Remark on the Cauchy problem for the evolution p-Laplacian equation

In this paper, we prove that the semigroup S(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S(t)$\end{document} generated by the Cauchy problem of the evolution p-Laplacian equation ∂u∂t−div(|∇u|p−2∇u)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{\partial u}{\partial t}-\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0$\end{document} (p>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p>2$\end{document}) is continuous form a weighted L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{\infty}$\end{document} space to the continuous space C0(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{0}(\mathbb{R}^{N})$\end{document}. Then we use this property to reveal the fact that the evolution p-Laplacian equation generates a chaotic dynamical system on some compact subsets of C0(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{0}(\mathbb{R}^{N})$\end{document}. For this purpose, we need to establish the propagation estimates and the space-time decay estimates for the solutions first.


Introduction
In this paper, we consider the Cauchy problems of the evolution p-Laplacian equation where p >  and the nonnegative initial value u  belongs to the weighted L ∞ space W σ (R N ) ≡ {ϕ; |x| σ ϕ(x) ∈ L ∞ (R N )} with the norm ϕ(·) W σ (R N ) = | · | σ ϕ(·) L ∞ (R N ) . The evolution p-Laplacian equation, as an important class of parabolic equations, comes from the compressible fluid flows in a homogeneous isotropic rigid porous medium. Comparing to the classical linear heat equation, this equation, to a certain extent, reflects even more exactly physical reality [, ]. So the studies of this equation have attracted a large number of mathematicians and remarkable progress has been achieved []. Among all of the progress, the semigroup method given by Bénilan and Véron [, ] is a successful and effective method to treat the evolution p-Laplacian equation.
Using the concepts of dynamical systems to study partial differential equations has also attracted much attention in recent decades. Such concepts, like orbit, ω-limit, attractor and chaos, were introduced to investigate the finite dimensional instances of dynamical systems of ordinary differential equations. In , it was Vázquez and Zuazua [] who first successfully used the ω-limit set of the rescaled solutions u(t equation by using the ω-limit set of the rescaled solutions t μ  u(t β ·, t) with  < μ, β < ∞.
Using ω-limit set to research other partial differential equations, one can refer to [-].
The theory of chaos on some partial differential equations has also been well developed since the pioneering work of Li, Wiggins, Shatah and McLaughlin; see [, ]. Li [] revealed the fact that around the Silnikov homoclinic orbits, the existence of chaos on Euler equations can be proved by constructing horseshoe. Cazenave, Dickstein and Weisslerit [] found that the discrete dynamical system generated by the heat equation in some sense is an example of chaos. In , Battellia and Fečkan proved the existence of homoclinic and chaotic solutions of beam equations in []. In , Lan and Li [] found that the numerical Melnikov integral can be used as a tool for both predicting and controlling chaos on Euler equations. The chaos theory on other partial differential equations, one can see [-].
Inspired by the above papers, especially by [], we focus our attention on the semigroup and the chaos theory for the evolution p-Laplacian equation. To overcome the difficulties caused by the degeneracy and nonlinearity of this equation, we first establish the propagation estimate and the decay estimate for the solutions of the problem of (.)-(.). By using the propagation estimate and the decay estimate of the solutions u(x, t), we see that the semigroup S(t) generated by the evolution p-Laplacian equation is continuous from with the weak-star topology of W σ (R N ). Then using the definition of chaos, the commutative relation between the delation operator D σ λ and the semigroup S(t), we find that, for any fixed λ > , the map Here the delation operator D σ λ is defined as . The rest of this paper is organized as follows. In the next section, we give some definitions and some propositions of the solutions for the problem (.)-(.). Section  is devoted to giving the propagation speed estimate and decay estimate for the solutions of problem (.)-(.). The continuity of the semigroup S(t) is consider in Section . We reveal the fact that the problem (.)-(.) generates a chaotic dynamical system on certain compact subsets of C  (R N ) in Section .

Preliminaries
In this section, we give some definitions and present some propositions of solutions for the problem (.)-(.). We first present the definition of chaos. Although there has been no universally accepted mathematical definition of chaos, the popular text by Devaney [] isolates three components as being the essential features of chaos. They are formulated for a continuous map F : X → X on some metric space (X, d). The first of Devaney's three conditions is that F is transitive; that is, for all non-empty open subsets U and V of X there exists a natural number k such that In a certain sense, transitivity is an irreducibility condition. The second of Devaney's conditions is that the periodic points of F form a dense subset of X. The final condition is called sensitive dependence on initial conditions; F verifies this property if there is a δ >  such that, for every point x ∈ X and every neighborhood of x, there exist a point y ∈ and a nonnegative integer k such that This sensitivity condition captures the idea that in chaotic systems minute errors in experimental readings eventually lead to large scale divergence. Sensitive dependence on initial conditions is thus widely understood as being the central idea in chaos.

Definition . (Devaney's definition of chaos []) Let (X, d) be a metric space. A continuous map
F : X → X is said to be chaotic on X if . F is transitive; . the periodic points of F are dense in X; . F has sensitive dependence on initial conditions.
To discuss chaotic dynamical system in the evolution p-Laplacian equation, we need to adopt some concepts as that in [, , ]. For f ∈ L  loc (R N ), we define The space X is given by with the norm ||| · |||  . Hence it is a Banach space. The space X  is defined as The weak solutions of the problem (.)-(.) is defined as follows.
and for every test function ϕ ∈ C ∞  (Q T ), the following identity holds: and u(x, t) satisfies the initial-value equation (.) in the following sense: The existence and uniqueness of weak solution of the problem (.)-(.) for the initial value u  ∈ X is shown in [, ], and these solutions satisfy the following proposition.
the solutions u(x, t) are Hölder continuous in Q T ≡ (, T) × R N and satisfy the following estimates: where B R is the closed ball in R N with the radius R.
From the above proposition, the following proposition can easily been proved; see [, ].

Proposition . ([])
If the initial value u  ∈ X  , one can easily see that T(u  ) = ∞ and thus these solutions are global. Moreover, the evolution p-Laplacian equation generates a bound semigroup in X  given by where S(t) is the semigroup given by (.) and the dilation D σ λ is given by In this paper, we consider our problem in the with the weak-star topology of W σ (R N ) is compact and separable. Thus it can be meterizable. We use the symbol d σ , * M to denote this metric. So, for all M ≥ , the metric space is compact, hence complete and separable. In the rest of this section, we study the relation between the semigroup operator S(t) and the dilation operator This mean that v(x, t) is a weak solution of the following Cauchy problem: Note that Then we get the following commutative relation between the semigroup operator S(t) and the dilation operator D σ λ : For any fixed λ >  and  < σ < N , we now define the map F σ λ :

Some estimates
In this section, we first estimate the propagation speed of solutions for Problem (.)-(.) with the nonnegative initial value u  ∈ W σ (R N ). For this purpose, we need some concepts. Let be the distance from x to the support of u  and let us introduce the following symbol to denote the positive set of u(x, t) at time t: We also define the ρ-neighborhood of the set (t) as where d(x, (t)) is the distance from x to (t).
Theorem . (Propagation estimate) Suppose  < σ < N and the initial value u  ∈ W + σ (R N ), i.e., u  ≥  and u  ∈ W σ (R N ). Let u(x, t) be a nonnegative weak solution of the problem (.)-(.). For any  ≤ t  < t  < ∞, then Proof Without loss of generality, we restrict our consideration to the case t  = . Assume If |x  | ≥ R, then, for any y ∈ B R (x  ), Combining this with (.) and (.), we have Therefore, Proposition . implies This means So we complete the proof of this theorem.
In the rest of this section, we pay our attention on the properties of the semigroup S(t).
Theorem . Let ω ∈ C(R N \ {}) be a homogeneous function of degree . Suppose  < σ < N and set u  (x) = |x| -σ ω(x). It follows that Proof From (.) and the definition of the initial value u  , we have First notice that, for  < σ < N , Therefore, for some  < α < ; see [, ]. In particular, as t → . Let So |y| → ∞ as t → . Therefore, as |y| → ∞. We complete the proof of this theorem. It follows from Theorem . that there exists a constant C such that So by (.), By the comparison principle [, ], we get So the proof of this theorem is complete.

Continuity of the semigroup
In this section, we first present the fact that the semigroup operator S(t) are continuous from the metric space B σ ,+ M to the space C  (R N ), which is the basis of the proof of our main result.

continuous. In particular, S() is a continuous map from the metric space
Therefore, Notice also that For any t, R > , let So, for all n ≥ , This means that the value of S(t)u n (x) in B R is only dependent on the initial value u n in B R  (t) . In other words, for x ∈ B R , For any > , taking the above R large enough, the inequality (.) clearly implies that, if |x| ≥ R, then By (.) and the hypothesis u n , u  ∈ W σ (R N ), we get, for  < q < N σ , So, In particular, From (.), we know that there exists a constant C(τ ) such that Therefore, From the regularity of the semigroup operator S(t), we obtain, for any fixed t > τ > , as n → ∞. This implies, via (.), that By (.), there exists an integer N such that if n > N , then So we complete the proof of this theorem.

Chaotic dynamical system
For any λ >  and M > , we recall that the map F σ λ : The ideas of the following theorem come from []. We also need a lemma which appeared in [].
Here we have used the facts that λ >  and S(t) is a bounded contractive continuous semigroup in L ∞ (R N ) for t >  (Proposition .). Therefore, Note that, for all k > , This means that Using Theorem ., we get This proves that the periodic points of where λ n = λ  n+ and χ n (x) is the cut-off function defined on the set A n ≡ {x ∈ R N ; λ -n σ (p-)+p < |x| < λ n σ (p-)+p } relative to the set A n- ≡ {x ∈ R N ; λ -n+ σ (p-)+p < |x| < λ n- σ (p-)+p }. Notice that, for all n ≥ , So, for n ≥ , from (.) and (.), we have Here we have used the fact that, if i > j ≥ , then Note also that These results mean that So we complete the proof of that F σ λ is topologically transitive. So the map F σ λ is chaotic by Devaney's definition of chaos and Lemma ., and the proof of this theorem is complete.

Conclusion
In this paper, our concern here is the properties of the solutions to the Cauchy problem of the evolution p-Laplacian equation with the initial value u  belonging to a weighted L ∞ spaces, and we get the following results: I. The propagation estimate and the decay estimate for the solutions of the problem of (.)-(.) have been established. II. The semigroup S(t) generated by the evolution p-Laplacian equation is continuous from the compact set B σ ,+ M to the space C  (R N ). III. The map F σ λ generated by the evolution p-Laplacian equation is chaotic on the compact subset S()B σ ,+ M of C  (R N ). In future work, we hope to continue to study the properties of solutions for the Cauchy problem of the evolution p-Laplacian equation with the initial value u  belonging to other Banach spaces, especially the unbounded spaces.