, L.—MURAT, F.—PUEL, J. P.: L^∞estimate for nonlinear elliptic partial differential equations and application to an existence result, SIAM J. Math. Anal. 23 (1992) no. 2, 326–333.
Search in Google Scholar Back to article
BOCCARDO, L.—DALL’AGLIO, A.—ORSINA. L.: Existence and regularity results for some elliptic equations with degenerate coercivity, AttiSemin. Mat. Fis. Univ. Modena Reggio Emilia 46, (1998), 51–81.
Search in Google Scholar Back to article
CROCE, G.: An elliptic problem with two singularities, Asymptotic Anal. 78 (2012), no. 1–2, 1–10.
Search in Google Scholar Back to article
DALL’AGLIO, A.—ORSINA, L.—PETITTA, F.: Existence of solutions for degenerate parabolic equations with singular terms, Nonlinear Analysis, 131 (2016), 273–288.
Search in Google Scholar Back to article
DE BONIS, I.—DE CAVE L. M.: Degenerate parabolic equations with singular lower order terms, Differential Integral Equations 27 (2014), 949–976.
Search in Google Scholar Back to article
DIBENEDETTO, E.: Degenerate Parabolic Equations. Springer-Verlag, Berlin, 1993.
Search in Google Scholar Back to article
EL OUARDY, M.—EL HADFI, Y. —A. SBAI, A.: On the existence and regularity of solutions to singular parabolic p-Laplacian equations with absorption term, Rend. Circ. Mat. Palermo II. Ser. 72 (2023), 4119–4147.
Search in Google Scholar Back to article
EL OUARDY, M.— EL HADFI, Y.—SBAI, A.: Existence of positive solutions to nonlinear singular parabolic equations with Hardy potential, J. Pseudo-Differ. Oper. Appl. 13 (2022), Paper no. 28, 32 pp.
Search in Google Scholar Back to article
EL OUARDY, M.—EL HADFI, Y.—SBAI, A.: Degenerate singular parabolic problems with natural growth, Opuscula Math. 44 (2024) no. 4, 471–503.
Search in Google Scholar Back to article
EL OUARDY, M.—EL HADFI, Y.—SBAI, A.: Existence and regularity results of parabolic problems with convection term and singular nonlinearity, Z. Anal. Anwendungen 43 (2024), no. 3, 299–328.
Search in Google Scholar Back to article
EL OUARDY, M.—EL HADFI, Y.—SBAI, A.: Existence and regularity results for some nonlinear singular parabolic problems with absorption terms, Monatsh. Math. 203 (2024), 583–612.
Search in Google Scholar Back to article
EL OUARDY, M.—EL HADFI, Y.—IFZARNE, A.: Existence end regularity results for a singular parabolic equations with degenerate coercivity, Discrete Contin. Dyn. Syst. Ser. S 15 (2022), no. 1, 117–141.
Search in Google Scholar Back to article
EL OUARDY, M.—EL HADFI, Y.: Some nonlinear parabolic problems with singular natural growth term, Results Math. 77 (2022) no. 3, Paper no. 95.
Search in Google Scholar Back to article
GRENON, N.—MERCALDO, A.: Existence and regularity results for solutions to nonlinear parabolic equations, Adv. Difference Equ. 10 (2005), no. 9, 1007–1034.
Search in Google Scholar Back to article
KHELIFI, H.: Regularity for entropy solutions of degenerate parabolic equations with L^mdata, Math. Model. Comput. 10 (2023), no. 1, 119–132.
Search in Google Scholar Back to article
KHELIFI, H.: Existence and regularity for a degenerate problem with singular gradient lower order term, Mem. Differ. Equ. Math. Phys. 90 (2023), 51–66.
Search in Google Scholar Back to article
KHELIFI, H.: Existence and regularity for solution to a degenerate problem with singular gradient lower order term, Moroccan Journal of Pure and Applied Analysis (MJPAA), 8 (2022), no. (3), 310–327.
Search in Google Scholar Back to article
KHELIFI, H.—ZOUATINI, M. A.: An nonlinear elliptic problem involves two types of terms: degenerate coercivity and singular nonlinearity, Bol. Soc. Paran. Mat. 43 (2025), 1–13. DOI: https://doi.org/10.5269/bspm.66340
Search in Google Scholar Back to article
LEONI, F.— PELLACCI. B.: Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data, J. Evol. Equ. 6 (2006), no. 1, 113–144.
Search in Google Scholar Back to article
MARTÍNEZ-APARICIO, P. J.—PETITTA, F.: Parabolic equations with nonlinear singularities, Nonlinear Anal. 74 (2011), 114–131.
Search in Google Scholar Back to article
MECHETER, R.—MOKHTARI. F.: Existence results for some nonlinear parabolic equations with degenerate coercivity and singular lower-order terms, Math. Bohem. 148 (2023), no. 4, 561–581.
Search in Google Scholar Back to article
MOKHTARI, F.—KHELIFI, H.: Regularity results for degenerate parabolic equations with L^mdata, Complex Var. Elliptic Equ. 68 (2023), no. 12, 2001–2015.
Search in Google Scholar Back to article
LIONS, J. L.: Quelques Méthodes de Résolution des Probèmes aux Limites. Dunod, Paris, 1969.
Search in Google Scholar Back to article
SOUILAH, R.: Existence and regularity results for some elliptic equations with degenerate coercivity and singular quadratic lower-order terms, Mediterr. J. Math. (2019) 16, no. 4, paper no. 87, 21 pp, DOI: 10.1007/s00009-019-1360-8.
Search in Google Scholar Back to article
SBAI, A.—EL HADFI, Y.—ZENG, S.: Nonlinear singular elliptic equations of p-Laplace type with superlinear growth in the gradient, Mediterr. J. Math. 20 (2023), no. 1, paper no. 32.
Search in Google Scholar Back to article
SIMON, J.: Compact sets in the space L^p(0, T; B), Annali di Matematica pura ed Applicata, 146 (1987), 65–96.
Search in Google Scholar Back to article
XIA, L.—YAO, Z.: Existence, uniqueness and asymptotic behavior of solutions for a singular parabolic equation, J. Math. Anal. Appl. 358 (2009), 182–188.
Search in Google Scholar Back to article
XIA, L.—YAO, Z.: Existence of the maximal weak solution for a class of singular parabolic equations, J. Math. Anal. Appl. 387 (2012), 439–446.
Search in Google Scholar Back to article
ZHANG, C.—ZHOU, S.: Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and L¹ data, J. Differential Equations 248 (2010), 1376–1400.
Search in Google Scholar Back to article
ZHOU, W.—WEI, X.: Some results on a singular parabolic equation in one dimension case, Math. Methods Appl. Sci. 36 (2013), 2576–2587.
Search in Google Scholar Back to article
ZIAD, Y.—KHELIFI, H.—BACHOUCHE, K.: Degenerate parabolic problem with singular lower order term and irregular data, Nonlinear Studies 32 (2025), no. 1, 129–150.
Search in Google Scholar Back to article

Boundedness and Regularity of Solutions for Nonlinear Parabolic Problem with two Singularities

Abstract

Paradigm

My account