1. Introduction

The Caginalp phase-field system

$\frac{\partial u}{\partial t}-\Delta u+f(u) = \theta,$

(1.1)

$\frac{\partial\theta}{\partial t}-\Delta\theta = -\frac{\partial u}{\partial t},$

(1.2)

has been introduced in ^[1] in order to describe the phase transition phenomena in certain class of material. In this context, $\theta$ denotes the relative temperature (relative to the equilibrium melting temperature), and $u$ is the phase-field or order parameter, $f$ is a given function (precisely, the derivaritve of a double-well potential $F$). This system has received much attention (see for example, ^[2], ^[3], ^[4], ^[5], ^[6], ^[7], ^[8], ^[10], ^[11], ^[12], ^[13], ^[14], ^[15], ^[16], ^[17], ^[18], ^[19], ^[22], ^[29], ^[33] and ^[41]). These equations can be derived by introducing the (total Ginzburg-Landau) free energy:

$\begin{equation}\label{e3} \psi = \int_{\Omega}\bigg(\frac{1}{2}|\nabla u|^2+F(u)-u\theta-\frac{1}{2}\theta^2\bigg)\, dx, \end{equation}$

(1.3)

where $\Omega$ is the domain occupied by the system (here, we assume that it is a bounded and smooth domain of $\mathbb{R}^n$, $n = 1, \, 2$ or $3$, with boundary $\partial \Omega$), and the enthalpy

$\begin{equation}\label{e4} H = u+\theta. \end{equation}$

(1.4)

Then, the evolution equation for the order parameter $u$ is given by:

$\begin{equation}\label{e5} \frac{\partial u}{\partial t} = -\delta_{u}\psi, \end{equation}$

(1.5)

where $\delta_{u}$ stands for the variational derivative with respect to $u$, which yields (1.1). Then, we have the energy equation

$\begin{equation}\label{e6} \frac{\partial H}{\partial t} = -\mbox{div}\, q, \end{equation}$

(1.6)

where $q$ is the heat flux. Assuming finally the classical Fourier law for heat conduction, which prescribes the heat flux as

$\begin{equation}\label{e7} q = -\nabla \theta, \end{equation}$

(1.7)

we obtain (1.2). Now, a well-known side effect of the Fourier heat law is the infinite speed of propagation of thermal disturbances, deemed physically unreasonable and thus called paradox of heat conduction (see, for example, ^[9]). In order to account for more realistic features, several variations of (1.7), based, for example, on the Maxwell-Cattaneo law or recent laws from thermomechanics, have been proposed in the context of the Caginalp phase-field system (see, for example, ^[19], ^[20], ^[21], ^[23], ^[24], ^[25], ^[26], ^[27], ^[28], ^[30], ^[31], ^[35], ^[36], ^[37], ^[38], ^[44], ^[45] and ^[46]).

A different approach to heat conduction was proposed in the Sixties (see, ^[47], ^[48] and ^[49]), where it was observed that two temperatures are involved in the definition of the entropy: the conductive temperature $\theta$, influencing the heat conduction contribution, and the thermodynamic temperature, appearing in the heat supply part. For time-independent models, it appears that these two temperatures coincide in absence of heat supply. Actually, they are different generally in the time depedent case see, for example, ^[19] and references therein for more discussion on the subject. In particular, this happens for non-simple materials. In that case, the two temperatures are related as follows (see ^[42], ^[43]):

$\begin{equation}\label{e8} \theta = \varphi-\Delta \varphi. \end{equation}$

(1.8)

Our aim in this paper is to study a generalization of the Caginalp phase-field system based on this two temperatures theory and the usual Fourier law with a nonlinear coupling.

The purpose of our study is the following initial and boundary value problem

$\frac{\partial u}{\partial t}-\Delta u+f(u) = g(u)\big(\varphi-\Delta \varphi\big) \label{e9},$

(1.9)

$\frac{\partial \varphi}{\partial t}-\Delta \frac{\partial \varphi}{\partial t}-\Delta \varphi = -g(u)\frac{\partial u}{\partial t}, \label{e10}$

(1.10)

$u = \varphi = 0 \mbox{on} \partial\Omega, \label{e11}$

(1.11)

$u|_{t = 0} = u_0, ~ \varphi|_{t = 0} = \varphi_0 \label{e12}.$

(1.12)

The paper is organized as follows. In Section 2, we give the derivation of the model. The Section 3 states existence, regularity and uniqueness results. In Section 4, we address the question of dissipativity properties of the system. The last section, analyzes the spatial behavior of solutions in a semi-infinite cylinder, assuming their existence.

Thoughout this paper, the same letters $c, \, c', \, c'', \,$ and sometimes $c'''$ denote constants which may change from line to line and also $\|.\|_{p}$ will denote the usual $L^{p}$ norm and $(., .)$ the usual $L^2$ scalar product. More generally, we will denote by $\|.\|_{X}$ the norm in the Banach space X. When there is no possible confusion, $\|.\|$ will be noted instead of $\|.\|_2$.

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2. Derivation of the model

In our case, to obtain equations (1.9) and (1.10), the total free energy reads in terms of the conductive temperature $\theta$,

$\begin{equation}\label{e13} \psi(u, \theta) = \int_{\Omega}\bigg(\frac{1}{2}|\nabla u|^2+F(u)-G(u)\theta-\frac{1}{2}\theta^2\bigg)\, dx, \end{equation}$

(2.1)

where $f = F'$ and $g = G'$, and (1.5) yields, in view of (1.8), the evolution equation for the order parameter (1.9). Furthermore, the enthalpy now reads

$\begin{equation}\label{e14} H = G(u)+\theta = G(u)+\varphi-\Delta\varphi, \end{equation}$

(2.2)

which yields thanks to (1.6), the energy equation,

$\begin{equation}\label{e15} \frac{\partial \varphi}{\partial t}-\Delta \frac{\partial \varphi}{\partial t}+\mbox{div}\, q = -g(u)\frac{\partial u}{\partial t}. \end{equation}$

(2.3)

Considering the usual Fourier law ($q = -\nabla\, \varphi$), one has (1.10).

Remark 2.1. We can note that we still have an infinite speed of propagation here.

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3. Existence and uniqueness of solutions

Before stating the existence result, we make some assumptions on nonlinearities $f$ and $g$:

$|G(s)|^2\leq c_1\, F(s)+c_2, \quad c_0, c_1, c_2\geq 0,$

(3.1)

$\begin{equation}\label{hyp2} |g(s)s|\leq c_{3}(|G(s)|^2+1), \quad c_{3}\geq 0, \end{equation}$

(3.2)

$\begin{equation}\label{hyp3} c_4\, s^{k+2}-c_5\leq F(s)\leq f(s)s+c_0\leq c_6\, s^{k+2}-c_7, \quad c_4, c_6>0, \, \, c_5, c_7\geq 0, \end{equation}$

(3.3)

$\begin{equation}\label{hyp4} |g(s)|\leq c_{8}(|s|+1), \quad |g^{\prime}(s)|\leq c_{9}\quad c_8, c_{9}\geq 0, \end{equation}$

(3.4)

$\begin{equation}\label{hyp5} |f^{\prime}(s)|\leq c_{10}(|s|^k+1), \quad c_{10}\geq 0, \end{equation}$

(3.5)

where $k$ is an integer.

Theorem 3.1. We assume that (3.1)-(3.4) hold true. For all initial data $(u_0, \, \varphi_0)\in H^1_0(\Omega)\cap L^{k+2}(\Omega)\times H^1_0(\Omega)\cap H^2(\Omega)$, the problem (1.9)-(1.12) possesses at least one solution $(u, \varphi)$ with the following regularity $u\in L^{\infty}(0, T;H^1_0(\Omega))\cap L^{k+2}(\Omega), \, \frac{\partial u}{\partial t}\in L^2(0, T;L^2(\Omega)), \, \varphi \in L^{\infty}(0, T;H^1_0(\Omega)\cap H^2(\Omega))$ and $\frac{\partial \varphi}{\partial t}\in L^2(0, T; H^1_0(\Omega))$.

Proof. The proof is based on the Galerkin scheme. Here, we just make formally computations to get a priori estimates, having in mind that these estimates can be rigourously justified using the Galerkin scheme see, for example, ^[10], ^[11] and ^[40] for details.

Multiplying (1.9) by $\frac{\partial u}{\partial t}$ and integrating over $\Omega$, we get

$\begin{equation}\label{e16} \frac{1}{2}\frac{d}{dt}\bigg(\|\nabla u\|^2+2\int_{\Omega}F(u)\, dx\bigg)+\left\|\frac{\partial u}{\partial t}\right\|^2 = \int_{\Omega}g(u)\frac{\partial u}{\partial t}(\varphi-\Delta \varphi)\, dx. \end{equation}$

(3.6)

Multiplying (1.10) by $\varphi-\Delta \varphi$ and integrating over $\Omega$, we have

$\begin{equation}\label{e17} \begin{split} \frac{1}{2}\frac{d}{dt}(\|\varphi\|^2+2\|\nabla\varphi\|^2+\|\Delta\varphi\|^2)+\|\nabla\varphi\|^2&+\|\Delta\varphi\|^2\\ & = -\int_{\Omega}g(u)\frac{\partial u}{\partial t}(\varphi-\Delta \varphi)\, dx. \end{split} \end{equation}$

(3.7)

Now, summing (3.6) and (3.7), we are led to,

$\begin{equation}\label{e18} \begin{split} \frac{d}{dt}\bigg(\|\nabla u\|^2+2\int_{\Omega}F(u)\, dx+\|\varphi\|^2&+2\|\nabla\varphi\|^2+\|\Delta\varphi\|^2\bigg)\\ &+2\bigg(\left\|\frac{\partial u}{\partial t}\right\|^2+\|\nabla\varphi\|^2+\|\Delta\varphi\|^2\bigg) = 0. \end{split} \end{equation}$

(3.8)

Multiplying (1.9) by $u$ and integrating over $\Omega$, we obtain

$\begin{equation}\label{e19} \frac{1}{2}\frac{d}{dt}\|u\|^2+\|\nabla u\|^2+\int_{\Omega}f(u)u\, dx = \int_{\Omega}g(u)u(\varphi-\Delta\varphi)\, dx. \end{equation}$

(3.9)

Using (3.2)-(3.3), (3.9) becomes

$\begin{equation}\label{e20} \begin{split} \frac{1}{2}\frac{d}{dt}\|u\|^2+\|\nabla u\|^2&+c\int_{\Omega}F(u)\, dx\\ &\leq c'\int_{\Omega}|G(u)|^2\, dx+\frac{1}{2}(\|\varphi\|^2+\|\Delta\varphi\|^2)+c''. \end{split} \end{equation}$

(3.10)

Adding (3.8) and (3.10), one has

$\begin{equation}\label{e21} \begin{split} \frac{d\, E_1}{dt}+2\bigg(\|\nabla u\|^2+c\int_{\Omega}F(u)\, dx&+\left\|\frac{\partial u}{\partial t}\right\|^2+\|\nabla\varphi\|^2\bigg)+\|\Delta\varphi\|^2\\ &\leq c'\int_{\Omega}|G(u)|^2\, dx+\|\varphi\|^2+c'', \end{split} \end{equation}$

(3.11)

where

$\begin{equation}\label{e22} E_1 = \|u\|^2+\|\nabla u\|^2+2\int_{\Omega}F(u)\, dx+\|\varphi\|^2+2\|\nabla\varphi\|^2+\|\Delta\varphi\|^2 \end{equation}$

(3.12)

enjoys

$\begin{equation}\label{e23} E_1\leq c\bigg(\|u\|^2_{H^1(\Omega)}+\|u\|^{k+2}_{k+2}+\|\varphi\|^2_{H^2(\Omega)}\bigg)-c' \end{equation}$

(3.13)

and

$\begin{equation}\label{e24} E_1\leq c''\bigg(\|u\|^2_{H^1(\Omega)}+\|u\|^{k+2}_{k+2}+\|\varphi\|^2_{H^2(\Omega)}\bigg)-c'''. \end{equation}$

(3.14)

Multiplying now (1.10) by $\frac{\partial \varphi}{\partial t}$ and integrating over $\Omega$, we have

$\begin{equation}\label{e25} \frac{1}{2}\frac{d}{dt}\|\nabla\varphi\|^2+\left\|\frac{\partial \varphi}{\partial t}\right\|^2+\left\|\nabla\frac{\partial \varphi}{\partial t}\right\|^2 = -\int_{\Omega}g(u)\frac{\partial u}{\partial t}\frac{\partial \varphi}{\partial t}\, dx. \end{equation}$

(3.15)

Taking into account (3.4) and using Hölder's inequality, we get

$\begin{equation}\label{e26} \frac{1}{2}\frac{d}{dt}\|\nabla\varphi\|^2+\frac{1}{2}\left\|\frac{\partial \varphi}{\partial t}\right\|^2+\left\|\nabla\frac{\partial \varphi}{\partial t}\right\|^2\leq c(\|\nabla u\|^2+1)\left\|\frac{\partial u}{\partial t}\right\|^2 \end{equation}$

(3.16)

and then, summing (3.11) and (3.16), we have

$\begin{equation}\label{e27} \begin{split} \frac{d\, E_2}{dt}&+2\bigg(\|\nabla u\|^2+c\int_{\Omega}F(u)\, dx+\left\|\frac{\partial u}{\partial t}\right\|^2\\ &+\|\nabla\varphi\|^2+\frac{1}{2}\|\Delta\varphi\|^2+\frac{1}{2}\left\|\frac{\partial \varphi}{\partial t}\right\|^2+\left\|\nabla\frac{\partial \varphi}{\partial t}\right\|^2\bigg)\\ &\leq c\int_{\Omega}|G(u)|^2\, dx+\|\varphi\|^2+c''(\|\nabla u\|^2+1)\left\|\frac{\partial u}{\partial t}\right\|^2+c''', \end{split} \end{equation}$

(3.17)

where

$\begin{equation}\label{e28} E_2 = E_1+\|\nabla \varphi\|^2 \end{equation}$

(3.18)

satisfies similar estimates as $E_1$.

We deduce from (3.1) and (3.17)

$\begin{equation}\label{e29} \frac{d\, E_2}{dt}+c\bigg(\left\|\frac{\partial\varphi}{\partial t}\right\|^2+\left\|\nabla\frac{\partial\varphi}{\partial t}\right\|^2\bigg)\leq c'E_2+c'', \end{equation}$

(3.19)

which achieve the proof.

For more regularity on solutions, we make following additional assumptions:

$\begin{equation}\label{hyp6} f(0) = 0 \mbox{and} f'(s)\geq -c, \, c\geq 0. \end{equation}$

(3.20)

We have:

Theorem 3.2. Under assumptions of Theorem 3.1 and assuming that (3.20) is satisfied. For every initial data $(u_0, \, \varphi_0)\in H^1_0(\Omega)\cap L^{k+2}(\Omega)\times H^1_0(\Omega)\cap H^2(\Omega)$, the problem (1.9)-(1.12) admits at least one solution $(u, \, \varphi)$ such that $u\in L^{\infty}(0, T;H^1_0(\Omega))\cap L^{k+2}(\Omega), \, \frac{\partial u}{\partial t}\in L^{\infty}(0, T;L^2(\Omega))\cap L^2(0, T; H^1_0(\Omega))$, $\varphi\in L^{\infty}(0, T; H^1_0(\Omega)\cap H^2(\Omega))$ and $\frac{\partial \varphi}{\partial t}\in L^2(0, T; H^1_0(\Omega)\cap H^2(\Omega))$.

Proof. As above proof, we focus on a priori estimates.

We multiply (1.10) by $-\Delta\frac{\partial \varphi}{\partial t}$ and have, integrating over $\Omega$,

$\begin{equation}\label{e30} \frac{1}{2}\frac{d}{dt}\|\nabla\varphi\|^2+\left\|\nabla\frac{\partial\varphi}{\partial t}\right\|^2+\left\|\Delta\frac{\partial\varphi}{\partial t}\right\|^2 = \int_{\Omega}g(u)\frac{\partial u}{\partial t}\Delta\frac{\partial \varphi}{\partial t}\, dx. \end{equation}$

(3.21)

Thanks to (3.4) and Hölder's inequality:

$\begin{equation}\label{e31} \begin{split} \int_{\Omega}g(u)\frac{\partial u}{\partial t}\Delta\frac{\partial \varphi}{\partial t}\, dx&\leq c\int_{\Omega}(|u|+1)\left|\frac{\partial u}{\partial t}\right|\left|\Delta\frac{\partial\varphi}{\partial t}\right|\, dx\\ &\leq c(\|\nabla u\|^2+1)\left\|\frac{\partial u}{\partial t}\right\|^2+\frac{1}{2}\left\|\Delta\frac{\partial\varphi}{\partial t}\right\|^2 \end{split} \end{equation}$

(3.22)

and then,

$\begin{equation}\label{e32} \frac{1}{2}\frac{d}{dt}\|\nabla\varphi\|^2+\left\|\nabla\frac{\partial\varphi}{\partial t}\right\|^2+\frac{1}{2}\left\|\Delta\frac{\partial\varphi}{\partial t}\right\|^2\leq c(\|\nabla u\|^2+1)\left\|\frac{\partial u}{\partial t}\right\|^2. \end{equation}$

(3.23)

Differentiating (1.9) with respect to time, we get

$\begin{equation}\label{e33} \frac{\partial^2u}{\partial t^2}-\Delta\frac{\partial u}{\partial t}+f'(u)\frac{\partial u}{\partial t} = g'(u)\frac{\partial u}{\partial t}(\varphi-\Delta\varphi)+g(u)\bigg(\frac{\partial\varphi}{\partial t}-\Delta\frac{\partial\varphi}{\partial t}\bigg). \end{equation}$

(3.24)

Multiplying (3.24) by $\frac{\partial u}{\partial t}$ and integrating over $\Omega$, we obtain

$\begin{equation}\label{e34} \begin{split} &\frac{1}{2}\frac{d}{dt}\left\|\frac{\partial u}{\partial t}\right\|^2+\left\|\nabla \frac{\partial u}{\partial t}\right\|^2+\int_{\Omega}f'(u)\left|\frac{\partial u}{\partial t}\right|^2\, dx\\ & = \int_{\Omega}g'(u)\left|\frac{\partial u}{\partial t}\right|^2(\varphi-\Delta\varphi)\, dx+\int_{\Omega}g(u)\frac{\partial u}{\partial t}\bigg(\frac{\partial\varphi}{\partial t}-\Delta\frac{\partial\varphi}{\partial t}\bigg)\, dx. \end{split} \end{equation}$

(3.25)

Using (1.10), we write,

$\begin{equation}\label{e35} \begin{split} \int_{\Omega}g(u)\frac{\partial u}{\partial t}\bigg(\frac{\partial\varphi}{\partial t}-\Delta\frac{\partial\varphi}{\partial t}\bigg)\, dx& = \int_{\Omega}g(u)\frac{\partial u}{\partial t}\bigg(-g(u)\frac{\partial u}{\partial t}+\Delta\varphi\bigg)\, dx\\ & = -\int_{\Omega}\left|g(u)\frac{\partial u}{\partial t}\right|^2\, dx+\int_{\Omega}g(u)\frac{\partial u}{\partial t}\Delta\varphi\, dx. \end{split} \end{equation}$

(3.26)

Owing to (3.26), (3.25) reads

$\begin{equation}\label{e36} \begin{split} &\frac{1}{2}\frac{d}{dt}\left\|\frac{\partial u}{\partial t}\right\|^2+\left\|\nabla \frac{\partial u}{\partial t}\right\|^2+\int_{\Omega}f'(u)\left|\frac{\partial u}{\partial t}\right|^2\, dx\\ & = \int_{\Omega}g'(u)\left|\frac{\partial u}{\partial t}\right|^2(\varphi-\Delta\varphi)\, dx+\int_{\Omega}g(u)\frac{\partial u}{\partial t}\Delta\varphi\, dx-\int_{\Omega}\left|g(u)\frac{\partial u}{\partial t}\right|^2\, dx, \end{split} \end{equation}$

(3.27)

since

$\begin{equation}\label{e37} \begin{split} \int_{\Omega}g'(u)\left|\frac{\partial u}{\partial t}\right|^2(\varphi-\Delta\varphi)\, dx&\leq c\int_{\Omega}\left|\frac{\partial u}{\partial t}\right|^2(|\varphi|+|\Delta\varphi|)\, dx\\ &\leq \frac{1}{2}\left\|\nabla\frac{\partial u}{\partial t}\right\|^2+c(\|\varphi\|^2+\|\Delta\varphi\|^2), \end{split} \end{equation}$

(3.28)

$\begin{equation}\label{e38} \int_{\Omega}g(u)\frac{\partial u}{\partial t}\Delta\varphi\, dx = -\int_{\Omega}g'(u)\nabla u\frac{\partial u}{\partial t}\nabla\varphi\, dx-\int_{\Omega}g(u)\nabla\frac{\partial u}{\partial t}\nabla\varphi\, dx \end{equation}$

(3.29)

and then,

$\begin{equation}\label{e39} \begin{split} \left|\int_{\Omega}g'(u)\nabla u\frac{\partial u}{\partial t}\nabla\varphi\, dx\right|&\leq c\int_{\Omega}|\nabla u|\left|\frac{\partial u}{\partial t}\right||\nabla\varphi|\, dx\\ &\leq \frac{1}{6}\left\|\nabla\frac{\partial u}{\partial t}\right\|^2+c\|\nabla u\|^2\|\Delta\varphi\|^2 \end{split} \end{equation}$

(3.30)

and

$\begin{equation}\label{e40} \begin{split} \left|\int_{\Omega}g(u)\nabla\frac{\partial u}{\partial t}\nabla\varphi\, dx\right|&\leq c\int_{\Omega}(|u|+1)\left|\nabla\frac{\partial u}{\partial t}\right||\nabla\varphi|\, dx\\ &\leq \frac{1}{6}\left\|\nabla\frac{\partial u}{\partial t}\right\|^2+c(\|\nabla u\|^2+1)\|\nabla\varphi\|^2. \end{split} \end{equation}$

(3.31)

Furthemore,

$\begin{equation}\label{e41} \begin{split} \int_{\Omega}\left|g(u)\frac{\partial u}{\partial t}\right|^2\, dx &\leq c\int_{\Omega}(|u|+1)^2\left|\frac{\partial u}{\partial t}\right|^2\, dx\\ &\leq c(\|\nabla u\|^2+\|u\|^2+1)\left\|\frac{\partial u}{\partial t}\right\|^2. \end{split} \end{equation}$

(3.32)

Now, collecting (3.27)–(3.32) and owing to (3.20), we are led to

$\begin{equation}\label{e42} \frac{d}{dt}\left\|\frac{\partial u}{\partial t}\right\|^2+c\left\|\nabla\frac{\partial u}{\partial t}\right\|^2\leq c'(\|u\|^2_{H^1(\Omega)}+1)\bigg(\left\|\frac{\partial u}{\partial t}\right\|^2+\|\varphi\|^2_{H^2(\Omega)}\bigg). \end{equation}$

(3.33)

Adding (3.19), $\varepsilon_1$(3.23) and $\varepsilon_2$(3.33), with $\varepsilon_i>0, \, i = 1, 2$, small enough, we obtain

$\begin{equation}\label{e43} \frac{d\, E_3}{dt}+c\bigg(\left\|\frac{\partial u}{\partial t}\right\|^2_{H^1(\Omega)}+\left\|\frac{\partial\varphi}{\partial t}\right\|^2_{H^2(\Omega)}\bigg)\leq c'E_3+c'', \end{equation}$

(3.34)

where

$\begin{equation}\label{e44} E_3 = E_2+\varepsilon_1\|\nabla\varphi\|^2+\varepsilon_2\left\|\frac{\partial u}{\partial t}\right\|^2 \end{equation}$

(3.35)

enjoys

$\begin{equation}\label{e45} E_3\geq c(\|u\|^2_{H^(\Omega)}+\|u\|^{k+2}_{k+2}+\|\varphi\|^2_{H^2(\Omega)})-c' \end{equation}$

(3.36)

and

$\begin{equation}\label{e46} E_3\leq c''(\|u\|^2_{H^(\Omega)}+\|u\|^{k+2}_{k+2}+\|\varphi\|^2_{H^2(\Omega)})-c'''. \end{equation}$

(3.37)

We complete the proof applying Gronwall's lemma.

We now give a uniqueness result

Theorem 3.3. Under assumptions of Theorem 3.2 and assuming that (3.5) holds true. The problem (1.9)-(1.12) has a unique solution $(u, \varphi)$, with the above regularity.

Proof. We suppose the existence of two solutions $(u_1, \varphi_1)$ and $(u_2, \varphi_2)$ to problem (1.9)-(1.11) associated to initial conditions $(u_{01}, \varphi_{01})$ and $(u_{02}, \varphi_{02})$, respectively. We then have

$\frac{\partial u}{\partial t}-\Delta u+f(u_1)-f(u_2) = g(u_1)\bigg(\varphi-\Delta\varphi\bigg)+\big(g(u_1)-g(u_2)\big)\bigg(\varphi_2-\Delta\varphi_2\bigg),$

(3.38)

$\frac{\partial \varphi}{\partial t}-\Delta\frac{\partial\varphi}{\partial t}-\Delta\varphi = -g(u_1)\frac{\partial u}{\partial t}-\big(g(u_1)-g(u_2)\big)\frac{\partial u_2}{\partial t},$

(3.39)

$u|_{\partial\Omega} = \varphi|_{\partial\Omega} = 0,$

(3.40)

$u|_{t = 0} = u_{01}-u_{02}, \, \varphi|_{t = 0} = \varphi_{01}-\varphi_{02},$

(3.41)

with $u = u_1-u_2$, $\varphi = \varphi_1-\varphi_2$, $u_0 = u_{01}-u_{02}$ and $\varphi_0 = \varphi_{01}-\varphi_{02}$.

Multiplying (3.38) by $\frac{\partial u}{\partial t}$ and integrating over $\Omega$, we have

$\begin{equation}\label{e51} \begin{split} &\frac{1}{2}\frac{d}{dt}\|\nabla u\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2+\int_{\Omega}\big(f(u_1-f(u_2))\big)\frac{\partial u}{\partial t}\, dx\\ & = \int_{\Omega}g(u_1)\big(\varphi-\Delta\varphi\big)\frac{\partial u}{\partial t}\, dx+\int_{\Omega}\big(g(u_1)-g(u_2)\big)\big(\varphi_2-\Delta\varphi_2\big)\frac{\partial u}{\partial t}\, dx. \end{split} \end{equation}$

(3.42)

Multiplying (3.39) by $\varphi$ and integrating over $\Omega$, one has

$\begin{equation}\label{e52} \begin{split} \frac{1}{2}\frac{d}{dt}(\|\varphi\|^2+\|\nabla\varphi\|^2)+\|\nabla\varphi\|^2 = &-\int_{\Omega}g(u_1)\frac{\partial u}{\partial t}\varphi\, dx\\ &-\int_{\Omega}\big(g(u_1)-g(u_2)\big)\frac{\partial u_2}{\partial t}\varphi\, dx. \end{split} \end{equation}$

(3.43)

Multiplying (3.39) by $-\Delta\varphi$ and integrating over $\Omega$, we obtain

$\begin{equation}\label{e53} \begin{split} \frac{1}{2}\frac{d}{dt}(\|\nabla\varphi\|^2+\|\Delta\varphi\|^2)+\|\Delta\varphi\|^2 = &\int_{\Omega}g(u_1)\frac{\partial u}{\partial t}\Delta\varphi\, dx\\ &+\int_{\Omega}\big(g(u_1)-g(u_2)\big)\frac{\partial u_2}{\partial t}\Delta\varphi\, dx. \end{split} \end{equation}$

(3.44)

Finally, adding (3.42), (3.43) and (3.44), we get

$\begin{equation}\label{e54} \begin{split} \frac{dE_4}{dt}+\left\|\frac{\partial u}{\partial t}\right\|^2+\|\nabla\varphi\|^2&+\|\Delta\varphi\|^2+\int_{\Omega}\big(f(u_1)-f(u_2)\big)\frac{\partial u}{\partial t}\, dx\\ & = \int_{\Omega}\big(g(u_1)-g(u_2)\big)\big(\varphi_2-\Delta\varphi_2\big)\frac{\partial u}{\partial t}\, dx\\ &-\int_{\Omega}\big(g(u_1)-g(u_2)\big)\big(\varphi-\Delta\varphi\big)\frac{\partial u_2}{\partial t}\, dx, \end{split} \end{equation}$

(3.45)

where

$\begin{equation}\label{e55} E_4 = \|\nabla u\|^2+\|\varphi\|^2+2\|\nabla\varphi\|^2+\|\Delta\varphi\|^2. \end{equation}$

(3.46)

Now, owing to (3.5), and applying Hölder's inequality for $k = 2$, when $n = 3$, we can write

$\begin{equation}\label{e56} \begin{split} \int_{\Omega}\big(f(u_1)-f(u_2)\big)\frac{\partial u}{\partial t}\, dx&\leq c\int_{\Omega}(|u_2|^k+1)|u|\left|\frac{\partial u}{\partial t}\right|\, dx\\ &\leq c(\|\nabla u_2\|^{2k}+1)\|\nabla u\|^2+\left\|\frac{\partial u}{\partial t}\right\|^2, \end{split} \end{equation}$

(3.47)

we also get, thanks to (3.4), and applying Hölder's inequality,

$\begin{equation}\label{e57} \begin{split} \int_{\Omega}\big(g(u_1)-g(u_2)\big)\big(\varphi_2-\Delta\varphi_2\big)\frac{\partial u}{\partial t}\, dx&\leq c\int_{\Omega}|u||\varphi_2-\Delta\varphi_2|\left|\frac{\partial u}{\partial t}\right|\, dx\\ &\leq c\|\nabla u\|^2\big(\|\varphi_2\|^2+\|\Delta\varphi_2\|^2\big)+\left\|\frac{\partial u}{\partial t}\right\|^2 \end{split} \end{equation}$

(3.48)

and

$\begin{equation}\label{e58} \begin{split} \int_{\Omega}\big(g(u_1)-g(u_2)\big)\big(\varphi-\Delta\varphi\big)\frac{\partial u_2}{\partial t}\, dx&\leq c\int_{\Omega}|u|\left|\frac{\partial u}{\partial t}\right||\varphi-\Delta\varphi|\, dx\\ &\leq c\left\|\frac{\partial u_2}{\partial t}\right\|^2\big(\|\varphi\|^2+\|\Delta\varphi\|^2\big)+\|\nabla u\|^2. \end{split} \end{equation}$

(3.49)

From (3.45)-(3.49), we deduce a differential inequality of the type

$\begin{equation} \frac{dE_4}{dt}+c\left\|\frac{\partial u}{\partial t}\right\|^2\leq c(\|\nabla u_2\|^{2k}+\left\|\frac{\partial u_2}{\partial t}\right\|^2+\|\varphi_2\|^2+\|\Delta\varphi_2\|^2+1)E_4. \end{equation}$

(3.50)

In particular,

$\begin{equation}\label{e59} \frac{dE_4}{dt}\leq cE_4 \end{equation}$

(3.51)

and then applying the Gronwall's lemma to (3.51), we end the proof.

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4. Dissipativity properties of the system

This section is devoted to the existence of bounded absorbing sets for the semigroup $S(t), \, t\geq 0$. To this end, we consider a more restrictive assumption on $G$, namely,

$\begin{equation}\label{4.1} \forall\, \epsilon>0, \, |G(u)|^2\leq \epsilon F(s)+c_{\epsilon}, \, s\in \mathbb{R}. \end{equation}$

(4.1)

We then have

Theorem 4.1. Under the assumptions of the Theorem 3.3 and assuming that (4.1) holds true. Then, $u\in L^{\infty}(\mathbb{R}^+; H^1_0(\Omega))\cap L^{k+2}(\Omega)$, $\varphi\in L^{\infty}(\mathbb{R}^+; H^1_0(\Omega)\cap H^2(\Omega))$.

Proof. Going from (3.8) and (3.10), we get, summing (3.8) and $\delta$(3.10), with $\delta>0$, as small as we need,

$\begin{equation}\label{4.3} \begin{split} &\frac{dE_5}{dt}+2\bigg(c\|\nabla u\|^2+\delta\int_{\Omega}F(u)\, dx+\left\|\frac{\partial u}{\partial t}\right\|^2+\|\nabla\varphi\|^2+\|\Delta\varphi\|^2\bigg)\\ &\leq 2c'\delta\int_{\Omega}|G(u)|^2\, dx+\delta\big(\|\varphi\|^2+\|\Delta\varphi\|^2\big)+c''\\ &\leq 2c'\delta\int_{\Omega}|G(u)|^2\, dx+\delta\big(c\|\nabla\varphi\|^2+\|\Delta\varphi\|^2\big)+c'', \end{split} \end{equation}$

(4.2)

where

$\begin{equation}\label{4.4} E_5 = \delta\|u\|^2+\|\nabla u\|^2+2\int_{\Omega}F(u)\, dx+\|\varphi\|^2+2\|\nabla\varphi\|^2+\|\Delta\varphi\|^2 \end{equation}$

(4.3)

satisfies

$\begin{equation}\label{4.5} E_5\geq c\bigg(\|u\|^2_{H^1(\Omega)}+\|u\|^{k+2}_{k+2}+\|\varphi\|^2_{H^2(\Omega)}\bigg)-c' \end{equation}$

(4.4)

and

$\begin{equation}\label{4.6} E_5\leq c''\bigg(\|u\|^2_{H^1(\Omega)}+\|u\|^{k+2}_{k+2}+\|\varphi\|^2_{H^2(\Omega)}\bigg)-c'''. \end{equation}$

(4.5)

From (4.2) and owing to (4.1), we obtain

$\begin{equation}\label{4.7} \begin{split} &\frac{dE_5}{dt}+2\bigg(c\|\nabla u\|^2+\delta\int_{\Omega}F(u)\, dx+\left\|\frac{\partial u}{\partial t}\right\|^2+\|\nabla\varphi\|^2+\|\Delta\varphi\|^2\bigg)\\ &\leq C_{\epsilon}\int_{\Omega}F(u)\, dx+\delta\big(c\|\nabla\varphi\|^2+\|\Delta\varphi\|^2\big)+C'_{\epsilon}, \end{split} \end{equation}$

(4.6)

where $C_{\epsilon}$ and $C'_{\epsilon}$ are positive constants which depend on $\epsilon$. Now, choosing $\epsilon$ and $\delta$ such that:

$\begin{equation} 2\delta\geq C_{\epsilon} \mbox{and} 2>c\delta, \end{equation}$

(4.7)

we then deduce from (4.6),

$\begin{equation}\label{4.8} \frac{dE_5}{dt}+c\bigg(E_5+\left\|\frac{\partial u}{\partial t}\right\|^2\bigg)\leq c', \end{equation}$

(4.8)

we complete the proof applying the Gronwall's lemma.

Remark 4.2. It follows from theorems 3.1, 3.2 and 4.1 that we can define the family solving operators:

$\begin{equation}\label{equ51} \begin{split} S(t): &\Phi\longrightarrow \Phi, \\ &(u_0, \varphi_0)\mapsto (u(t), \varphi(t)), \, \forall\, t\geq 0, \end{split} \end{equation}$

(4.9)

where $\Phi = H^1_0(\Omega)\times H^1_0(\Omega)\cap H^2(\Omega)$, and $(u, \varphi)$ is the unique solution to the problem (1.9)-(1.12). Moreover, this family of solving operators forms a continuous semigroup i.e., $S(0) = Id$ and $S(t+\tau) = S(t)\circ S(\tau), \, \forall \, t, \, \tau\geq 0$. And then, it follows from (4.8) that $S(t)$ is dissipative in $\Phi$, it means that it possesses a bounded absorbing set $\mathbb{B}_0\subset \Phi$ i.e., $\forall\, B\subset \Phi (bounded), \exists\, t_0 = t_0(B) ~such that~ t\geq t_0 ~implies~ S(t)B\subset \mathbb{B}_0.$ (see, e.g., ^[32], ^[34] for details).

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5. Spatial behavior of solutions

The aim of this section is to study the spatial behavior of solutions in a semi-infinite cylinder, assuming that such solutions exist. This study is motivated by the possibility of extending results obtained above to the case of unbounded domains like semi-infinite cylinders. To do so, we will study the behavior of solutions in a semi-infinite cylinder denoted $R = (0, +\infty)\times D$, where $D$ is a smooth bounded domain of $\mathbb{R}^{n-1}$, $n$ being the space dimension. We then consider the problem defined by the system (1.9)-(1.10) in the semi-infinite $R$, with $n = 3$. Furthermore, we endow to this system following boundary conditions:

$\begin{equation}\label{1} u = \varphi = 0 \mbox{on} (0, \, +\infty)\times \partial D\times (0, T) \end{equation}$

(5.1)

and

$\begin{equation}\label{2} u(0, x_2, x_3;t) = h(x_2, x_3;t), \, \varphi(0, x_2, x_3;t) = l(x_2, x_3;t) \mbox{on} \{0\}\times D\times (0, T), \end{equation}$

(5.2)

where $T>0$ is a given final time.

We also consider following initial data

$\begin{equation}\label{3} u|_{t = 0} = \varphi|_{t = 0} = 0 \mbox{on} R. \end{equation}$

(5.3)

Let us suppose that such solutions exist. We consider the function

$\begin{equation}\label{4} F_{w}(z, t) = \int^t_0\int_{D(z)}e^{-ws}\bigg(u_su_{, 1}+\varphi(\varphi_{, 1}+\varphi_{, 1s})+\varphi_s\varphi_{, 1}\bigg)\, dads, \end{equation}$

(5.4)

where $D(z) = \{x\in R: x_1 = z\}$, $u_{, 1} = \frac{\partial u}{\partial x_1}$, $u_s = \frac{\partial u}{\partial s}$ and $w$ is a positive constant. Using the divergence theorem and owing to (5.1), we have

$\begin{equation}\label{5} \begin{split} &F_{w}(z+h, t)-F_w(z, t) = \frac{e^{-wt}}{2}\int_{R(z, z+h)}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dx\\ &+\int^t_0\int_{R(z, z+h)}e^{-ws}\bigg(|u_s|^2+|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dxds\\ &+\frac{w}{2}\int^t_0\int_{R(z, z+h)}e^{-ws}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dxds, \end{split} \end{equation}$

(5.5)

where $R(z, z+h) = \{x\in R: z<x_1<z+h\}$.

Hence,

$\begin{equation}\label{6} \begin{split} &\frac{\partial F_w}{\partial t}(z, t) = \frac{e^{-wt}}{2}\int_{D(z)}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, da\\ &+\int^t_0\int_{D(z)}e^{-ws}\bigg(|u_s|^2+|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dads\\ &+\frac{w}{2}\int^t_0\int_{D(z)}e^{-ws}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dads. \end{split} \end{equation}$

(5.6)

We consider a second function, namely,

$\begin{equation}\label{7} G_w(z, t) = \int^t_0\int_{D(z)}e^{-ws}\bigg(u_su_{, 1}+\varphi(\theta_{, 1}+\varphi_{, 1s})\bigg)\, dads, \end{equation}$

(5.7)

where $\theta = \int^t_0\varphi(s)\, ds$.

Similarly, we have

$\begin{equation}\label{8} \begin{split} &G_w(z+h, t)-G_w(z, t) = \frac{e^{-wt}}{2}\int_{R(z, z+h)}(|u|^2+|\nabla\theta|^2)\, dx\\ &+\int^t_0\int_{R(z, z+h)}e^{-ws}\bigg(|\nabla u|^2+f(u)u+u\Delta\varphi+|\varphi|^2+|\nabla\varphi|^2\bigg)\, dxds\\ &+\frac{w}{2}\int^t_0\int_{R(z, z+h)}e^{-ws}(|u|^2+|\nabla\theta|^2)\, dxds\\ &+\int^t_0\int_{R(z, z+h)}e^{-ws}\big(G(u)-g(u)u\big)\varphi\, dxds \end{split} \end{equation}$

(5.8)

and then

$\begin{equation}\label{9} \begin{split} &\frac{\partial G_w}{\partial t}(z, t) = \frac{e^{-wt}}{2}\int_{D(z)}(|u|^2+|\nabla\theta|^2)\, da\\ &+\int^t_0\int_{D(z)}e^{-ws}\bigg(|\nabla u|^2+f(u)u+u\Delta\varphi+|\varphi|^2+|\nabla\varphi|^2\bigg)\, dads\\ &+\frac{w}{2}\int^t_0\int_{D(z)}e^{-ws}(|u|^2+|\nabla\theta|^2)\, dads\\ &+\int^t_0\int_{D(z)}e^{-ws}\big(G(u)-g(u)u\big)\varphi\, dads. \end{split} \end{equation}$

(5.9)

We choose $\tau$ large enough such as

$\begin{equation}\label{10} 2F(u)+\tau u^2\geq C_1u^2, C_1>0. \end{equation}$

(5.10)

Now, we focus on the nonliear part i.e.,

$\begin{equation}\label{11} w\big(F(u)+\frac{\tau}{2}|u|^2\big)+\tau f(u)u+\tau\big(G(u)-g(u)u\big)\varphi+\frac{w}{2}|\varphi|^2. \end{equation}$

(5.11)

We assume that $G(s)-g(s)s\leq c(|s|^{k+2}+s^2)$.

For $\tau$ large enough, we have $F(u)+\frac{\tau}{2}|u|^2\geq C_2(|u|^{k+2}+|u|^2), \, C_2>0$. Thus, for $w\gg \tau$, we deduce that

$\begin{equation}\label{12} \begin{split} w\big(F(u)+\frac{\tau}{2}|u|^2\big)+\tau f(u)u&+\tau\big(G(u)-g(u)u\big)\varphi\\ &+\frac{w}{2}|\varphi|^2\geq C_3\big(|u|^2+|\varphi|^2+|\Delta\varphi|^2\big). \end{split} \end{equation}$

(5.12)

Taking into account previous choices, it clearly appears that the following function

$\begin{equation}\label{13} H_w = F_w+\tau G_w \end{equation}$

(5.13)

satisfies

$\begin{equation}\label{14} \begin{split} \frac{\partial H_w}{\partial t}(z, t)\geq C_4\int^t_0\int_{D(z)}e^{-ws}\bigg(|u|^2&+|\nabla u|^2+|u_s|^2\\ &+|\varphi|^2+|\nabla\varphi|^2+|\Delta\varphi|^2+|\nabla\theta|^2\bigg)\, dads. \end{split} \end{equation}$

(5.14)

We give now an estimate of $|H_w|$ in terms of $\frac{\partial H_w}{\partial t}$. Applying Cauchy-Schwarz's inequality, one has

$\begin{equation}\label{15} \begin{split} &|F_w|\leq \bigg(\int^t_0\int_{D(z)}e^{-ws}u^2_s\, dads\bigg)^{1/2}\bigg(e^{-ws}u^2_{, 1}\bigg)^{1/2}\\ &+\bigg(\int^t_0\int_{D(z)}e^{-ws}\varphi^2\, dads\bigg)^{1/2}\bigg(e^{-ws}\varphi^2_{, 1}\bigg)^{1/2}\\ &+\bigg(\int^t_0\int_{D(z)}e^{-ws}\varphi^2\, dads\bigg)^{1/2}\bigg(e^{-ws}\varphi^2_{, 1s}\bigg)^{1/2}\\ &+\bigg(\int^t_0\int_{D(z)}e^{-ws}\varphi^2_s\, dads\bigg)^{1/2}\bigg(e^{-ws}\varphi^2_{, 1}\bigg)^{1/2}\\ &\leq C_5\int^t_0\int_{D(z)}e^{-ws}\bigg(|\nabla u|^2+|u_s|^2+|\varphi|^2+|\nabla\varphi|^2\\ &+|\varphi_s|^2+|\nabla\varphi_s|^2\bigg)\, dads, \, C_5>0. \end{split} \end{equation}$

(5.15)

Similarly,

$\begin{equation}\label{16} \begin{split} &|G_w|\leq \bigg(\int^t_0\int_{D(z)}e^{-ws}u^2\, dads\bigg)^{1/2}\bigg(\int^t_0\int_{D(z)}e^{-ws}u^2_{, 1}\, dads\bigg)^{1/2}\\ &+\bigg(\int^t_0\int_{D(z)}e^{-ws}\varphi^2\, dads\bigg)^{1/2}\bigg(\int^t_0\int_{D(z)}e^{-ws}\theta^2_{, 1}\, dads\bigg)^{1/2}\\ &+\bigg(\int^t_0\int_{D(z)}e^{-ws}\varphi^2_s\, dads\bigg)^{1/2}\bigg(\int^t_0\int_{D(z)}e^{-ws}\varphi^2_{, 1}\, dads\bigg)^{1/2}\\ &\leq C_6\int^t_0\int_{D(z)}e^{-ws}\bigg(|u|^2+|\nabla u|^2+|\varphi|^2\\ &+|\nabla\varphi|^2+|\nabla\theta|^2\bigg)\, dads, \, C_6>0. \end{split} \end{equation}$

(5.16)

We then deduce the existence of a positive constant $C_7 = \frac{C_5+\tau C_6}{C_4}$ such that

$\begin{equation}\label{17} |H_w|\leq C_7\frac{\partial H_w}{\partial z}. \end{equation}$

(5.17)

Remark 5.1. The inequality (5.17) is well known in the study of spatial estimates and leads to the Phragmén-Lindelöf alternative (see, e.g., ^[9], ^[39]).

In particular, if there exist $z_0\geq 0$ such that $F_w(z_0, t)>0$, then the solution satisfies

$\begin{equation}\label{18} H_w(z, t)\geq H_w(z_0, t)e^{C^{-1}_7(z-z_0)}, \, z\geq z_0. \end{equation}$

(5.18)

The estimate (5.18) gives information in terms of measure defined in the cylinder. Actually, from (5.18), we deduce that

$\begin{equation}\label{19} \begin{split} &\frac{e^{-wt}}{2}\int_{R(0, z)}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dx\\ &+\tau\frac{e^{-wt}}{2}\int_{R(0, z)}\big(|u|^2+|\nabla\theta|^2\big)\, dx\\ &+\int^t_0\int_{R(0, z)}e^{-ws}\bigg(|u_s|^2+|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dxds\\ &+\tau\int^t_0\int_{R(0, z)}e^{-ws}\bigg(|\nabla u|^2+f(u)u+g(u)u\Delta\varphi+|\varphi|^2+2|\nabla\varphi|^2\bigg)\, dxds\\ &+\frac{w}{2}\int^t_0\int_{R(0, z)}e^{-ws}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dxds\\ &+\tau\frac{w}{2}\int^t_0\int_{R(0, z)}e^{-ws}\big(|u|^2+|\nabla\theta|^2\big)\, dx\\ &+\tau\int^t_0\int_{R(0, z)}e^{-ws}\big(G(u)-g(u)u\big)\varphi\, dxds \end{split} \end{equation}$

(5.19)

tends to infinity exponentially fast. On the other hand, if $H_w(z, t)\leq 0$, for every $z\geq 0$, we deduce that the solution decreases and we get an inequality of the type

$\begin{equation}\label{20} -H_w(z, t)\leq -H_w(0, t)e^{C^{-1}_7z}, \, z\geq 0, \end{equation}$

(5.20)

where

$\begin{equation}\label{21} \begin{split} &E_w(z, t) = \frac{e^{-wt}}{2}\int_{R(z)}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dx\\ &+\tau\frac{e^{-wt}}{2}\int_{R(z)}\big(|u|^2+|\nabla\theta|^2\big)\, dx\\ &+\int^t_0\int_{R(z)}e^{-ws}\bigg(|u_s|^2+|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dxds\\ &+\tau\int^t_0\int_{R(z)}e^{-ws}\bigg(|\nabla u|^2+f(u)u+g(u)u\Delta\varphi+|\varphi|^2+2|\nabla\varphi|^2\bigg)\, dxds\\ &+\frac{w}{2}\int^t_0\int_{R(z)}e^{-ws}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dxds\\ &+\tau\frac{w}{2}\int^t_0\int_{R(z)}e^{-ws}\big(|u|^2+|\nabla\theta|^2\big)\, dx\\ &+\tau\int^t_0\int_{R(z)}e^{-ws}\big(G(u)-g(u)u\big)\varphi\, dxds \end{split} \end{equation}$

(5.21)

and $R(z) = \{x\in R: x_1>z\}$.

Finally, setting

$\begin{equation}\label{22} \begin{split} &\mathcal{E}_w(z, t) = \frac{1}{2}\int_{R(z)}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dx\\ &+\tau\frac{1}{2}\int_{R(z)}\big(|u|^2+|\nabla\theta|^2\big)\, dx\\ &+\int^t_0\int_{R(z)}\bigg(|u_s|^2+|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dxds\\ &+\tau\int^t_0\int_{R(z)}\bigg(|\nabla u|^2+f(u)u+g(u)u\Delta\varphi+|\varphi|^2+2|\nabla\varphi|^2\bigg)\, dxds\\ &+\frac{w}{2}\int^t_0\int_{R(z)}\bigg(|\nabla u|^2+2F(u)+|\varphi|^2+2|\nabla\varphi|^2+|\Delta\varphi|^2\bigg)\, dxds\\ &+\tau\frac{w}{2}\int^t_0\int_{R(z)}\big(|u|^2+|\nabla\theta|^2\big)\, dx\\ &+\tau\int^t_0\int_{R(z)}\big(G(u)-g(u)u\big)\varphi\, dxds. \end{split} \end{equation}$

(5.22)

We have the following result

Theorem 5.2. Let $(u, \varphi)$ be a solution to the problem given by (1.9)-(1.10), boundary conditions (5.1)-(5.2) and initial data (5.3). Then, either this solution satisfies (5.18), or it satisfies

$\begin{equation}\label{23} \mathcal{E}_w(z, t)\leq E_w(0, t)e^{wt-C^{-1}_7z}, \, z\geq 0, \end{equation}$

(5.23)

where the energy $\mathcal{E}_w$ is given by (5.22).

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Acknowledgments

The author would like to thank Alain Miranville for his advices and for his careful reading of this paper.

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Conflict of interest

The author declares no conflicts of interest in this paper.

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