Existence of global attractor in 𝑳𝒒 for a nonlinear
parabolic equation with m-Laplacian type
Naimah Aris
Department of Mathematics, Hasanuddin University
Jl. Perintis Kemerdekaan KM. 10 Makassar 90245, Indonesia
e-mail: imaaris@yahoo.com
1. Introduction
`Global attractor' is a basic concept and tool to study asymptotic behaviours of solutions of
nonlinear evolution equations and now we have standard monographs on this theory (cf. [2], [12],
etc.). A global attractor is an invariant compact set absorbing all of the bounded sets as time goes to
infinity and by use of this method we can reduce problems for motions in infinite dimensional
spaces to some finite dimensional or `finite dimensional like' ones.
In this paper, we consider global attractor for a typical strongly nonlinear parabolic
equation:
𝑢𝑡 − 𝑑𝑖𝑣(𝜎(|∇𝑢|2 )∇𝑢) + 𝑔(𝑢) = 𝑓(𝑥)
𝑡 > 0, 𝑥 ∈ Ω,
𝑢(𝑥, 𝑜) = 𝑢0 (𝑥), 𝑥 ∈ Ω; 𝑢(𝑥, 𝑡) = 0, 𝑥 ∈ 𝜕Ω, 𝑡 > 0.
(1.1)
(1.2)
Where σ(v2) is a function like σ(v2) = |𝑣|𝑚 and g(u) is a globally Lipshitz function.
This is one of the most typical nonlinear parabolic equations and investigated from various
points of view like in [3], [1], [9], [5], [6] and [10]). In [4], it is shown the existence of a global
attractor A in 𝐿2 (𝛺).
The main purpose of this paper is to study the regularity of A. Under an additional
assumption on the forcing term f we shall show that the global attractor A is in fact a bounded set in
𝑊01,∞ (Ω). For the proof, Moser's technique as in [7], [8] is frequently used in deriving estimates.
Theoritical considerations of the following section are mostly based on paper [10].
2. Results
In this selection it should prove that the global attractor A⊂ 𝐿2 (𝛺)constructed in [4] is in
fact included in 𝑊 1,∞ under an additional regularity assumption on f. Here, we assume 𝜕𝛺 is C2class.
Under the following assumptions,
Assumption A. σ(.) is differentiable on R+ = [0, ∞] and satisfies the condition:
𝑘0 |𝑣|𝑚 ≤ 𝜎(𝑣 2 ) 𝑎𝑛𝑑 𝑘0 |𝑣|𝑚 ≤ 𝜎 𝑓 (𝑣 2 )𝑣 2 ,
And
𝑣2
2
2
𝑘1 𝜎(𝑣 )𝑣 ≤ ∫ 𝜎(𝜂)𝑑𝜂.
0
For v ∈ R, m ≥ 0 and k0, k1 are some positive constants.
Assumption B. g(u) is a globally Lipschitz function on R with g(0) = 0.
Assumption C. f belongs to 𝐿∞ ([0, ∞); 𝑊01,∞ (Ω)).
Our result is:
Theorem 1. Under the assumptions A, B, and C the global attractor A of the problem (1,1)-(1, 2) is
a bounded set in 𝑊01,∞ (Ω) and it holds that
‖𝑢‖𝑊 1,∞ ≤ 𝐶(𝑀1 ) < ∞𝑖𝑓 𝑢 ∈ 𝐴,
0
where M1 = ‖𝑓‖𝑊1,∞ .
0
To proof the theorem, some estimates to approximate solutions need to derive. The
solutions are in fact given as limits of smooth solutions of appropriate approximate equations and
for the argument the solutions under consideration are sufficiently smooth may assume.
Estimates for ‖𝑢(𝑡)‖2 and ‖𝑢(𝑡)‖∞ .
Proposition 1 Let u(t) be a solution of the problem (4.1)-(4.2). Then we have
‖𝑢(𝑡)‖2 ≤ 𝐶(𝑀0 , ‖𝑢0 ‖2 ),
0 ≤ 𝑡 < ∞,
(2.1)
‖𝑢(𝑡)‖∞ ≤ 𝐶(𝑀0 , ‖𝑢0 ‖2 )𝑡 −𝜆 ,
0 < 𝑡 ≤ 1.
(2.2)
And
Proposition 2 Let u(t) be a solution of problem (1.1)-(1.2), obtain the equation
‖𝑢(𝑡)‖∞ ≤ 𝐶(𝑀0 , ‖𝑢0 ‖2 ),
𝑡 ≥ 1.
(2.3)
Estimate for ‖∇𝑢(𝑡)‖𝑚+2
Proposition 3 Let u(t) be a solution of problem (1.1)-(1.2). Then we have
‖∇𝑢(𝑡)‖𝑚+2 ≤ 𝐶(𝑀0 , ‖𝑢0 ‖2 )𝑡 −𝜆 ,
0 < 𝑡 ≤ 1,
(2.4)
𝑡 ≥ 1.
(2.5)
for a certain λ > 0, and
‖∇𝑢(𝑡)‖𝑚+2 ≤ 𝐶(𝑀0 , ‖𝑢0 ‖2 ) < ∞,
Estimate for ‖∇𝑢(𝑡)‖𝑝
Proposition 4 For p > m + 2, the estimate is
‖∇𝑢(𝑡)‖𝑝 ≤ 𝐶(𝑀1 , ‖𝑢0 ‖2 )𝑡 −𝜇 ,
p > 𝑚 + 2,
(2.6)
with a certain μ > 0.
Similarly:
Proposition 5
‖∇𝑢(𝑡)‖𝑝 ≤ 𝐶(𝑝, 𝑀1 ) < ∞, 𝑡 ≥ 1, 𝑓𝑜𝑟 𝑎𝑛𝑦 p > 𝑚 + 2.
(2.7)
Estimate for ‖∇𝑢(𝑡)‖∞ by using results above, we obtain
Proposition 6
‖∇𝑢(𝑡)‖∞ ≤ 𝐶(𝑀1 , ‖𝑢0 ‖2 )𝑡 −𝜇̅ ,
0 < 𝑡 ≤ 1,
(2.8)
with 𝜇̅ > 0.
The final estimate is following
Proposition 7
‖∇𝑢(𝑡)‖∞ ≤ 𝐶(𝑀1 , ‖𝑢(1)‖∞ ) < ∞, 𝑡 ≥ 1.
(2.9)
Proof of Theorem 1.
The regularity of the global attractor A constructed in [4] which also indicates that A is
bounded in 𝑊01,∞ will be explained below.
From estimation for ‖𝑢(𝑡)‖2 in equation (2.1) we get A ∈ L2 and A bounded in L2.
If u0 ∈ A,
‖𝑢(𝑡)‖2 ≤ 𝐶(𝑀0 ).
(2.10)
Indeed, from equations (2.8) for t = 0, the solutions will blow up hence by taking t = 1, we
get from (2.8) and (2.10) that
‖∇𝑢(1)‖∞ ≤ 𝐶(𝑀1 , 𝐶(𝑀0 )).
Since 𝑀1 = ‖∇𝑓‖𝑊1,∞ and 𝑀0 = ‖∇𝑓‖, we have that M1 more uniformly constant than M0
‖∇𝑢(1)‖∞ ≤ 𝐶(𝑀1 ).
(2.11)
Substitute equation (2.11) to (2.9) obtain the following result
‖∇𝑢(𝑡)‖∞ ≤ 𝐶(𝑀1 , 𝐶(𝑀1 )) < ∞,
𝑡 ≥ 1.
Since A invariant and bounded in L2 we see that for any u0 ∈ A,
‖∇𝑇(𝑡)𝑢0 ‖2 ≤ 𝐶(𝑀1 ) < ∞,
𝑡 ≥ 1.
For u0∈A. Since T(t)A = A we conclude that A is include in the ball in 𝑊 1,∞ with the radius C(M1).
3. Conclusion
In conclusion of this paper we would like to remark that under additional regularity
assumption f(x) and an estimate of solutions u(t), t > 0, in 𝑊01,∞, the global attractor A is in fact a
bounded set of 𝑊01,∞ which is also proved the regularity of the global attractor A.
Acknowledgement
I would like to thank professor Nakao Mitsuhiro for his guidance during my research.
References
[1]
A. Haraux, Nonlinear evolution equations - Global behaviour of solutions, Lecture Notes in
Mathematics, 841, Springer, 1981.
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
A.V. Bavin, M. I. Vishik, Attractors of evolution equations, North-Holland, Amsterdam,
1992.
H. Brezis, Operateurs maximaaux monotones et semi-groupes de contractions dans les
espaces de Hilbert, North-Holland, Amsterdam, 1973.
J. W. Cholewa, T. Dlotko, Global attractors in abstract parabolic problems, Cambridge
Univ. Press., 2000.
J. W. Cholewa, T. Dlotko, "Global attractor for sectorial evolutionary equations",
J.Differential Equations, Vol.125, pp.27-39, 1996.
M. Nakao, "Lp estimates of solutions of some nonlinear degenerate diffusion equations",
J.Math. Soc. Japan, Vol. 37, pp. 41-63, 1985.
M. Nakao, C. Chen, "Global existence and gradient estimate for the quasilinear parabolic
equations of m-Laplacian type with a nonlinear convection term", J. Diff. Eqns. Vol. 162,
pp. 224-250, 2000.
M .Nakao, Y. Ohara, "Gradient estimate of periodic solution for some quasilinear parabolic
equations", J. Math. Anal. Appl. , Vol. 204, pp.868-883, 1996.
M.Tsutsumi, "Existence and nonexistence of global solutions for nonlinear parabolic
equations", Publ. RIMS, Vol. 17, pp.211-229, 1993.
M. Nakao, N. Aris, "On global attractor for nonlinear parabolic equations of m-Laplacian
type", J. Math. Anal. App., Vol. 331, pp. 793-809, 2007.
R.A. Adams, Sobolev space, 2nd edition, Academic Press, New York, 2003.
R. Temam, Infinite dimensional dynamical systems in mechanics and physics, Springer,
New York, 1997.