Existence for a Nonlocal Porous Medium Equations of Kirchhoff Type with Logarithmic Nonlinearity

Abstract

We study the Dirichlet problem for the nonlocal parabolic equation of the Kirchhoff type \[ u_{t}-a\left(\|u\|_{L^{p}(\Omega)}^{p}\right)\sum\limits_{i=1}^{n}D_{i}\left( \left\vert u\right\vert ^{p-2}D_{i}u\right) +b(x,t) \left\vert u \right\vert ^{\alpha \left( x,t\right) -2}u\log|u|=f\left( x,t\right) \quad \text{in $Q_T=\Omega \times (0,T)$}, \] where $p\geq2$, $T>0$, $\Omega \subset \mathbb{R}^{n}$, $n\geq 2$, is a smooth bounded domain. The coefficient $a(\cdot)$ is real-valued function defined on $\mathbb{R}_+$. It is shown that the problem has a weak solution under appropriate and general conditions on $a(\cdot)$, $\alpha(\cdot,\cdot)$ and $b(\cdot)$.

Keywords

• Kirchhoff-type equation
• nonlocal
• existence
• logarithmic nonlinearity

References

1. Ackleh, AS., Ke, L., Existence-uniqueness and long time behavior for a class of nonlocal nonlinear parabolic evolution equations, Proceedings of the American Mathematical Society, 128(12)(2000), 3483–3492.
2. Antontsev, S., Shmarev, S., Evolution PDEs with Nonstandard Growth Conditions, Atlantis Studies in Differential Equations. Paris: Atlantis Press, 2015.
3. Bebernes, J., Eberly, D., Mathematical Problems From Combustion Theory, Applied Mathematical Sciences, New York, USA: Springer-Verlag, 1989.
4. Boudjeriou, T., Global existence and blow-up for the fractional p-Laplacian with logarithmic nonlinearity, Mediterr. J. Math., 17(5)(2020), 162.
5. Bu,W., An, T., Li, Y., He, J., Kirchhoff-type problems involving logarithmic nonlinearity with variable exponent and nonvection term, Mediterranean Journal of Mathematics, 20(2)(2023), 77.
6. Chen, S., Zhang, B., Tang X., Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity. Advances in Nonlinear Analysis, 9(1)(2020), 148–167.
7. Chen, H., Luo, P., Liu, G, Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, J. Math. Anal. Appl., 422(1)(2015), 84–98.
8. Chen, H., Tian, S., Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258(12)(2015), 4424–4442.

9. Chipot, M., Lovat, B., On the asymptotic behaviour of some nonlocal problems. Positivity, 3(1)(1999), 65–81.
10. Chipot, M., Lovat, B., Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems, Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis, 8(1)(2001), 35–51.
11. Chipot, M., Molinet, L., Asymptotic behaviour of some nonlocal diffusion problems, Applicable Analysis, An International Journal, 80(3-4)(2021), 279–315.
12. Diaz, J.I., Nagai, T., Shmarev, S., On the interfaces in a nonlocal quasilinear degenerate equation arising in population dynamics. Japan Journal of Industrial and Applied Mathematics, 13(3)(1996) 385–415.
13. Diening, L., Harjulehto, P., H¨ast¨o P., Ruˇziˇcka M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Heidelberg: Springer, 2011.
14. Ding, H., Zhou, J., Global existence and blow-up for a parabolic problem of Kirchhoff type with logarithmic nonlinearity, Appl. Math. Optim., 83(2021), 1651–1707.
15. Enqvist, K., McDonald, J., Q-balls and baryogenesis in the mssm, Physics Letters B, 425(1998), 309–321.
16. Furter, J., Grinfeld, M., Local vs. nonlocal interactions in population dynamics, Journal of Mathematical Biology, 27(1)(1989), 65–80.
17. Hu, B., Yin, H.M., Semilinear parabolic equations with prescribed energy, Rend. Circ. Mat. Palermo (2), 44(3)(1995), 479–505.
18. Ji, S., Yin, J., Cao, Y., Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 261(10)(2016), 5446–5464.
19. Kalashnikov, A.S., Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russian Mathematical Surveys, 42(2)(1987), 169–222.
20. Kirchhoff, G., Vorlesungen uber Mechanik, BG Teubner, 1883.
21. Nhan, Le C., Le, X.T., Global solution and blow-up for a class of p-Laplacian evolution equations with logarithmic nonlinearity, Acta Appl. Math., 151(2017), 149–169.
22. Lions, JL. Quelques M´ethodes de ´esolution des Probl`emes aux Limites non Lin´eaires, Paris, Dunod: Gauthier-Villars, 1969.
23. Pişkin, E., Cömert T., Qualitative analysis of solutions for a parabolic type Kirchhoff equation with logarithmic nonlinearity, Open J. Discrete Appl. Math., 4(2)(2021), 1–10.
24. Sert, U., On a class of nonlocal porous medium equations of Kirchhoff type, Turkish J. Math., 46(6)(2022), 2231–2249.
25. Sert U., Shmarev, S., On a degenerate nonlocal parabolic equation with variable source, Journal of Mathematical Analysis and Applications 2020; 484 (1): 123695.
26. Sert, U., Soltanov, K., On solvability of a class of nonlinear elliptic type equation with variable exponent, The Journal of Applied Analysis and Computation, 7(3)(2017), 1139–1160.
27. Shao, X., Global existence and blow-up for a Kirchhoff-type hyperbolic problem with logarithmic nonlinearity, Appl. Math. Optim., 84(2)(2021), 2061–2098.
28. Soltanov, K., Sert, U., Certain results for a class of nonlinear functional spaces, Carpathian Mathematical Publications, 12(1)(2020), 208–228.
29. Soltanov, KN., Some embedding theorems and their applications to nonlinear equations, Differentsial’nye Uravneniya, 20 (12)(1984), 2181–2184.
30. Soltanov, KN., On some modification Navier-Stokes equations, Nonlinear Analysis Theory, Methods & Applications, An International Multidisciplinary Journal, 52(3)(2003), 769–793.
31. Soltanov, KN., Some nonlinear equations of the nonstable filtration type and embedding theorems, Nonlinear Analysis Theory, Methods & Applications, An International Multidisciplinary Journal, 65(11)(2006), 2103–2134.
32. Soltanov, KN., Sprekels, J., Nonlinear equations in non-reflexive Banach spaces and strongly nonlinear differential equations, Advances in Mathematical Sciences and Applications, 2(1999), 939–972.
33. Xiang, M., Yang, D., Zhang, B., Degenerate kirchhoff-type fractional diffusion problem with logarithmic nonlinearity, Asymptotic Analysis, 118(2020), 313–329.
34. Vazquez, JL., The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007.
35. Xiang, M., Radulescu, V.D., Zhang, B., Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions, Nonlinearity, 7(2018), 3228–3250.
36. Yan, L., Yang, Z., Blow-up and non-extinction for a nonlocal parabolic equation with logarithmic nonlinearity, Bound. Value Probl., 1(2018), 1–11.
37. Zeng, F., Shi, P., Jiang, M., Global existence and finite time blow-up for a class of fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity, AIMS Math., 6(2021), 2559–2578.