## Existence of positive solutions for a new class of Kirchhoff parabolic systems.(English)Zbl 1443.35075

Summary: We study the existence of weak positive solutions for a new class of Kirchhoff parabolic systems in bounded domains with multiple parameters taking into account the symmetry conditions and the right-hand side defined as a multiplication of two separate functions. Our results are natural extensions of previous results in the field, which used the same method for some classical elliptic equations.

### MSC:

 35K57 Reaction-diffusion equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35D30 Weak solutions to PDEs 35J62 Quasilinear elliptic equations
Full Text:

### References:

 [1] G. A. Afrouzi, N. T. Chung, and S. Shakeri, “Existence and non-existence results for nonlocal elliptic systems via sub-supersolution method”, Funkcial. Ekvac. 59:3 (2016), 303-313. · Zbl 1365.35028 [2] C. O. Alves and F. J. S. A. Correa, “On existence of solutions for a class of problem involving a nonlinear operator”, Comm. Appl. Nonlinear Anal. 8:2 (2001), 43-56. · Zbl 1011.35058 [3] N. Azzouz and A. Bensedik, “Existence results for an elliptic equation of Kirchhoff-type with changing sign data”, Funkcial. Ekvac. 55:1 (2012), 55-66. · Zbl 1248.35065 [4] Y. Bouizem, S. Boulaaras, and B. Djebbar, “Some existence results for an elliptic equation of Kirchhoff-type with changing sign data and a logarithmic nonlinearity”, Math. Methods Appl. Sci. 42:7 (2019), 2465-2474. · Zbl 1417.35031 [5] S. Boulaaras, “A well-posedness and exponential decay of solutions for a coupled Lamé system with viscoelastic term and logarithmic source terms”, Appl. Anal. (online publication August 2019). [6] S. Boulaaras and A. Allahem, “Existence of positive solutions of nonlocal $$p(x)$$-Kirchhoff evolutionary systems via sub-super solutions concept”, Symmetry 11:2 (2019), art. id. 253. · Zbl 1416.35133 [7] S. Boulaaras and R. Guefaifia, “Existence of positive weak solutions for a class of Kirrchoff elliptic systems with multiple parameters”, Math. Methods Appl. Sci. 41:13 (2018), 5203-5210. · Zbl 1397.35096 [8] S. Boulaaras, R. Guefaifia, and S. Kabli, “An asymptotic behavior of positive solutions for a new class of elliptic systems involving of $$(p(x), q(x))$$-Laplacian systems”, Bol. Soc. Mat. Mex. $$(3) 25$$:1 (2019), 145-162. · Zbl 1419.35029 [9] C. Chen, “On positive weak solutions for a class of quasilinear elliptic systems”, Nonlinear Anal. 62:4 (2005), 751-756. · Zbl 1130.35044 [10] N. T. Chung, “Multiple solutions for a $$p(x)$$-Kirchhoff-type equation with sign-changing nonlinearities”, Complex Var. Elliptic Equ. 58:12 (2013), 1637-1646. · Zbl 1281.35034 [11] N. T. Chung and H. Q. Toan, “Multiple solutions for a class of $$N$$-Kirchhoff type equations via variational methods”, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 109:1 (2015), 247-256. · Zbl 1315.35082 [12] R. Guefaifia and S. Boulaaras, “Existence of positive solutions for a class of $$(p(x),q(x))$$-Laplacian systems”, Rend. Circ. Mat. Palermo $$(2) 67$$:1 (2018), 93-103. · Zbl 1390.35082 [13] D. D. Hai and R. Shivaji, “An existence result on positive solutions for a class of $$p$$-Laplacian systems”, Nonlinear Anal. 56:7 (2004), 1007-1010. · Zbl 1330.35132 [14] X. Han and G. Dai, “On the sub-supersolution method for $$p(x)$$-Kirchhoff type equations”, J. Inequal. Appl. (2012), art. id. 283. · Zbl 1284.35175 [15] G. · JFM 09.0597.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.