Received April 26, 2023. Published online February 29, 2024.
Abstract: This paper is concerned with the study of a nonlocal nonlinear parabolic problem associated with the equation $u_t-M(\int_{\Omega}\phi u {\rm d}x){\rm div} (A(x,t,u)\nabla u)=g(x,t,u)$ in $\Omega\times(0,T)$, where $\Omega$ is a bounded domain of $\mathbb{R}^n$ $(n\geq1)$, $T>0$ is a positive number, $A(x,t,u)$ is an $n\times n$ matrix of variable coefficients depending on $u$ and $M \mathbb{R}\rightarrow\mathbb{R}$, $\phi \Omega\rightarrow\mathbb{R}$, $g \Omega\times(0,T)\times\mathbb{R}\rightarrow\mathbb{R}$ are given functions. We consider two different assumptions on $g$. The existence of a weak solution for this problem is proved using the Schauder fixed point theorem for each of these assumptions. Moreover, if $A(x,t,u)=a(x,t)$ depends only on the variable $(x,t)$, we investigate two uniqueness theorems and give a continuity result depending on the initial data.
References: [1] C. O. Alves, T. Boudjeriou: Existence of solution for a class of nonlocal problem via dynamical methods. Rend. Circ. Mat., Palermo (2) 71 (2022), 611-632. DOI 10.1007/s12215-021-00644-4 | MR 4453341 | Zbl 1497.35209
[2] C. O. Alves, D.-P. Covei: Existence of solution for a class of nonlocal elliptic problem via sub-supersolution method. Nonlinear Anal., Real World Appl. 23 (2015), 1-8. DOI 10.1016/j.nonrwa.2014.11.003 | MR 3316619 | Zbl 1319.35057
[3] C. T. Anh, L. T. Tinh, V. M. Toi: Global attractors for nonlocal parabolic equations with a new class of nonlinearities. J. Korean Math. Soc. 55 (2018), 531-551. DOI 10.4134/JKMS.j170233 | MR 3800554 | Zbl 1415.35055
[4] D. Arcoya, T. Leonori, A. Primo: Existence of solutions for semilinear nonlocal elliptic problems via a Bolzano theorem. Acta Appl. Math. 127 (2013), 87-104. DOI 10.1007/s10440-012-9792-1 | MR 3101144 | Zbl 1280.35050
[5] M. Bousselsal, E. Zaouche: Existence of solution for nonlocal heterogeneous elliptic problems. Mediterr. J. Math. 17 (2020), Article ID 129, 10 pages. DOI 10.1007/s00009-020-01564-w | MR 4122738 | Zbl 1448.35185
[6] M. Chipot: Elements of Nonlinear Analysis. Birkhäuser Advanced Texts. Birkhäuser, Basel (2000). DOI 10.1007/978-3-0348-8428-0 | MR 1801735 | Zbl 0964.35002
[7] M. Chipot, B. Lovat: Some remarks on non local elliptic and parabolic problems. Nonlinear Anal., Theory Methods Appl. 30 (1997), 4619-4627. DOI 10.1016/S0362-546X(97)00169-7 | MR 1603446 | Zbl 0894.35119
[8] M. Chipot, B. Lovat: On the asymptotic behaviour of some nonlocal problems. Positivity 3 (1999), 65-81. DOI 10.1023/A:1009706118910 | MR 1675465 | Zbl 0921.35071
[9] M. Chipot, B. Lovat: Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 8 (2001), 35-51. MR 1820664 | Zbl 0984.35066
[10] M. Chipot, L. Molinet: Asymptotic behavior of some nonlocal diffusion problems. Appl. Anal. 80 (2001), 279-315. DOI 10.1080/00036810108840994 | MR 1914683 | Zbl 1023.35016
[11] F. J. S. A. Corrêa: On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Anal., Theory Methods Appl., Ser. A 59 (2004), 1147-1155. DOI 10.1016/j.na.2004.08.010 | MR 2098510 | Zbl 1133.35043
[12] R. Dautray, J.-L. Lions: Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 5. Evolution Problems I. Springer, Berlin (1992). MR 1156075 | Zbl 0755.35001
[13] S. B. de Menezes: Remarks on weak solutions for a nonlocal parabolic problem. Int. J. Math. Math. Sci. 2006 (2006), Article ID 82654, 10 pages. DOI 10.1155/IJMMS/2006/82654 | MR 2172802 | Zbl 1136.35407
[14] J. K. Hale, S. M. Verduyn Lunel: Introduction to Functional Differential Equations. Applied Mathematical Sciences 99. Springer, New York (1993). DOI 10.1007/978-1-4612-4342-7 | MR 1243878 | Zbl 0787.34002
[15] R. Jiang, C. Zhai: Properties of unique positive solutions for a class of nonlocal semilinear elliptic equations. Topol. Methods Nonlinear Anal. 50 (2017), 669-682. DOI 10.12775/TMNA.2017.036 | MR 3747033 | Zbl 1387.35240
[16] F. Jin, B. Yan: Existence and global behavior of the solution to a parabolic equation with nonlocal diffusion. AIMS Math. 6 (2021), 5292-5315. DOI 10.3934/math.2021313 | MR 4236733 | Zbl 1484.35263
[17] B. Lovat: Études de quelques problèmes paraboliques non locaux: Thèse. Université de Metz, Metz (1995). (In French.) Available at https://hal.univ-lorraine.fr/tel-01777092.
[18] B. Yan, D. Wang: The multiplicity of positive solutions for a class of nonlocal elliptic problem. J. Math. Anal. Appl. 442 (2016), 72-102. DOI 10.1016/j.jmaa.2016.04.023 | MR 3498319 | Zbl 1344.35043
[19] E. Zaouche: Nontrivial weak solutions for nonlocal nonhomogeneous elliptic problems. Appl. Anal. 101 (2022), 1261-1270. DOI 10.1080/00036811.2020.1778674 | MR 4408272 | Zbl 1489.35091
[20] E. Zaouche: Existence theorems of nontrivial and positive solutions for nonlocal inhomogeneous elliptic problems. Ric. Mat. 72 (2023), 949-960. DOI 10.1007/s11587-021-00612-1 | MR 4649474 | Zbl 1525.35125
Affiliations: Tayeb Benhamoud, Elmehdi Zaouche (corresponding author), Department of Mathematics, University of El Oued, El Oued 39000, Algeria, e-mail: tayeb06@yahoo.fr, elmehdi-zaouche@univ-eloued.dz, elmehdizaouche45@gmail.com; Mahmoud Bousselsal, Department of Mathematics, Laboratoire des équations aux dérivées partielles et H.M., École Normale Supérieure, Vieux-Kouba, Algiers, Algeria, e-mail: bousselsal55@gmail.com
