The long-time behavior of weighted p-Laplacian equations

Shan Ma, Hongtao Li



In this work we study weighted $p$-Laplacian equations in abounded domain with a variable and generally non-smooth diffusion coefficient having at most a finite number of zeroes. The main attention is focused on the case that the diffusion coefficient $a(x)$ in such equations satisfies the inequality $\liminf\limits_{x\to z}|x-z|^{-p}a(x)> 0$ for every $ z\in \overline\Omega$. We show the existence of weak solutions and global attractors in $L^2(\Omega)$, $L^q(\Omega)(q\geq 2)$ and $D_0^{1,p}(\Omega)$, respectively.


Global existence of solutions; global attractors; weighted $p$-Laplacian equations

Full Text:



C.T. Anh, N.M. Chuong and T.D. Ke, global attractors for the m-semiflow generated by a quasilinear degenerate parabolic equations, J. Math. Anal. Appl. 363 (2010), 444–453.

C.T. Anh and P.Q. Hung, global attractors for a class of degenerate parabolic equations, Acta Math. Vietnam. 34 (2009), 213–231.

C.T. Anh and T.D. Ke, Long-time behavior for quasilinear parabolic equations involving weighted p-Laplacian operators, Nonlinear Anal. 71 (2009), 4415–4422.

A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear Differential Equations Appl. 7 (2000), 187–199.

J.W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.

R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. I. Physical Origins and Classical Methods, Springer–Verlag, Berlin, 1985.

E. Dibenedetto, Degenerate Parabolic Equations, Springer–Verlag, NewYork, 1993.

N.I. Karachalios and N.B. Zographopoulos, Convergence towards attractors for a degenerate Ginzburg–Landau equation, Z. Angew. Math. Phys. 56 (2005), 11–30.

N.I. Karachalios and N.B. Zographopoulos, On the dynamics of a degenerate parabolic equation global bifurcation of stationary states and convergence, Calc. Var. 25 (2006), no. 3, 361–393.

N.I. Karachalios and N.B. Zographopoulos, Global attractors and convergence to equilibrium for degenerate Ginzburg–Landau and parabolic equations, Nonlinear Anal. 63 (2005), e1749–e1768.

A.Kh. Khanmamedov, Global attractors for one dimensional p-Laplacian equation, Nonlinear Anal. 71 (2009), no. 1–2, 155–171.

H. Li and S. Ma, Asymptotic behavior of a class of degenerate parabolic equations, Abstr. Appl. Anal. (2012), Art. ID 673605, 15 pp.

H. Li, S. Ma and C. Zhong, Long-time behavior for a class of degenerate parabolic equations, Discrete Contin. Dyn. Syst. 34 (2014), no. 7, 2873–2892.

Q.F. Ma, S.H. Wang and C.K. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana University Math. J. 51 (2002), no. 6, 1541–1557.

I. Peral, Multiplicity of Solutions for the p-Laplacian, International Center for Theoretial Phusics Trieste, 1997.

J.C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Berlin, Springer, 1997.

M. Yang, Ch. Sun and Ch.Zhong, Existenceof a global attractor for a p-Laplacian equation in RN , Nonlinear Anal. 66 (2007), 1–13.

J.X. Yin and C.P. Wang, Evolutionary weighted p-laplacian equation, J. Differential Equations 237 (2007), 421–445.

C.K. Zhong, M.H. Yang and C.Y. Sun, The existence of global attractors for the normto-weak continuous semigroup and its application to the nonlinear reaction-diffusion equations, J. Differential Equations 223 (2006), 367–399.


  • There are currently no refbacks.

Partnerzy platformy czasopism