Abstract
The asymptotic behavior for a class of parabolic p-Laplace equations in an open bounded or unbounded domain of \({\mathbb{R}}^N\) is investigated. Based on a general condition on the nonlinearity f(x, u) and the invading domain technique, the global well-posedness of the equation is established. By proving the \(\omega\)-limit compactness of the continuous semigroup, the existence of the global attractor for the equation is obtained. Besides, in case of bounded domains, we also get estimates of the finite fractal dimension of the global attractor based on the classical method of \(\ell\)-trajectories.
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1 Introduction
In this paper, we study the existence of a global attractor for the following evolutionary partial differential equation:
where \(\Omega \subseteq {\mathbb {R}}^N\) is an arbitrary bounded or unbounded smooth domain, \(\Delta _{p}u=\text{ div }(|\nabla u|^{p-2}\nabla u)\) is the so-called p-Laplacian, \(\mu >0, 1<p<N\), and \(q>2\). Moreover, the external force f(x, u) is a Carathéodory function, i.e., \(f(\cdot , u)\) is measurable and \(f(x, \cdot )\) is continuous, whose properties will be specified later, and the external source \(g\in L^2(\Omega )\cap L^\infty (\Omega )\) is a given function independent of time.
The equation of type (1.1) is usually called a parabolic p-Laplace equation, which appears in many applications in mechanics, physics and biology (non-Newtonian fluids, gas flows in porous media, spread of biological populations, etc., see [6, 9, 10, 18] and the references therein). In the past decades, the well-posedness and the asymptotic behavior for the p-Laplace equation have been thoroughly investigated (see [3,4,5, 11,12,13, 16, 17, 22, 23, 29, 30, 33,34,37]). In particular, Geredeli [16] considered the existence of the global attractor for the equation
where \(\Omega \subset {\mathbb {R}}^N\) is bounded domain and the nonlinearity f satisfies a very general condition, namely,
for some \(\ell \ge 0.\) This condition implies that \(-\frac{\ell }{2} |u|^2 \le F(u)\le uf(u) + \frac{\ell }{2}|u|^2\) where \(F(u)=\int _0^u f(s)\,\text{d}s.\)
As in the previous results, for (weighted) p-Laplace equations, nonlinear reaction-diffusion equations or nonlinear wave equations, the existence of a global attractor is usually proved when the nonlinear external force satisfies a doubly bounded condition
for some \(\alpha _1, \alpha _2>0\) and \(\beta \ge 0\). In particular, Yang-Sun and Zhong in [34] considered the existence of a global attractor for a p-Laplace equation in \({\mathbb {R}}^N\) under a condition similar to both (1.3) and (1.4). There are many papers referring to conditions either of the type (1.3) or (1.4), see e.g. [25, 29, 35]. As pointed out in [16], a main difficulty arises in the proof of the well-posedness for (1.2) when we simplify the condition of f only involving (1.3). The critical issue in the proof of well-posedness is to deal with the limiting procedure on f which is overcome by weak convergence techniques in Orlicz spaces. However, the boundedness of the domain \(\Omega \subset {\mathbb {R}}^N\) plays an essential role in the proof, since the author of [16] relies on the Poincaré inequality to get the well-posedness of (1.2) (global estimates of solutions) and an absorbing set for the semigroup. With a doubly bounded condition on f, the authors of [34] also obtain the global well-posedness and global attractor for the p-Laplace equation in unbounded domains (or \({\mathbb {R}}^N\)) without the help of the Poincaré inequality. Moreover, the authors in [4] also considered quasilinear parabolic equations with weighted p-Laplacian operators of the type \(-\text{ div }(\sigma (x)|\nabla u|^{p-2}\nabla u)\) on unbounded domains. In that situation, they find suitable conditions on the weight function \(\sigma (x)\) to obtain global existence of solutions. However, in our situation, we cannot use this property to get the desired result. Finally, we also point out that the condition \(f^{\prime }(u)\ge -\ell\) also helps to get uniqueness of solutions; actually, it is another form of the estimate
In this paper, we intend to generalize the conditions (1.3) and (1.5), and assume
where \(f_1, f_2\) and the weight function m(x) satisfy the following conditions:
- (\(H_m\))::
-
\(m\in L^{\frac{N}{p}}(\Omega )\cap L^{\infty }(\Omega )\), \(m\ge 0\) and, if \(\Omega\) is unbounded, then additionally \(m\not \equiv 0.\)
- (\(H^1_{f}\))::
-
There exists a constant \(a^*, \nu ,\nu _- \in (0,\infty )\) such that, with \(p^*=\frac{pN}{N-p}\) and \(2<q_-<q\),
$$\begin{aligned}&\liminf _{|u|\rightarrow 0}\frac{f_1(u)u}{|u|^2}\ge a^*,\quad \underset{u\ne 0}{{\text {ess}}\inf \,} \frac{f_1(u)u}{|u|^{q_-}}\ge -\nu _-, \\&\underset{u\ne 0}{{\text {ess}}\inf \,} \frac{f_1(u)u}{|u|^{q}}\ge -\nu \ \text{ and } \ \liminf _{|u|\rightarrow 0}\frac{f_2(u)u}{|u|^{p^*}}\ge -a^*. \end{aligned}$$(1.7) - (\(H_{f}^2\))::
-
The functions \(f_1\) and \(f_2\) are continuous on \({\mathbb {R}}\), \(f_1(0)=f_2(0)=0\) and there holds for all \(u,v\in {\mathbb {R}}\)
$$\begin{aligned}&\left| f_1(u)-f_1(v)\right| \le \nu (|u-v|+ |u-v|^{q-1}) , \end{aligned}$$(1.8)$$\begin{aligned} 2< p<N:&\left| f_2(u)-f_2(v)\right| \le \lambda |u-v|^{p-1}, \end{aligned}$$(1.9)$$\begin{aligned} 1<p\le 2:&\left| f_2(u)-f_2(v)\right| \le \lambda \min \big \{|u-v|,|u-v|^{p-1}\big \}. \end{aligned}$$(1.10) - (\(H_c\))::
-
The coefficients \(\nu ,\lambda\) in \(H_f^2\) satisfy the conditions
$$\begin{aligned} \lambda< \min (1,c_p) \lambda _1 \quad \text { and }\; \nu <\min (1,c_q')\mu , \end{aligned}$$where \(\lambda _1=\lambda _1(\Omega )>0\) is the first eigenvalue of a modified eigenvalue problem of \(-\Delta _p\) on \(\Omega\). Moreover, \(c_p>0\) is the constant in the strong monotonicity estimate of \(-\Delta _p\), i.e., \((|\nabla u|^{p-2}\nabla u - |\nabla v|^{p-2}\nabla v, \nabla (u-v)) \ge c_p|\nabla u-\nabla v|^{\max (p,2)}\) and \(c_q'>0\) is an analogous constant in an estimate for the operator defined by \(|u|^{q-2}u\); for estimates of that type see [27, Lemma 5.1.19]. We note that the proof of an absorbing set in Sect. 4 requires a further smallness assumption on \(\lambda\), namely
$$\begin{aligned} \frac{\lambda }{\lambda _1} < \Big (\frac{p}{p+k_0(q-2)}\Big )^p (k_0(q-2)+1) \end{aligned}$$(1.11)where \(k_0\in {\mathbb {N}}\) is defined to satisfy the assumption \(k_0(q-2)+2>\frac{N(p+q-2)}{N-p}\), see (4.18) and (4.28) in the proof of Proposition 4.4.
The first (principal) eigenvalue \(\lambda _1\) of the eigenvalue problem
is positive, see [15] and, for more details, Proposition 2.7 below. For a bounded smooth domain \(\Omega\), thanks to the pioneering work [2, 8, 19], there is a largest constant \(\lambda _1=\lambda _1(\Omega )>0\) such that
for all functions u vanishing on \(\partial \Omega\). In other words, \(\lambda _1\) is the smallest eigenvalue of the eigenvalue problem \(-\Delta _p u = \lambda |u|^{p-2}u\) on \(\Omega\) with \(u=0\) on \(\partial \Omega\). However, as in the proof of Theorem 3.2 below, when \(\Omega\) is increasing to an unbounded domain, \(\lambda _1(\Omega )\) will tend to 0 as is well-known in the case \(p=2\). To this aim, \(\lambda _1\) is defined as the principal eigenvalue of (1.12) where the assumptions on m imply that the first eigenvalue of the operator \(-\Delta _p\) is positive. This allows for estimates of Poincaré type (1.13) on a sequence of bounded invading domains \(\Omega _R\subset \Omega\) which are independent of R.
Our main results can be summarized in the following theorem.
Theorem 1.1
Let \(1<p<N, q>2\) and let \(\Omega\) be an arbitrary (bounded or unbounded) smooth domain in \({\mathbb {R}}^{N}, N\ge 2\). Assume further that (\(H_m\)), (\(H^1_{f}\)) and (\(H_{f}^2\)) hold.
(1) For each \(g\in L^{2}(\Omega )\cap L^{\infty }(\Omega )\) and \(u_0\in L^{2}(\Omega )\) problem (1.1) has a unique global weak solution u such that
(2) Further assume that \(p>\frac{2N}{N+2}\) and \(\lambda \in (0, \lambda _1)\) is sufficiently small in (\(H_{f}^2\)). Then the nonlinear operator semigroup \((S(t))_{t\ge 0}\) on \(L^{2}(\Omega )\) defined by problem (1.1), i.e., \(S(t)u_0=u(t)\), possesses an invariant compact global attractor \(\mathcal {A}\) in \(L^{2}(\Omega )\cap L^{q}(\Omega )\).
(3) If \(\Omega\) is a bounded domain and \(p\ge 2\), then the global attractor has finite fractal dimension.
The main achievements of this paper are as follows:
(1) We get the existence of global-in-time solutions and of a global attractor for (1.1) when the nonlinear term f(u) does not satisfy the doubly bounded condition (1.4), even not in case of unbounded domains. Actually, our results in this paper improve the disadvantages of the above-mentioned results of [16, 34]. On one hand, we extend the result of [16] to unbounded domains. On the other hand, by comparing with the result of [34], our main progress is to get the global attractor for Eq. (1.1) in an unbounded domain with a generalized condition on f(x, u) related to the critical exponent.
(2) In the case of unbounded domains, usually the Poincaré inequality or a geometric property of the domain is exploited to get a damping impact; see, for example, Abergel [1] and Marín-Rubio, Real [28] who studied the 2D Navier–Stokes equations on a strip in \({\mathbb {R}}^2\) and arbitrary domains of \({\mathbb {R}}^2\) satisfying the Poincaré inequality, respectively. Especially, the existence of a global attractor is obtained for equations with a damping term (see [14] for the whole space case). In this paper, we not only remove any property of the shape of the domain and the Poincaré inequality, but also weaken the globally acting damping to a local one; actually, only the term \(f_1(u)\) is assumed to have a local damping near the origin.
(3) Furthermore, based on the method of \(\ell\)-trajectories introduced by Málek and Pra\(\check{\text{ z }}\)ák [26], we also get estimates of the finite fractal dimension of the global attractor when \(\Omega\) is a bounded domain. From our proof, one can see that we also obtain an exponential attractor for (1.1).
We will divide the proof of Theorem 1.1 into several sections. For convenience of the reader, we recall the definition of continuous semigroups, the \(\omega\)-limit compactness of semigroups, describe an abstract result on the existence of a global attractor, and discuss the eigenvalue problem (1.12). The existence of global solutions of (1.1) will be proved in Sect. 3. With a further condition on \(\lambda\) and p, we get the existence of the global attractor by proving the \(\omega\)-limit compactness of the semigroup in Sect. 4. In the last section, using the method of \(\ell\)-trajectories, we estimate the finite fractal dimension of the global attractor when \(\Omega\) is a bounded domain.
2 Preliminary results
Let \(\Omega\) be a domain of \({\mathbb {R}}^N, N\ge 2\), and let \(\Omega _T\) be the space-time cylinder \(\Omega \times [0,T]\). The symbols \( L^q (\Omega ), L^r (0,T ;L^q (\Omega )),\) and so forth, denote the usual Lebesgue and Bochner spaces. The homogeneous Sobolev space \(\mathcal {D}^{1,p}_0 (\Omega )\) is defined as the closure of \(\mathcal {C}_0^\infty (\Omega )\) in the norm
Let \(\mathcal {D}^{-1,p^\prime } (\Omega )\) be the dual space of \(\mathcal {D}_0^{1,p} (\Omega )\). Furthermore, we denote
where \(p^\prime\) and \(q^{\prime }\) are the conjugates of p and q, respectively, i.e., \(1/p+1/p^{\prime }=1\) and \(1/q+1/q^{\prime }=1\). In order to simplify the notation, we will write \(L^q(\Omega )\), \(L^r(0,T;L^q(\Omega ))\) and \(\mathcal D^{1,p}_0(\Omega )\) as \(L^q\), \(L^r(0,T;L^q)\) and \(\mathcal D^{1,p}_0\), respectively, except for special circumstances. The letter C denotes a generic constant which may vary from line to line.
Next, we recall the basic concept of Kuratowski’s measure of non-compactness which will be used to establish the \(\omega\)-limit compactness of semigroup. The following definitions and propositions can be found in [24, 37].
Definition 2.1
Let M be a complete metric space. A one-parameter family \(\{S(t)\}_{t\ge 0}\) of operators \(S(t): M\rightarrow M, t\ge 0\), is called a \(C^0\) or continuous semigroup if
-
1.
\( S(0) \text{ is } \text{ the } \text{ identity } \text{ map } \text{ on } M,\)
-
2.
\( \ S(t+s)=S(t)S(s)\ \text{ for } \text{ all } t,s\ge 0,\)
-
3.
\( \text{ for } \text{ each } x\in M \text{ the } \text{ function } S(t)x \text{ is } \text{ continuous } \text{ in } t\ge 0\).
Definition 2.2
Let M be a complete metric space and A be bounded subset of M. The measure of non-compactness \(\kappa (A)\) of A is defined by
Definition 2.3
A continuous semigroup \(\{S(t)\}_{t\ge 0}\) in a complete metric space M is called \(\omega\)-limit compact, if for any bounded set \(B\subset M\) and any \(\varepsilon >0\), there exists a time \(t^*\ge 0\) such that
Proposition 2.4
Let M be an infinite dimensional Banach space, and \(B(\varepsilon ) \subset M\) a ball of radius \(\varepsilon >0\). Then \(\kappa (B(\varepsilon ))=2\varepsilon .\)
Proposition 2.5
Let \(\{S(t)\}_{t\ge 0}\) be a continuous semigroup in a complete metric space \((M,\rho )\). Then \(\{S(t)\}_{t\ge 0}\) has a global attractor in M if and only if
-
1.
there is a bounded absorbing set \(B\subset M\), and
-
2.
\(\{S(t)\}_{t\ge 0}\) is \(\omega\)-limit compact.
Moreover, we recall the Uniform Gronwall lemma:
Lemma 2.6
(Uniform Gronwall Lemma) Let g, h and y be non-negative locally integrable functions on \([0,\infty [\) such that the differential inequality \( \frac{{\text {d}}y(t)}{{\text {d}}t}\le g(t)y(t)+h(t)\) holds for a.a. \(t\ge 0\) and
where \(a_1, a_2\) and \(a_3\) are positive constants. Then
For the following results we refer to J. Fleckinger-Pellé et al., see [15, Theorem 4.1, Proposition 5.2].
Proposition 2.7
(Principal Eigenvalue) Let \(\Omega \subset {\mathbb {R}}^N\) be an unbounded domain with smooth boundary and let m satisfy \((H_m)\). Then the equation
has a smallest positive principal eigenvalue \(\lambda _1=\lambda _1(\Omega )>0\) and a corresponding eigenfunction \(u_1 \in W^{1,p}_m(\Omega ) = \overline{C_0^\infty (\Omega )}^{\Vert \cdot \Vert _{W^{1,p}_m}}\) where
Moreover,
As a consequence we mention that for any smooth bounded subdomain \(\Omega '\subset \Omega\) the positive principal eigenvalue \(\lambda _1(\Omega ')\) for the eigenvalue problem (2.1) in \(\Omega '\) satisfies
Indeed, first we note that Proposition 2.7 also holds for the domain \(\Omega '\). Now, given any \(u\in C_0^\infty (\Omega ')\) with \(\Vert m^{1/p} u\Vert _{L^p(\Omega ')}=1\), there holds
where \({\overline{u}}\) denotes the extension of u from \(\Omega '\) to \(\Omega\) by 0. Since \(\Vert m^{1/p} \,{\overline{u}}\Vert _{L^p(\Omega )}=1\) as well, we get (2.3).
3 Global well-posedness
In this section, we prove the global well-posedness of Eq. (1.1).
Definition 3.1
A function u(t, x) is called a weak solution of (1.1) on [0, T] iff \(u\in V,\) \( u_t\in V^*\) and \(u|_{t=0}=u_0\) almost everywhere in \(\Omega\) such that
for all test functions \(\xi \in V\).
Let us define the nonlinear operator
It is well-known that the operator A is bounded, monotone and hemicontinuous. Next, we prove the existence of the weak solution to (1.1) by Galerkin’s method.
Theorem 3.2
Let the assumptions \((H^1_{f}),(H^2_{f})\) and \((H_m)\) hold. Then for any \(u_0\in L^2\) and any fixed \(T>0,\) there exists a unique weak solution to (1.1).
Proof
In the first part of the proof we assume that \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain. For the final part of the proof concerning unbounded domains it will be crucial that estimates are independent of the domain \(\Omega\). Consider approximate solutions \(u_n(t)\) of the form
where \(\{e_j\}_{j=1}^\infty\) is an orthonormal basis of \(L^2\) and has a dense linear hull in \(\mathcal {D}^{1,p}\cap L^{q}\cap L^{2}\). We get \(u_n\) from solving the nonlinear ODE problem
By the boundedness, monotonicity and hemicontinuity of \(A: \mathcal {D}_0^{1,p}\rightarrow \mathcal {D}^{-1,p^\prime }\), it follows that the operator A is demicontinuous (see [31, Lemma 2.1 and Lemma 2.2, p. 38]). So, since \(\langle e_j, e_k\rangle )=\delta _{j,k}\) and \(f(x,\cdot )\) is a continuous function, the Peano existence theorem yields at least one local solution \(u_n\) to (3.2) in some interval \([0, T_n)\). Multiplying Eq. (3.2)\(_k\) by the function \(u_{nk}(t)\) for each k, and adding these relations for \(k= 1,\ldots ,n\), we have
Next, we prove that we can extend the approximate solution \(u_n\) from \([0, T_n)\) to the interval [0, T], for every \(T > 0\). Firstly, by (1.7) and (1.9), one has
so that
where \(\lambda _0 := 2a^* \sigma _0^{p^*-p} + \lambda\). Since \(\lambda < \lambda _1(\Omega )\) we find \(\sigma _0>0\) small enough such that \(\lambda _0 < \lambda _1(\Omega )\). Then the property of the first eigenvalue \(\lambda _1(\Omega )\) of \(-\Delta _q\), see Proposition 2.7, and Hölder’s inequality imply that
where we note that \(m\ge 0\). Concerning \(f_1\), by (1.7) and (1.8), we have
These estimates can be summarized to the pointwise lower bound
Since the second last term \(\frac{a^*}{2\delta _0^{q-2}} \Vert u_n\Vert _{L^q}^q\) will finally appear as a problematic term on the right-hand side of the a priori estimate and has to be absorbed, we also derive a more localized version of (3.7). To this aim, we fix a cut-off function \(a\in C^\infty _0({\mathbb {R}}^N)\) such that \(0\le a\le a^*\) and \(\Vert a\Vert _{L^\infty } = a^*\) and obtain that
Integrating (3.7) and (3.8) over \(\Omega\) we get the estimates
respectively. For the second term on the right-hand side of (3.10), we use the compactness of the support of a and deduce from Young’s inequality that
Moreover, it is easy to see that
On the one hand, by combining (3.4), (3.6), (3.9), and (3.12), we infer that
where \(\widetilde{\lambda } = \frac{2}{\lambda _1} (\lambda _1-\lambda _0)\) and \(\widetilde{\mu }:=\mu -\nu >0.\) On the other hand, by (3.4), (3.6), (3.10), (3.11), (3.12), and even omitting the integral \(\int _{{\Omega }} |a(x)||u_n|^2\,\text{d}x\), we get the modified estimate
Next we choose \(\mu _0>0\) small enough such that \(\frac{\mu _0a^*}{\delta _0^{q-2}}\le \frac{\widetilde{\mu }}{2}\), multiply (3.13) by \(\mu _0\) and add (3.14) to see that
Then by Gronwall’s inequality, we obtain
where \(\Vert u_n(0)\Vert _{L^{2}}\le \Vert u(0)\Vert _{L^{2}}\). Therefore, we can extend the approximate solution to the interval [0, T], for every \(T > 0.\) On the other hand, by (3.15), we also have
for any \(t\in [0,T]\). We note that the generic constant C so far depends only on the constants in the assumptions \((H_m), (H_f^1),(H_f^2)\), but neither on the solution, its initial value, the time interval nor the domain \(\Omega\).
Now, multiplying Eq. (3.2)\(_k\) by the function \(u^\prime _{nk}(t)\), adding these relations for \(k= 1,\ldots ,n\), integrating over (s, T) and taking into account (3.3), we have
where \(F(x,u)=\int _0^uf(x,\xi )\,\text{d}\xi\). By conditions (1.8) and (1.9), we see that
By using the Sobolev embedding \(\mathcal {D}_0^{1,p}\hookrightarrow L^{r}\) with \(\min (2,p)\le r \le p^*\) and then integrating (3.18) over (0, T) with respect to the variable s, we have from (3.6)–(3.19)
Note that so far all constants are independent of the bounded domain \(\Omega\).
In the second step of the proof we show that the sequence \((u_n)\) converges to a weak solution of (1.1). The boundedness of A, (3.17) and (3.20) imply the existence of functions \(u\in L^{\infty }(0,T;L^{2})\) and \(\chi \in \ L^{p^\prime }(0,T; \mathcal {D}^{-1,p^\prime })\) such that—ignoring the notion of subsequences
as \(n\rightarrow \infty\) where \(\chi\) will later shown to satisfy \(\chi =Au.\) From \((3.21)_{1,2}\) we deduce that
Since \(\Omega\) is bounded, an Aubin-type compact embedding theorem ( [32, Corollary 4]), yields the precompactness of the set \(\{u_n: n\in {\mathbb {N}}\}\) in \(L^{p}(\epsilon , T; L^{r})\) for every \(\epsilon \in (0,T)\) and \(\min (2,p)\le r<p^*\). Hence, given any sequence \(\{\epsilon _k\}_k\) with \(\epsilon _k \searrow 0\), there exist subsequences \(\{u_{n_m}^{(k)}\}_{m=1}^\infty \subset \{u_{n_m}^{(k-1)}\}_{m=1}^\infty \subset \cdots \subset \{u_{n_m}\}_{m=1}^\infty\) such that for each \(k\in {\mathbb {N}}\)
Now, by Cantor’s diagonal argument, we obtain (up to a subsequence \(\{u_{n_k}^{(k)}\}\)) for \(\min (2,p)\le r<p^*\)
Hence \(f(x,u_n)\rightarrow f(x,u)\) a.e. in \(\Omega _T\). Moreover, the boundedness of \(\{u_n\}\) in \( L^{p}(0,T; L^{p^{*}})\cap L^{q}(0,T; L^{q})\) and assumption (1.8) and (1.9) imply that \(\{f(x,u_n)\}\) is bounded in \(L^{\rho _1}(0,T; L^{\rho _2})\), where
Consequently, \(f(x,u_n)\rightarrow \psi\) weakly in \(L^{\rho _1}(0,T; L^{\rho _2})\). Thus \(\psi =f(x,u)\) thanks to [21, Lemma 1.3]. By the same process, we see that \(|u_n|^{q-2}u_n\rightarrow |u|^{q-2}u\) weakly in \(L^{q^\prime }(\Omega _T)\). Therefore, we have
Note that \(V^* + L^{\rho _1}(0,T; L^{\rho _2}) \subseteq L^{1}\big (0,T; \mathcal {D}^{-1,p^\prime } + L^{q^{\prime }}+L^{\rho _2}\big )\), so that we get \(u\in \mathcal {C}([0,T];\mathcal {D}^{-1,p^\prime }+L^{q^{\prime }}+L^{\rho _2})\). Because of \(u\in L^{\infty }(0,T; L^{2})\) and \(L^2 \hookrightarrow L^{q'}\), [20, Lemma 8.1, p. 275] implies that
i.e., u is continuous in time with respect to the weak topology of \(L^2\). From (3.21) and (3.24), expressing \(u_n(0)\) and u(0) as integrals with respect to time of \(u_n,\partial _tu_n\) and \(u,\partial _tu\), respectively, we have
and then, by (3.3), \(u(0)=u_0\). Hence, taking into account (3.21) and passing to the limit in (3.17) when \(n\rightarrow \infty\), we get at first that \(\Vert u(t)\Vert _{L^{2}}^2\le \Vert u_0\Vert _{L^{2}}^2 +Ct\) and hence
Moreover, by (3.25), \(\Vert u_0\Vert _{L^{2}} \le \liminf _{t\rightarrow 0} \Vert u(t)\Vert _{L^{2}}\) so that with the above inequality u(t) is continuous at \(t=0\) in \(L^2\). By (3.22) we conclude that
Next, we prove \(\chi =Au\). It suffices to prove that \(\limsup _{n\rightarrow \infty }\int _0^T\langle Au_n, u_n\rangle \,\text{d}t\le \int _0^T \langle \chi ,u\rangle \,\text{d}t\) (see [31, Lemma 2.1, p. 38]. From (3.4) we have
Note that due to (3.3) \(u_{n}(0)\rightarrow u_0\) in \(L^{2}\), whereas, by weak convergence properties,
For the proof of \((3.28)_3\) we use \((H_f^1)\), \((H_f^2)\) to get on the one hand that \(f_1(x,v)v + \nu _-|v|^{q_-}\ge 0\) and \(f_2(v)v + \lambda |v|^p\ge 0\) for all \(v\in {\mathbb {R}}\). On the other hand, by \((H_f^1)\), \((H_f^2)\), the non-negative sequence \(\{f_1(x,u_n)u_n + \nu _-|u_n|^{q_-} + m(x)(f_2(u_n)u_n + \lambda |u_n|^p)\}_n\) is uniformly bounded in \(L^1((0,T)\times \Omega )\). Hence Fatou’s Lemma and the pointwise convergence properties in (3.23) imply that
Furthermore, the sequences \(\{|u_n|^{q_-}\}_n\) and \(\{m(x)|u_n|^{p}\}_n\) are converging pointwise a.e. to \(|u|^{q_-}\) and \(m(x)|u|^p\), respectively, and are uniformly bounded in the space \(L^{r}((0,T)\times \Omega )\) for some \(1<r<\infty\), since \(q_-<q\) and \(p<p^*\). This, by the weak convergence result of [21, Lemma 1.3] and using the admissible test function \(1\in L^{r'}\),
as \(n\rightarrow \infty\). Similarly, we obtain that the space-time integral of \(\{m(x)|u_n|^{p}\}\) converges to the corresponding integral of \(m(x)|u|^p\). Therefore, we are allowed to remove the artificial terms \(\nu _-|u_n|^{q_-}\), \(\lambda m(x)|u_n|^p\), and their limits, in (3.29) to get \((3.28)_3\). Finally, writing \(u_n(T)-u_n(t)\) and \(u(T)-u(t)\) as integrals on [t, T] using time derivatives, \((3.21)_2\) and (3.23) imply that
In view of (3.24), we have
Hence (3.27)–(3.31) imply that
Finally, we prove the uniqueness of the solution. Let u and v be the weak solutions of (1.1), with initial data \(u_0\) and \(v_0\), respectively. Denoting \(\omega =u-v\), we have
Multiplying (3.33) by \(\omega\), using (1.8) and the strong monotonicity of the p-Laplacian, we get
so that the assumptions \((H_c)\) on \(\nu ,\lambda\) imply that
Now it is obvious to obtain \(\omega =0\) if \(u_0=v_0\). Therefore, the solution is unique.
In the final part of the proof let \(\Omega \subseteq {\mathbb {R}}^N\) be an unbounded domain. We will use the method of invading domains to get the desired result following the idea of Babin-Vishik (see [7] for details). For the convenience of the reader, we give the details below. Let \(({\tilde{u}}_{R}),\, R\rightarrow +\infty ,\) be a sequence of solutions of problem (1.1) on the bounded open sets \({\Omega }_R=\{x\in \Omega : |x|<R\}\) with initial data \(u_{0,R}=\psi _R(|x|)u_0(x)\) and boundary value \(\tilde{u}_R=0\) on \(\partial \Omega _R\), where \(\psi _R\in C^\infty ({\mathbb {R}};[0,1])\) is a cut-off function satisfying \(\psi _R(\xi )=1\) for \(|\xi |\le R-1\) and \(\psi _R(\xi )=0\) for \(|\xi |\ge R\); moreover, let \(\Vert \psi _R^{(\ell )}\Vert _ {L^\infty } \le c_\ell <\infty\) for \(\ell \in {\mathbb {N}}\) and all \(R\ge 2\). Note that
Since \(\Vert u_{0,R}\Vert _{L^2(\Omega _R)} \le \Vert u_0\Vert _{L^2(\Omega )}\), the estimates (3.4)–(3.20) hold uniformly with respect to R for \({\tilde{u}}_R\). To extend the function \({\tilde{u}}_R\) from \(\Omega _R\) to \({\mathbb {R}}^N\) we multiply \({\tilde{u}}_R\) by \(\psi _R(|x|)\) and extend the product by zero for \(|x|\ge R\), to get the function by \(u_R\). To show that \(u_R\) is a solution of an equation of type (1.1) on \(\Omega\) we note that by Poincaré’s inequality
due to the special structure of the set \(\Omega _R\setminus \Omega _{R-1}\) of “thickness” 1. From (3.4) to (3.20) it follows that there exists a subsequence \(\{u_j\}_j=\{u_{R_j}\}\) of \(\{u_R\}_R\), which is weakly convergent in \(L^p([0,T]; \mathcal {D}_0^{1,p}({\Omega }))\) and weakly\(-*\) convergent in \(L^{\infty }(0,T; L^{2}({\Omega }))\). Denote the limit of \(\{u_j\}\) by \(u_\infty =u_{\infty }(t,x)\).
For fixed \(R=R_k\), let \(u_{kj}:=\psi _k(x)u_j\) for \(j>k\). Via arguments as above for \(\{u_R\}\), the sequence \(\{\psi _ku_j\}_j\), for fixed \(k\in {\mathbb {N}}\), satisfies the estimates (3.4)–(3.20) uniformly with respect to j (\(j>k)\) and with norms evaluated on the domain \({\Omega }_{R_k}\). Exploiting Cantor’s method there exists a subsequence \(\{u_{j'}\}_{j'}\) of \(\{u_{j}\}_{j}\) such that for each \(k\in {\mathbb {N}}\) the sequence \(\{\psi _k u_{j'}\}_{j'}\) is weakly convergent in the above mentioned spaces on \(\Omega _k\). Obviously, due to pointwise- a.e. convergence, see (3.23), the limit \(u_{k\infty }\) satisfies the identity
As in the case of a bounded domain, we prove that \(\psi _ku_\infty\) is a weak solution of (1.1) in \([0,T]\times \Omega _{R_k}\) so that for any \(v\in \mathcal {C}_0^\infty ([0,T]\times {\Omega }_{ R_k-1})\)
Since \(R=R_k\) is arbitrary, we deduce that (3.36) is fulfilled for arbitrary \(v\in \mathcal {C}_0^\infty ([0,T]\times \Omega )\), and \(u_\infty\) is a solution of (1.1). By (3.35), the solution satisfies (1.1)\(_2\). In the estimates (3.4)–(3.20) we pass to the limit as \(R\rightarrow \infty\) and we find that these estimates hold for \(u=u_\infty\).
The uniqueness of the solution \(u_\infty\) is proved as in the bounded domain case. Now the proof of this theorem is finished. \(\square\)
Thus, by Theorem 3.2, the solution operator \(S(t)u_0 = u(t)\) of problem (1.1) generates a continuous semigroup in \(L^{2}\).
4 Existence of the global attractor
We begin this section with the existence of an absorbing set for the semigroup \({S(t)}_{t\ge 0}\).
Proposition 4.1
Under the assumption of Theorem 3.2, the semigroup \({S(t)}_{t\ge 0}\) associated with problem (1.1) admits an absorbing set in \(\mathcal {D}_0^{1,p}\cap L^{2}\cap L^{q}\); i.e., there is a bounded set \(B_0\subset \mathcal {D}_0^{1,p}\cap L^{2}\cap L^{q}\) such that, for any bounded set B in \(L^{2}\), there exists a \(T_1>0\), depending only on B, such that
Proof
Firstly, we multiply (1.1) by u and get that
Considering the a priori estimate (3.16) on any interval [0, t) and passing to the limit \(n\rightarrow \infty\), we get an absorbing set in \(L^{2}({\mathbb {R}}^2)\); namely, for any bounded set \(B\subset L^2\) there exists a \(t_1=t_1(B)>0\) such that
where \(C>0\) is a constant independent of the initial data \(u_0\in B\). Moreover, by (3.14), we see that
Let
where \(F(x,u)=\int _0^u f(x,\xi )\,\text{d}\xi\). By condition (1.8)–(1.10), (3.19) and (4.3), we see that
Multiplying (1.1) by \(u_t\) we find
so that for any \(t\ge s\ge 0\)
A further integration of the previous inequality over (0, t) with respect to the variable s, and (4.2)–(4.4) imply that
where the estimate \(\int _0^t \Vert u\Vert _{L^{p^*}}^{p} \,\text{d}s \le c \int _0^t\Vert \nabla u\Vert _{L^{p}}^{p}\) was used. To get a lower bound for the left-hand side of (4.6), the estimate \(\int _\Omega gu\,\text{d}x \ge -\Vert g\Vert _{L^2}\Vert u\Vert _{L^2} \ge -C\) is obvious. Concerning the estimate for the integral of F consider the term involving \(f_1\). By (1.7) and for \(u>0\)
It is easy to see that the same estimate holds when \(u<0\). Since \(2<q_-<q\), by Hölder’s and Young’s inequality, the term \(\frac{\nu _-}{q_-}\Vert u\Vert _{L^{q_-}}^{q_-}\) can be estimated by \(\frac{\mu }{2q}\Vert u\Vert _{L^q}^q +C\). For the integral with \(f_2\) the assumption (1.7) implies for \(u>0\) that
The same estimate holds for negative u. Summarizing the last two estimates we conclude that
where \(1-\frac{\lambda }{\lambda _1}>0\). Together with (4.6) we obtain the final pointwise estimate
In view of (4.7) there exists an absorbing set in \(\mathcal {D}_0^{1,p}\cap L^{q}\) for any \(t\ge T_1\) for some \(T_1\ge t_1\) large enough. This completes the proof. \(\square\)
Proposition 4.2
Under the assumption of Theorem 3.2, the semigroup \({S(t)}_{t\ge 0}\) associated with problem (1.1) admits an absorbing set in \( L^{2q-2}\) if \(\lambda >0\) in \((H_f^2)\) is sufficiently small; i.e., there is a bounded set \(B_0\subset L^{2q-2}\) such that, for any bounded set \(B \subset L^{2}\), there exists a \(T_2>0\) depending only on B such that
Proof
Similarly to Proposition 4.1, we multiply (1.1) by \(|u|^{q-2}u\) to get
We note that
and rewrite (4.8) in the form
With (1.8), (1.9), and (3.5), (3.6), and by applying (2.2) to \(|u|^{\frac{q+p-2}{p}}\) we deduce that
We choose \(\lambda\) small enough such that
Moreover, note that
Therefore, with \({\widetilde{\mu }}=\mu -\nu\), we obtain from (4.9), (4.10) and (4.11) that
Integrating (4.12) on \([t, t+1]\) with \(t\ge T_1\), Proposition 4.1 implies that
Now, we multiply (1.1) by \(|u|^{2(q-2)}u\), and by a procedure similar as for (4.8)–(4.10) we arrive at the estimate
For the first integral on the right-hand side we use Hölder’s inequality to get that
where the term \(\frac{{\widetilde{\mu }}}{2} \Vert u\Vert _{L^{3q-4}}^{3q-4}\) can be absorbed. The second integral on the right-hand side is rewritten in the form \(\lambda \int _\Omega m(x)\big (|u|^{\frac{p+2(q-2)}{p}}\big )^p\,\text{d}x\) and estimated with the help of (2.2) and finally absorbed by the integral \((2q-3) \int _{\Omega } |\nabla u|^p|u|^{2(q-2)}\,\text{d}x\) in (4.14). For this final step we need the smallness assumption \( \frac{\lambda }{\lambda _1} < \big (\frac{p}{p+2(q-2)}\big )^p (2q-3).\) This gives, cf. (4.12),
Applying the Uniform Gronwall Lemma 2.6, (4.13) and (4.14), we finally obtain
and we let \(T_2=T_1+1\) to finish the proof of this proposition. \(\square\)
Remark 4.3
By using of (4.14) and (4.15), one also has
Multiplying (1.1) by \(|u|^{3(q-2)}u\), we have
A further argument in this step is the absorption of the integral \(\int _\Omega m(x) |u|^{p+3(q-2)}\,\text{d}x\) with the help of (2.2) which requires the smallness \(\frac{\lambda }{\lambda _1} < \Big (\frac{p}{p+3(q-2)}\Big )^p (3(q-2)+1).\) With the same arguments as before, we conclude that
Finally, adopting the smallness assumption on \(\lambda\) step by step, we obtain for a finite number \(k\in {\mathbb {N}}\) the uniform bound
provided
Proposition 4.4
Under the assumptions of Theorem 3.2additionally suppose that \(\frac{2N}{N+2}<p<N\). Then for any \(\varepsilon >0\) and any bounded subset \(B\subset L^{2}\), there exist constants \(T_3=T_3(B,\varepsilon )>0\) and \(R=R(\varepsilon )>0\) such that
where the constant C is independent of \(\varepsilon\) and B.
Proof
Fix any \(\vartheta \in \mathcal {C}^{\infty }({\mathbb {R}})\) such that \(0\le \vartheta \le 1\),
and define \(\vartheta _{R}(x)=\vartheta (\frac{|x|^2}{R^2})\). Now, multiplying (1.1) by \(\vartheta _{R}^{\beta }|u|^{\alpha -1}u\) with \(\alpha =1\) and \(\alpha =q-1\), \(\beta >p\) and integrating in \(\Omega\), one has
Concerning \(f(x,u) = f_1(u)+mf_2(u)\) where \(0\le m=m(x)\) satisfies \((H_m)\), one gets, similar to the estimates of (3.6), (3.7), (3.8) and (3.9), (3.10),
Moreover, for the integral involving \(f_1\),
as well as with the cut-off function \(a\in C^\infty _0({\mathbb {R}}^N)\) satisfying \(0\le a\le a^*=\Vert a\Vert _{L^\infty }\),
Finally, the first term on the right hand side of (4.20) is given by
where \(A_{R}\) denotes the “annulus” \(A_{R}=\{x\in \Omega : R\le |x|\le 2R\}\).
Thus, by (4.21), (4.22) and (4.21), (4.23) respectively, we obtain two different estimates as follows:
as well as
Since \(a(x)\in C_0^\infty ({\mathbb {R}}^N)\) and \(m(x)\in L^{\frac{N}{p}}\cap L^{\infty }\), for any \(\varepsilon >0\), there exists \(R_0\) sufficiently large such that for any \(R\ge R_0\), one has \(\int _{{\Omega }} \vartheta _{R}^{\beta } a^{\frac{q+\alpha -1}{q-q_-}}\,\text{d}x = 0\) and
We also note that
Since \(p>\frac{2N}{N+2}\), we find \(k_0\in {\mathbb {N}}\) such that
Then by an interpolation inequality and uniform estimates of \(\Vert u(t)\Vert _{L^{2}}\) and \(\Vert u(t)\Vert _{L^{k_0(q-2)+2}}\) with respect to time t, see (4.7), (4.17), we obtain in view of Remark 4.3 with an adequate \(\theta \in (0,1)\) the estimate
Therefore, we deduce from (4.25)–(4.29) for \(t\ge T_1+k_0-1\) and \(R\ge R_0\) that
Similar to the proof of (3.13)–(3.15), we obtain, choosing \(\mu _0>0\) sufficiently small, that
Firstly, we take \(\alpha =1\), apply (4.17) in Remark 4.3 with \(k_0\) as in (4.28) so that \(k_0(q-2)+2> p\), and Hölder’s inequality to get with a suitable number \(\theta \in (0,1)\) that
By using Proposition 4.1 and (4.17) and choosing \(R_0\) sufficiently large, it follows that
Moreover, we have for any \( t\ge T_1\) and \(R\ge R_0\)
This implies that for any \(t\ge T_1+k_0-1 \text{ and } R\ge R_0\)
Applying Gronwall’s Lemma, we have
for some \(\widehat{T}_3>T_1+k_0-1\). Next, we insert (4.35) into (4.34) and integrate on \([t, t+1]\) with \(t\ge \widehat{T}_3\) to get the estimate
Secondly, we take \(\alpha =q-1\) in (4.32), and have
Applying (4.17) in Remark 4.3 with \(k_0\) as above, i.e. \((k_0+1)(q-2)+2> p+q-2\), we get with some \(\theta \in (0,1)\) that
Using Proposition 4.1 and (4.37) and choosing \(R_1>R_0\) sufficiently large, it follows that
Moreover, by (4.15), we also get for any \( t\ge T_1+1\) and \(R\ge R_1\)
Therefore,
Now the Uniform Gronwall Lemma 2.6, (4.36) and (4.38) yield the inequality
which finishes the proof of this proposition with \(T_3:=\max \{T_1+k_0,\widehat{T}_3\}\). \(\square\)
Theorem 4.5
Under the assumptions of Theorem 1.1(1), (2) and for \(\frac{2N}{N+2}<p<N\) the semigroup \({S(t)}_{t\ge 0}\) generated by (1.1) with initial data \(u_0\in L^2\) has an \((L^2,L^2\cap L^q)\)-global attractor \(\mathcal {A}\), that is, \(\mathcal {A}\) is compact, invariant in \(L^2\cap L^q\) and attracts every bounded subset of \(L^2\) in the topology of \(L^2\cap L^q\).
Proof
Let \(B\subset L^2(\Omega )\) be bounded. It suffices to verify that the semigroup \({S(t)}_{t\ge 0}\) is \(\omega\)–limit compact. For any fixed R, by Proposition 4.1 and Proposition 4.2, there exists \(T_2=T_2(B,R)\) such that
Then by the compact embedding \( W^{1,p}(\Omega _{2R})\hookrightarrow L^{2}(\Omega _{2R})\) and the embedding \(L^2\cap L^{2q-2} \hookrightarrow L^q\) we obtain that \(\bigcup \nolimits _{t\ge T_2}\bigcup \nolimits _{u_0\in B}(1-\vartheta _R)S(t)u_0\) is precompact in \(L^{2}(\Omega _{2R})\cap L^{q}(\Omega _{2R})\); hence
On the other hand, due to Proposition 4.4, there exists positive constants \(T_3=T_3(B,\varepsilon )\) and \(R=R(\varepsilon )\) such that
so that in view of Proposition 2.4
Choosing \(T_3>T_2\), we see that
from which we conclude that the semigroup \(\{S(t)\}_{t\ge 0}\) is \(\omega\)–limit compact. Therefore, the existence of a global attractor attracting every bounded subset \(B\subset L^2\) in the topology of \(L^2\cap L^q\) is proved. \(\square\)
5 Dimension of the global attractor
In this section, we use the method of \(\ell\)-trajectories (see [26]) to estimate the finite dimension of the global attractor in case of a bounded domain \(\Omega \subset {\mathbb {R}}^N\) and \(p\ge 2\). We first recall the definition of the fractal dimension of a compact subset \(\mathscr {C}\) of a metric space \(\mathscr {X}\), denoted by
where \(N_{\varepsilon }^{\mathscr {X}}(\mathscr {C})\) is the minimal number of \(\varepsilon\)-ball (with respect to the metric of \(\mathscr {X}\)) needed to cover \(\mathscr {C}\).
Lemma 5.1
[26, Lemma 1.3] Let \(\mathscr {X},\mathscr {Y}\) by normed spaces such that \(\mathscr {Y}\hookrightarrow \hookrightarrow \mathscr {X}\) and \(\mathscr {C}\subset \mathscr {X}\) is bounded. Assume that there exists a mapping \(\mathscr {L}: \mathscr {X} \mapsto \mathscr {Y}\) such that \(\mathscr {L}\mathscr {C}=\mathscr {C}\) and \(\mathscr {L}\) is Lipschitz continuous on \(\mathscr {C}\). Then \(d^{\mathscr {X}}_{f}(\mathscr {C})\) is finite.
Remark 5.2
In view of the proof in [26], the same result as Lemma 5.1 holds if \(\mathscr {L}:\mathscr {X}\mapsto \mathscr {Y}\) is Hölder continuous on \(\mathscr {C}\).
For convenience, we introduce the space of trajectories
where \(\ell >0\) is a constant to be fixed, which we endow with the topology of \(\mathcal {X}_{\ell }:=L^{2}(0,\ell ; L^{2})\). Then we define the semigroup \(L_t\) on \(\mathcal {X}_\ell\) by
where u is the unique solution on \([0,\ell +t]\) such that \(u|_{[0,\ell ]}=\omega\). We take
where \(B_0\) is bounded absorbing set in \(L^{2}\) and \(T_1\) is given in Proposition 4.1 such that \(S(t)B_0\subset B_0\) for all \(t\ge T_1\); the closure is taken in the weak topology of \(L^{2}\). Furthermore, let
Theorem 5.3
The dynamical system \((L_t, \mathcal {X}_{\ell })\) possesses a global attractor \(\mathscr {A}_{\ell }\).
Proof
From Proposition 4.1, \(\mathscr {B}_{\ell }\) is positively invariant with respect to \(L_t\). In view of the proof in Theorem 3.2, the solutions to (1.1) are compact in the following sense:
Applying (CP), we have
In fact, let \(\{\omega _n\}\subset \mathscr {B}_{\ell }\) be a sequence of trajectories such that \(\omega _n\rightarrow \omega\) in \(\mathcal {X}_{\ell }\). Then, by (CP), \(\omega\) is also a trajectory, i.e. \(\omega \in X_\ell\). It remains to show that \(\omega (0)\in \mathcal {B}_0\). Firstly, by Proposition 4.1, \(\omega _{n}(t)\in \mathcal {B}_0\) for all t. Secondly, at least for a subsequence, by (3.23), \(\omega _n(t)\rightarrow \omega (t)\) in \(L^{2}\) for almost all t, and since \(\mathcal {B}_0\) is closed, \(\omega (t)\in \mathcal {B}_0\) for almost all \(t\in (0,\ell )\). In particular, \(\omega (t)\in \mathcal {B}_0\) for points t arbitrarily close to 0, which by the continuity of \(\omega :[0,\ell ]\mapsto L^{2}\) and the closedness of \(\mathcal {B}_0\) implies that \(\omega (0)\in \mathcal {B}_0\). Therefore, Theorem 3.2, Proposition 4.1 and (5.1) guarantee that all the assumptions of [26, Lemma 1.1] are satisfied with \(\Sigma _t=L_t, \mathscr {X}=\mathcal {X}_\ell\) and \(\mathscr {B}^{1}=\overline{L_{t}\mathscr {B}_{\ell }}^{\mathcal {X}_\ell }\), and we finish the proof of this theorem. \(\square\)
Lemma 5.4
Let \(u_1\) and \(u_2\) be two solutions of (1.1) starting from \(\mathcal {B}_0\). Then, there exists a constant \(\ell >0\) such that
where the constant \(C>0\) is independent of \(u_1(0)\) and \(u_2(0)\).
Proof
We set \(v(t)=u_1(t)-u_2(t)\). We have
Similarly to (3.34), we obtain the inequality
For \(\ell >0\), one has
We fix \(s\in (0,\ell )\) and integrate (5.4) over \(\tau \in [s,2\ell ]\) to obtain
From above inequality, we have
where C depends on \(\widetilde{\lambda }\) and \(\widetilde{\mu }\). Finally, we finish the proof by integrating (5.7) over \(s\in (0,\ell ).\) \(\square\)
Lemma 5.5
Let \(p\ge 2\) and let \(u_1, u_2\) be as in Lemma 5.4. Then \(v=u_1-u_2\) satisfies the estimate
where \(W^*=\mathcal {D}^{-1,p^\prime }+ L^{q^\prime }\) is the dual of the space \(W=\mathcal {D}_0^{1,p}\cap L^{q}\).
Proof
We begin with (5.3) written in the form
Let us estimate the norm of I, II, and III considered as functionals on W. Concerning I we have with \(c_p=(p-1)\max (1,2^{p-1})\)
Since \(u_1\) and \(u_2\) are solutions to (1.1), from (5.2) we conclude that
For II, since
by (5.2) we get that
Finally, by (1.8) and (1.9), we deduce
Now it follows from (5.2) that
Since \(q>2\) and \(\Omega\) is bounded, we see that \(\mathcal {D}_0^{1,p}\cap L^{2}\cap L^{q}=\mathcal {D}_0^{1,p}\cap L^{q}\), and hence \(W^*+L^{2}=W^*\). Besides, it is obvious that
By (5.3), (5.8), (5.10), (5.11), we get the result of the lemma. \(\square\)
Theorem 5.6
The fractal dimension of the global attractor \(\mathscr {A}_\ell \subset \mathcal {X}_\ell\) introduced in Theorem 5.3is finite, i.e., \(d_{f}^{\mathcal {X}_\ell }(\mathscr {A}_\ell )\) is finite.
Proof
We set
and note the compact embedding \(\mathcal {W}_\ell \hookrightarrow \hookrightarrow \mathcal {X}_{\ell }\), since \(\mathcal {D}_{0}^{1,p}\cap L^{q}\hookrightarrow \hookrightarrow L^{2}\). Then by Lemmas 5.4 and 5.5, we obtain with \(\mathcal {L}:=L_{\ell }\) for all \(\omega _1,\omega _2\in \mathscr {B}_{\ell }\) that
and
Therefore, \(\mathcal {L}:\mathcal {X}_{\ell }\mapsto \mathcal {X}_{\ell }\) is Hölder continuous on \(\mathscr {A}_{\ell }\). We apply Lemma 5.1 with \(\mathscr {X}=\mathcal {X}_{\ell }, \mathscr {Y}=\mathcal {W}_{\ell }, \mathscr {L}=\mathcal {L}\) and \(\mathscr {C}=\mathscr {A}_\ell\). Since \(\mathscr {A}_\ell \subset \mathcal {X}_\ell\), \(L_{t}\mathscr {A}_\ell =\mathscr {A}_\ell\), we finish the proof using Remark 5.2. \(\square\)
Finally, for all \(\tau >0\), we apply (5.5) to get that \(S_t: L^{2}\mapsto L^{2}\) is uniformly (with respect to \(t\in [0,\tau ]\)) Lipschitz continuous on \(L^{2}\). Then, by [26, Lemma 2.1], the operator \(L_t: \mathcal {X}_{\ell }\mapsto \mathcal {X}_{\ell }\) is uniformly Lipschitz continuous in \(t\in [0,\tau ]\) on \(\mathscr {B}_{\ell }\). Hence, also the operator \(e :\mathcal {X}_{\ell }\rightarrow L^{2}\) defined by \(e(v) = v(l)\) i.e., e maps an l-trajectory onto its endpoint, is Lipschitz continuous on \(\mathscr {B}_\ell\). Therefore, by [26, Theorem 2.4], the global attractor \(\mathscr {A}_\ell\) is finite dimensional.
Besides, by (4.5), we see that the unique solution u satisfies \(u^{\prime }\in L^{2}(0,\ell ; L^{2})\). Then, by [26, Lemma 2.2], the mapping \(t\mapsto L_t\omega\) has the following property: there exists \(c>0\) such that for all \(\omega \in \mathscr {B}_{\ell }\) and \(t_1, t_2\in [0,\tau ]\) it holds
Thus, it can be proved (see [26] for details of the proof) that the dynamical system \((L_t, \mathcal {X}_{\ell })\) possesses an exponential attractor \(\mathscr {E}_{\ell }\). Since e is Lipschitz, it follows that \(\mathscr {E}=\mathscr {E}_{\ell }\) is an exponential attractor for S(t) on \(L^{2}\). Since an exponential attractor has, by definition, finite fractal dimension and always contains the global attractor, in view of Theorem 5.6, we get the following theorem.
Theorem 5.7
If \(\Omega\) is a bounded domain of \({\mathbb {R}}^N\) and \(p\ge 2\), then the global attractor associated with (1.1) has finite fractal dimension.
Change history
13 October 2021
A Correction to this paper has been published: https://doi.org/10.1007/s41808-021-00126-9
References
Abergel, F.: Attractor for Navier–Stokes flow in an unbounded domain. RAIRO Modél. Math. Anal. Numér. 23, 359–370 (1989)
Anane, A.: Simplicité et isolation de la première valeur propre du \(p\)-Laplacien avec poids. C. R. Acad. Sci. Paris Sér. I Math. 305, 725–728 (1987)
Anh, C.T., Ke, T.D.: Long-time behavior for quasilinear parabolic equations involving weighted \(p\)-Laplacian operators. Nonlinear Anal. 71(10), 4415–4422 (2009)
Anh, C.T., Ke, T.D.: On quasilinear parabolic equations involving weighted \(p\)-Laplacian operators. Nonlinear Differ. Equ. Appl. 17, 195–212 (2010)
Antontsev, S., Öztürk, E.: Well-posedness and long time behavior for \(p\)-Laplacian equation with nonlinear boundary condition. J. Math. Anal. Appl. 472, 1604–1630 (2019)
Aronson, D.G., Caffarelli, L.A.: The initial trace of a solution of the porous medium equation. Trans. Am. Math. Soc. 280, 351–366 (1983)
Babin, A.V., Vishik, M.I.: Attractors of partial differential evolution equations in an unbounded domain. Proc. R. Soc. Edinburgh Sect. A 116, 221–243 (1990)
Bhattacharya, T.: Some results concerning the eigenvalue problem for the \(p\)-Laplacian. Ann. Acad. Sci. Fenn. Math. 14, 325–343 (1989)
Caffarelli, L.A., Wolanski, N.: \(\cal{C}^{1,\alpha }\) regularity of the free boundary for the Ndimensional porous media equation. Comm. Pure Appl. Math. 43, 885–902 (1990)
Caffarelli, L.A., Vázquez, J.L., Wolanski, N.I.: Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation. Indiana Univ. Math. J. 36, 373–401 (1987)
Caraballo, T., Cobos, M.H., Rubio, P.M.: Asymptotic behaviour of nonlocal \(p\)-Laplacian reaction-diffusion problems. J. Math. Anal. Appl. 459, 997–1015 (2018)
Caraballo, T., Cobos, M.H., Rubio, P.M.: Global attractor for a nonlocal \(p\)-Laplacian equation without uniqueness of solution. Discret. Cont. Dyn. Syst. Ser. B 22, 1801–1816 (2017)
Chipot, M., Savitska, T.: Nonlocal \(p\)-Laplace equations depending on the \(L^p\) norm of the gradient. Adv. Diffe. Equ. 19, 997–1020 (2014)
Efendiev, M.A., Zelik, S.V.: The attractor for a nonlinear reaction-diffusion system in an unbounded domain. Commun. Pure Appl. Math. LIV, 625–688 (2001)
Fleckinger-Pellé, J., Gossez, J.-P., de Thélin, F.: Principal eigenvalue in an unbounded domain and a weighted Poincaré inequality. Prog. Nonlinear Differ. Equ. Appl. 66, 283–296 (2005)
Geredeli, P.G.: On the existence of regular global attractor for \(p\)-Laplacian evolution equation. Appl. Math. Optim. 71, 517–532 (2015)
Geredeli, P.G., Khanmamedov, A.: Long-time dynamics of the parabolic \(p\)-Laplacian equation. Commun. Pure Appl. Anal. 12, 735–754 (2013)
Ladyzenskaya, O.A.: New equations for the description of motions of viscous incompressible fluids and solvability in the large for their boundary values problems. Proc. Steklov Inst. Math. 102, 95–118 (1967)
Lindqvist, P.: On the equation \(\text{ div }(|\nabla u|^{p-2}\nabla u)+ \lambda |u|^{p-2}u=0\). Proc. Am. Math. Soc. 109, 157–164 (1990)
Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. I. Springer, New York (1972)
Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)
Li, F., You, B., Zhong, C.: Multiple equilibrium points in global attractors for some \(p\)-Laplacian equations. Appl. Anal. 97, 1591–1599 (2018)
Liu, Y., Yang, L., Zhong, C.: Asymptotic regularity for \(p\)-Laplacian equation. J. Math. Phys. 51, 052702 (2010)
Ma, Q.F., Wang, S.H., Zhong, C.K.: Necessary and sufficient conditions for the existence of global attractors for semigroups and applications. Indiana Univ. Math. J. 51, 1541–1559 (2002)
Ma, S., Li, H.: Global attractors for weighted \(p\)-Laplacian equations with boundary degeneracy. J. Math. Phys. 53, 012701 (2012)
Málek, J., Pražák, D.: Large time behavior via the method of \(\ell\)-trajectories. J. Differ. Equ. 181, 243–279 (2002)
Málek, R., Nečas, J., Rokyta, J., R\(\mathring{u}\)žička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation, 13. Chapman & Hall, London (1996)
Marín-Rubio, P., Real, J.: Attractors for 2D-Navier–Stokes equations with delays on some unbounded domains. Nonlinear Anal. 67, 2784–2799 (2007)
Niu, W., Zhong, C.: Global attractors for the \(p\)-Laplacian equations with nonregular data. J. Math. Anal. Appl. 392, 123–135 (2012)
Qian, C., Shen, Z.: Existence of global solutions and attractors for the parabolic equation with critical Sobolev and Hardy exponent in \({{\mathbb{R}}}^{N}\). Nonlinear Anal. RWA 42, 290–307 (2018)
Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys Monographs, 49. American Mathematical Society, Providence (1997)
Simon, J.: Compact sets in the space \(L^p(0, T;B)\). Ann. Mat. Pura Appl. 146, 65–96 (1987)
Simsen, J., Valero, J.: Global attractors for \(p\)-Laplacian differential inclusions in unbounded domains. Discret. Contin. Dyn. Syst. Ser. B 21, 3239–3267 (2016)
Yang, M., Sun, C., Zhong, C.: Existence of a global attractor for a \(p\)-Laplacian equation in \({{\mathbb{R}}}^n\). Nonlinear Anal. 66, 1–13 (2007)
Yang, M., Sun, C., Zhong, C.: Global attractors for \(p\)-Laplacian equation. J. Math. Anal. Appl. 327, 1130–1142 (2007)
Zhao, W.: Long-time random dynamics of stochastic parabolic \(p\)-Laplacian equations on \({{\mathbb{R}}}^N\). Nonlinear Anal. 152, 196–219 (2017)
Zhong, C., Yang, M., Sun, C.: The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction–diffiusion equations. J. Differ. Equ. 223(2), 367–399 (2006)
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The research is partly supported by NSFC 11501517.
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Farwig, R., Qian, C. The global existence and attractor for p-Laplace equations in unbounded domains. J Elliptic Parabol Equ 6, 311–342 (2020). https://doi.org/10.1007/s41808-020-00067-9
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DOI: https://doi.org/10.1007/s41808-020-00067-9