On the Existence of Regular Global Attractor for $p$-Laplacian Evolution Equation

Abstract

In this study, we consider the nonlinear evolution equation of parabolic type

$$\begin{aligned} u_t-\mathrm{div}\big (|\nabla u|^{p-2}\nabla u\big )+f(u)=g. \end{aligned}$$

We analyze the long time dynamics (in the sense of global attractors) under very general conditions on the nonlinearity $f$. Since we do not assume any polynomial growth condition on it, the main difficulty arises at first in the proof of well-posedness. In fact, the very first contribution to this problem is a pioneering paper (Efendiev and Ôtano, Differ Int Equ 20:1201–1209, 2007) where the well-posedness result has been shown by exploiting the technique from the theory of maximal monotone operators. However, from some physical aspects, to obtain the solution in variational sense might be demanding which requires limiting procedure on the approximate solutions. In this work, we are interested in variational (weak) solution. The critical issue in the proof of well-posedness is to deal with the limiting procedure on $f$ which is overcome utilizing the weak convergence tecniques in Orlicz spaces (Geredeli and Khanmamedov, Commun Pure Appl Anal 12:735–754, 2013; Krasnosel’skiĭ and Rutickiĭ, Convex functions and Orlicz spaces, 1961). Then, proving the existence of the global attractors in $L^2(\Omega )$ and in more regular space $W_0^{1,p}(\Omega )$, we show that they coincide. In addition, if $f$ is monotone and $g=0$, we give an explicit estimate of the decay rate to zero of the solution.

Appendix

We report two technical lemmas needed in the investigation. The first can be inferred from [12], whereas the second one is a variant of the uniform Gronwall lemma [23], and is implicitly contained in [19]. For the reader’s convenience, we provide a short proof of both.

Lemma 6.1

Let $u_n:\Omega _T\rightarrow \mathbb {R}$ be a sequence of functions, and let $h\in {\mathcal C}(\mathbb {R})$ be strictly increasing with $h(0)=0$. Assume that the convergence $u_n(x,t)\rightarrow u(x,t)$ holds for almost every $(x,t)\in \Omega _T=\Omega \times (0,T)$, and

$$\begin{aligned} \int _{\Omega _T} h(u_n)u_n\,\mathrm{d}x\mathrm{d}t\le a, \end{aligned}$$

(6.1)

for some $a\ge 0$ independent of $n$. Then both $h(u_n)$ and $h(u)$ belong to $L^1(\Omega _T)$, and for every test function $\zeta \in {\mathcal C}^\infty _0([0,T],W)$ we have the convergence

$$\begin{aligned} \int _{\Omega _T} h(u_n)\zeta \,\mathrm{d}x\mathrm{d}t\rightarrow \int _{\Omega _T} h(u)\zeta \,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

(6.2)

Proof

We begin to show that

$$\begin{aligned} \int _{\Omega _T} |h(u_n)|\,\mathrm{d}x\mathrm{d}t\le b, \end{aligned}$$

where $b=a+|\Omega _T|\sup _{|s|\le 1}|h(s)|$. In turn, by the Fatou lemma we readily obtain

$$\begin{aligned} \int _{\Omega _T} |h(u)|\,\mathrm{d}x\mathrm{d}t\le b. \end{aligned}$$

To this end, for a fixed $n$, let us decompose $\Omega _T$ as $\Omega _T^+\cup \Omega _T^-$, where $\Omega _T^+=\Omega _T\cap \{|u_n|\ge 1\}$ and $\Omega _T^-=\Omega _T\cap \{|u_n|< 1\}$. Since $h(u_n)u_n$ is positive, we get

$$\begin{aligned} \int _{\Omega _T} |h(u_n)|\,\mathrm{d}x\mathrm{d}t&\le \int _{\Omega _T^+} |h(u_n)u_n|\,\mathrm{d}x\mathrm{d}t+\int _{{\Omega _T}^-} |h(u_n)|\,\mathrm{d}x\mathrm{d}t\\&\le \int _{\Omega _T} h(u_n)u_n\,\mathrm{d}x\mathrm{d}t+|\Omega _T|\sup _{|u_n|< 1}|h(u_n)|\le b. \end{aligned}$$

By (6.1), it follows that

$$\begin{aligned} \int _{\Omega _T} h(u_n^{+})u_n^{+}\mathrm{d}x\mathrm{d}t + \int _{\Omega _T} h(u_n^{-})u_n^{-}\mathrm{d}x\mathrm{d}t \le a, \end{aligned}$$

where $u_n^{+}= \max \{u_n,0\}$ and $u_n^{-}=\min \{u_n,0\}$. Taking into account the properties of $h$, we can define the $N$-function (see [1, 17] for the definition)

$$\begin{aligned} H(\xi )=\int _{0}^{\xi } h^{-1}(s)\mathrm{d}s,\quad \xi \ge 0. \end{aligned}$$

It is then apparent that

$$\begin{aligned} \int _{\Omega _T} H(h(u_{n}^{+}))\,\mathrm{d}x\mathrm{d}t \le \int _{\Omega _T} h(u_{n}^{+})u_{n}^{+}\,\mathrm{d}x\mathrm{d}t \le a. \end{aligned}$$

Introducing the Orlicz space $L_{H}({\Omega _T})$ (see [1, 17]), which is a Banach space with respect to the so-called Luxemburg norm

$$\begin{aligned} \Vert v\Vert _{L_{H}({\Omega _T})}=\inf \left\{ k>0:\,\int _{\Omega _T}H\big (k^{-1}|v|\big ) \,\mathrm{d}x\mathrm{d}t\le 1\right\} , \end{aligned}$$

the latter inequality provides the bound²

$$\begin{aligned} \Vert h(u_n^{+})\Vert _{L_{H}({\Omega _T})} \le 1+a. \end{aligned}$$

Since $u_n(x,t)\rightarrow u(x,t)$ for almost every $(x,t)\in {\Omega _T}$ and $h$ is continuous, we infer that $h(u_n^{+})\rightarrow h(u^{+})$ in measure. Appealing to [17, Theorem 14.6], these two facts imply the convergence

$$\begin{aligned} \int _{\Omega _T} h(u_n^{+})\zeta \,\mathrm{d}x\mathrm{d}t\rightarrow \int _{\Omega _T} h(u^{+})\zeta \,\mathrm{d}x\mathrm{d}t, \end{aligned}$$

for every $\zeta \in {\mathcal C}_0^{\infty }([0,T],W)$. By a similar argument, making use of the Orlicz space $L_{G}({\Omega _T})$ relative to the function

$$\begin{aligned} G(\xi )=\int _{0}^{\xi } -h^{-1}(-s)\mathrm{d}s,\quad \xi \ge 0, \end{aligned}$$

we can prove that

$$\begin{aligned} \int _{\Omega _T} h(u_n^{-})\zeta \,\mathrm{d}x\mathrm{d}t\rightarrow \int _{\Omega _T} h(u^{-})\zeta \,\mathrm{d}x\mathrm{d}t. \end{aligned}$$

Recalling that $h(0)=0$, the claimed convergence (6.2) is established. $\square$

Lemma 6.2

Let $\phi$ be a nonnegative absolutely continuous function on $[0,\infty )$ satisfying for some $\varepsilon >0$ and some nonnegative function $\psi$ the differential inequality

$$\begin{aligned} \phi '+\varepsilon \phi \le \psi . \end{aligned}$$

If $\int _0^1\phi (\tau )+\psi (\tau )\,\mathrm{d}\tau \le m$ for some $m\ge 0$, then for every $t>0$ we have the estimate

$$\begin{aligned} \phi (t)\le \frac{1+t}{t}m\mathrm{e}^{-\varepsilon (t-1)}+\int _0^t\mathrm{e}^{-\varepsilon (t-\tau )}\psi (\tau )\,\mathrm{d}\tau . \end{aligned}$$

In particular, if $\psi$ is summable the last term can be simply replaced by $\int _0^\infty \psi (\tau )\,\mathrm{d}\tau$.

Proof

Multiplying the differential inequality by $\tau$ we get

$$\begin{aligned}{}[\tau \phi (\tau )]'\le \phi (\tau )+\psi (\tau ), \end{aligned}$$

and integrating on $(0,t)$ with $t\le 1$ we readily obtain

$$\begin{aligned} \phi (t) \le \frac{m}{t},\quad \forall t\le 1. \end{aligned}$$

Next, we apply the Gronwall lemma on $(1,t)$ with $t>1$ to the original inequality. This entails

$$\begin{aligned} \phi (t)\le \phi (1)\mathrm{e}^{-\varepsilon (t-1)}+\int _1^t\mathrm{e}^{-\varepsilon (t-\tau )}\psi (\tau )\,\mathrm{d}\tau ,\quad \forall t>1. \end{aligned}$$

Since $\phi (1)\le m$, collecting the two formulae we are done. $\square$