In this paper, we perform a Lie group classification of the quasilinear evolution equation $ u_t = \bigl(\Phi(u)(u_x)^n\bigr)_x+F(u), \ n\ne -1, \, 0, \, 1, $ where $\Phi(u)$ is a state-dependent diffusivity and $F(u)$ is an arbitrary reaction term. The kernel Lie algebra admitted for arbitrary nonconstant $\Phi(u)$ and $F(u)$ is identified. All symmetry extensions beyond the kernel algebra are then classified with respect to the arbitrary elements, and the corresponding infinitesimal generators are explicitly constructed. For each classified symmetry algebra, one-dimensional optimal systems of subalgebras are obtained and used to derive similarity reductions of the governing equation. Several reduced ordinary differential equations are solved explicitly, leading to exact invariant solutions for selected nonlinear models in the classification. In addition, conservation laws are constructed by the multiplier method for the admissible reaction case, yielding explicit conserved vectors.
Citation: Khalid Ali Alanezy, Ali Raza. Lie group classification and invariant solutions of a class of quasilinear evolution equations with nonlinear gradient-power flux and state-dependent diffusivity[J]. AIMS Mathematics, 2026, 11(9): 28009-28054. doi: 10.3934/math.20261118
In this paper, we perform a Lie group classification of the quasilinear evolution equation $ u_t = \bigl(\Phi(u)(u_x)^n\bigr)_x+F(u), \ n\ne -1, \, 0, \, 1, $ where $\Phi(u)$ is a state-dependent diffusivity and $F(u)$ is an arbitrary reaction term. The kernel Lie algebra admitted for arbitrary nonconstant $\Phi(u)$ and $F(u)$ is identified. All symmetry extensions beyond the kernel algebra are then classified with respect to the arbitrary elements, and the corresponding infinitesimal generators are explicitly constructed. For each classified symmetry algebra, one-dimensional optimal systems of subalgebras are obtained and used to derive similarity reductions of the governing equation. Several reduced ordinary differential equations are solved explicitly, leading to exact invariant solutions for selected nonlinear models in the classification. In addition, conservation laws are constructed by the multiplier method for the admissible reaction case, yielding explicit conserved vectors.
| [1] |
S. D. Koval, R. O. Popovych, Point and generalized symmetries of the heat equation revisited, J. Math. Anal. Appl., 527 (2023), 127430. https://doi.org/10.1016/j.jmaa.2023.127430 doi: 10.1016/j.jmaa.2023.127430
|
| [2] | G. W. Bluman, J. D. Cole, The general similarity solution of the heat equation, J. Math. Mech., 18 (1969), 1025–1042. |
| [3] | W. I. Fushchych, W. M. Shtelen, M. I. Serov, R. O. Popovych, Q-conditional symmetry of the linear heat equation, Proc. Acad. Sci. Ukraine, 4 (2022), 479–485. |
| [4] |
R. O. Popovych, Reduction operators of linear second-order parabolic equations, J. Phys. A Math. Theor., 41 (2008), 185202. https://doi.org/10.1088/1751-8113/41/18/185202 doi: 10.1088/1751-8113/41/18/185202
|
| [5] |
R. O. Popovych, M. Kunzinger, N. M. Ivanova, Conservation laws and potential symmetries of linear parabolic equations, Acta Appl. Math., 100 (2008), 113–185. https://doi.org/10.1007/s10440-007-9178-y doi: 10.1007/s10440-007-9178-y
|
| [6] |
S. C. Anco, G. Bluman, Direct construction method for conservation laws of partial differential equations. Part Ⅰ: Examples of conservation law classifications, Eur. J. Appl. Math., 13 (2002), 545–566. https://doi.org/10.1017/S095679250100465X doi: 10.1017/S095679250100465X
|
| [7] |
N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A Math. Theor., 44 (2011), 432002. https://doi.org/10.1088/1751-8113/44/43/432002 doi: 10.1088/1751-8113/44/43/432002
|
| [8] |
R. Cherniha, J. R. King, S. Kovalenko, Lie symmetry properties of nonlinear reaction-diffusion equations with gradient-dependent diffusivity, Commun. Nonlinear Sci. Numer. Simul., 36 (2016), 98–108. https://doi.org/10.1016/j.cnsns.2015.11.023 doi: 10.1016/j.cnsns.2015.11.023
|
| [9] |
S. Opanasenko, V. Boyko, R. O. Popovych, Enhanced group classification of nonlinear diffusion-reaction equations with gradient-dependent diffusivity, J. Math. Anal. Appl., 484 (2020), 123739. https://doi.org/10.1016/j.jmaa.2019.123739 doi: 10.1016/j.jmaa.2019.123739
|
| [10] |
V. A. Galaktionov, S. R. Svirshchevskii, Invariant solutions of nonlinear diffusion equations with maximal symmetry algebra, J. Nonlinear Math. Phys., 18 (2011), 107–121. https://doi.org/10.1142/S1402925111001301 doi: 10.1142/S1402925111001301
|
| [11] |
D. Min, L. Guo, Existence and nonexistence of nontrivial solutions for fractional advection-dispersion equation with instantaneous and non-instantaneous impulses, Fractal Fract., 9 (2025), 571. https://doi.org/10.3390/fractalfract9090571 doi: 10.3390/fractalfract9090571
|
| [12] | L. V. Ovsiannikov, Group analysis of differential equations, New York: Academic Press, 1982. https://doi.org/10.1016/C2013-0-07470-1 |
| [13] | G. W. Bluman, S. Kumei, Symmetries and differential equations, New York: Springer, 1989 https://doi.org/10.1007/978-1-4757-4307-4 |
| [14] | P. J. Olver, Applications of Lie groups to differential equations, New York: Springer, 1993. https://doi.org/10.1007/978-1-4612-4350-2 |
| [15] | N. H. Ibragimov, CRC handbook of Lie group analysis of differential equations, Boca Raton: CRC Press, 1993. https://doi.org/10.1201/9781003419808 |
| [16] | P. E. Hydon, Symmetry methods for differential equations: a beginner's guide, Cambridge: Cambridge University Press, 2000. https://doi.org/10.1017/CBO9780511623967 |
| [17] |
A. H. Kara, S. Naseer, A. Raza, F. D. Zaman, Optimal system and conservation laws for the generalized Fisher equation in cylindrical coordinates, Quaest. Math., 47 (2024), 1369–1386. https://doi.org/10.2989/16073606.2024.2321261 doi: 10.2989/16073606.2024.2321261
|
| [18] |
J. Yu, Y. Feng, Symmetry analysis, optimal system, conservation laws and exact solutions of time fractional diffusion-type equation, Int. J. Geom. Methods Mod. Phys., 22 (2025), 2450286. https://doi.org/10.1142/s0219887824502864 doi: 10.1142/s0219887824502864
|
| [19] |
Z. Cao, L. Zhang, Symmetries and conservation laws of a time dependent nonlinear reaction-convection-diffusion equation, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2703–2717. https://doi.org/10.3934/dcdss.2020218 doi: 10.3934/dcdss.2020218
|
| [20] |
J. Yu, Y. Feng, Lie symmetry, exact solutions and conservation laws of some fractional partial differential equations, J. Appl. Anal. Comput., 13 (2023), 1872–1889. https://doi.org/10.11948/20220268 doi: 10.11948/20220268
|
| [21] |
A. Tomar, H. Kumar, M. Ali, H. Gandhi, D. Singh, G. Pathak, Application of symmetry analysis and conservation laws to a fractional-order nonlinear conduction-diffusion model, AIMS Math., 9 (2024), 17154–17170. https://doi.org/10.3934/math.2024833 doi: 10.3934/math.2024833
|
| [22] |
L. Guo, C. Li, N. Qiao, J. Zhao, Convergence analysis of positive solution for Caputo-Hadamard fractional differential equation, Nonlinear Anal. Model. Control, 30 (2025), 212–230. https://doi.org/10.15388/namc.2025.30.38509 doi: 10.15388/namc.2025.30.38509
|
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Khalid Ali Alanezy, Ali Raza. Lie group classification and invariant solutions of a class of quasilinear evolution equations with nonlinear gradient-power flux and state-dependent diffusivity[J]. AIMS Mathematics, 2026, 11(9): 28009-28054. doi: 10.3934/math.20261118
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