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

Yıl 2023, , 247 - 257, 31.12.2023

Uğur Sert

https://doi.org/10.47000/tjmcs.1260780

Öz

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 $b(\cdot)$.

Anahtar Kelimeler

Kirchhoff-type equation, nonlocal, existence, logarithmic nonlinearity

Kaynakça

• 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.
• Antontsev, S., Shmarev, S., Evolution PDEs with Nonstandard Growth Conditions, Atlantis Studies in Differential Equations. Paris: Atlantis Press, 2015.
• Bebernes, J., Eberly, D., Mathematical Problems From Combustion Theory, Applied Mathematical Sciences, New York, USA: Springer-Verlag, 1989.
• Boudjeriou, T., Global existence and blow-up for the fractional p-Laplacian with logarithmic nonlinearity, Mediterr. J. Math., 17(5)(2020), 162.
• 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.
• 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.
• 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.
• 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.
• Chipot, M., Lovat, B., On the asymptotic behaviour of some nonlocal problems. Positivity, 3(1)(1999), 65–81.
• 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.
• Chipot, M., Molinet, L., Asymptotic behaviour of some nonlocal diffusion problems, Applicable Analysis, An International Journal, 80(3-4)(2021), 279–315.
• 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.
• Diening, L., Harjulehto, P., H¨ast¨o P., Ruˇziˇcka M., Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Heidelberg: Springer, 2011.
• 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.
• Enqvist, K., McDonald, J., Q-balls and baryogenesis in the mssm, Physics Letters B, 425(1998), 309–321.
• Furter, J., Grinfeld, M., Local vs. nonlocal interactions in population dynamics, Journal of Mathematical Biology, 27(1)(1989), 65–80.
• Hu, B., Yin, H.M., Semilinear parabolic equations with prescribed energy, Rend. Circ. Mat. Palermo (2), 44(3)(1995), 479–505.
• 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.
• 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.
• Kirchhoff, G., Vorlesungen uber Mechanik, BG Teubner, 1883.
• 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.
• Lions, JL. Quelques M´ethodes de ´esolution des Probl`emes aux Limites non Lin´eaires, Paris, Dunod: Gauthier-Villars, 1969.
• 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.
• Sert, U., On a class of nonlocal porous medium equations of Kirchhoff type, Turkish J. Math., 46(6)(2022), 2231–2249.
• Sert U., Shmarev, S., On a degenerate nonlocal parabolic equation with variable source, Journal of Mathematical Analysis and Applications 2020; 484 (1): 123695.
• 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.
• Shao, X., Global existence and blow-up for a Kirchhoff-type hyperbolic problem with logarithmic nonlinearity, Appl. Math. Optim., 84(2)(2021), 2061–2098.
• Soltanov, K., Sert, U., Certain results for a class of nonlinear functional spaces, Carpathian Mathematical Publications, 12(1)(2020), 208–228.
• Soltanov, KN., Some embedding theorems and their applications to nonlinear equations, Differentsial’nye Uravneniya, 20 (12)(1984), 2181–2184.
• Soltanov, KN., On some modification Navier-Stokes equations, Nonlinear Analysis Theory, Methods & Applications, An International Multidisciplinary Journal, 52(3)(2003), 769–793.
• 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.
• 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.
• Xiang, M., Yang, D., Zhang, B., Degenerate kirchhoff-type fractional diffusion problem with logarithmic nonlinearity, Asymptotic Analysis, 118(2020), 313–329.
• Vazquez, JL., The Porous Medium Equation: Mathematical Theory, Oxford University Press, 2007.
• Xiang, M., Radulescu, V.D., Zhang, B., Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions, Nonlinearity, 7(2018), 3228–3250.
• Yan, L., Yang, Z., Blow-up and non-extinction for a nonlocal parabolic equation with logarithmic nonlinearity, Bound. Value Probl., 1(2018), 1–11.
• 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.