1. Introduction

G. Caginalp proposed in ^[3] and ^[4] two phase-field system, namely,

$\dfrac{\partial u}{\partial t}-\Delta u+f(u)=T, $

(1.1)

$\dfrac{\partial T}{\partial t}-\Delta T=-\dfrac{\partial u}{\partial t}, $

(1.2)

called nonconserved system, and

$\dfrac{\partial u}{\partial t}+\Delta^{2} u-\Delta f(u)=-\Delta T, $

(1.3)

$\dfrac{\partial T}{\partial t}-\Delta T=-\dfrac{\partial u}{\partial t}, $

(1.4)

called concerved system (in the sense that, when endowed with Neumann boundary conditions, the spacial average of u is conserved). In this context, u is the order parameter, T is the relative temperature (defined as $ T=\tilde{T}-T_E$, where $\tilde{T}$ is the absolute temperature and $T_E$ is the equilibrium melting temperature) and $f$ is the derivative of a double-well potential $F$ (a typical choice is $F (s)=\frac{1}{4}(s^{2}-1)^{2}$, hence the usual cubic nonlinear term $f (s)=s^{3}-s$). Furthermore, we have set all physical parameters equal to one. These systems have been introduced to model phase transition phenomena, such as melting-solidication phenomena, and have been much studied from a mathematical point of view. We refer the reader to, e.g., ^{[3,4,5,8,9,10,12,13,14,15,16,18,19,21,22,23,25]}.

Both systems are based on the (total Ginzburg-Landau) free energy

$\Psi_{GL}=\int_{\Omega}(\dfrac{1}{2}\vert\nabla u\vert^{2}+F(u)-uT-\frac{1}{2}T^{2})\mathrm{d}x, $

(1.5)

where $\Omega$ is the domain occupied by the system (we assume here that it is a bounded and regular domain of $\mathbb{R}^{3}$, with boundary $\Gamma$), and the enthalpy

$H=u+T.$

(1.6)

As far as the evolution equations for the order parameter are concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one)

$\dfrac{\partial u}{\partial u}=-\dfrac{D\Psi_{GL}}{Du}, $

(1.7)

for the nonconserved model, and

$\dfrac{\partial u}{\partial u}=\Delta\dfrac{D\Psi_{GL}}{Du}, $

(1.8)

for the conserved one, where $\dfrac{D}{Du}$ denotes a variational derivative with respect to u, which yields (1.1) and (1.3), respectively. Then, we have the energy equation

$\dfrac{\partial H}{\partial t}=-\mbox {divq}, $

(1.9)

where q is the heat flux. Assuming finally the usual Fourier law for heat conduction,

$q=-\nabla T, $

(1.10)

we obtain (1.2).

In (1.5), the term $\vert\nabla u\vert^{2}$ models short-ranged interactions. It is however interesting to note that such a term is obtained by truncation of higher-order ones; it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions ^[11].

G. Caginalp and Esenturk recently proposed in ^[6] (see also ^[20]) higher-order phase-field models in order to account for anisotropic interfaces (see also ^[7] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these autors proposed the following modified (total) free energy

$\Psi_{HOGL}=\int_{\Omega}(\dfrac{1}{2}\sum_{i=1}^{k}\sum_{\vert\beta\vert=i}a_{\beta}\vert\mathcal{D}^{\beta} u\vert^{2}+F(u)-uT-\frac{1}{2}T^{2})\mathrm{d}x, \quad k\in\mathbb{N}, $

(1.11)

where, for $\beta=(k_{1}, k_{2}, k_{3})\in (\mathbb{N}\cup\lbrace 0\rbrace)^{3}$,

$ \vert\beta\vert=k_{1}+k_{2}+k_{3} $

and, for $\beta\neq (0, 0, 0)$,

$ \mathcal{D}^{\beta}=\dfrac{\partial^{\vert\beta\vert}}{\partial x_{1}^{k_{1}}\partial x_{2}^{k_{2}}\partial x_{3}^{k_{3}}} $

(we agree that $\mathcal{D}^{(0, 0, 0)}v=v)$.

A. Miranville studied in ^[17] the corresponding nonconserved higher-order phase-field system.

As far as the conserved case is concerned, the above generalized free energy yields, procceding as above, the following evolution equation for the order parameter u:

$\dfrac{\partial u}{\partial t}-\Delta\sum_{i=1}^{k}(-1)^{i}\sum_{\vert\beta\vert=i}a_{\beta}\mathcal{D}^{2\beta}u-\Delta f(u)=-\Delta(\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}), $

(1.12)

In particular, for k = 1 (anisotropic conserved Caginalp phase-field), we have an equation of the form

$ \dfrac{\partial u}{\partial t}+\Delta\sum_{i=1}^{3}a_i\dfrac{\partial^{2} u}{\partial x_i^{2}}-\Delta f(u)=-\Delta(\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}) $

and, for k = 2 (fourth-order anisotropic conserved Caginalp phase-field system), we have an equation of the form

$ \dfrac{\partial u}{\partial t}-\Delta\sum_{i, j=1}^{3}a_{ij}\dfrac{\partial^{4} u}{\partial x_i^{2}\partial x_j^{2}}+\Delta\sum_{i=1}^{3}b_i\dfrac{\partial^{2} u}{\partial x_i^{2}}-\Delta f(u)=-\Delta(\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}). $

L. Cherfils A. Miranville and S. Peng have studied in ^[8] the corresponding higher-order isotropic equation (without the coupling with the temperature), namely, the equation

$ \dfrac{\partial u}{\partial t}-\Delta P(-\Delta )u-\Delta f(u)=0, $

where

$ P(s)=\sum_{i=1}^{k}a_is^{i}, \quad a_k>0, \quad k\geqslant 1, $

endowed with the Dirichlet/Navier boundary conditions

$ u=\Delta u=...=\Delta^{k}u=0\quad on\quad \Gamma. $

Our aim in this paper is to study the model consisting of the higher-order anisotropic equation (1.12) and the temperature equation

$\dfrac{\partial^{2} \alpha}{\partial t^{2}}-\Delta\dfrac{\partial^{2} \alpha}{\partial t^{2}}-\Delta\dfrac{\partial \alpha}{\partial t}-\Delta \alpha=-\dfrac{\partial u}{\partial t}.$

(1.13)

In particular, we obtain the existence and uniqueness of solutions.

---

2. Setting of the problem

We consider the following initial and boundary value problem, for $k\in\mathbb{N}$, $k\geqslant 2$ (the case k = 1 can be treated as in the original conserved system; see, e.g., ^[23]):

$\dfrac{\partial u}{\partial t}-\Delta\sum_{i=1}^{k}(-1)^{i}\sum_{\vert\beta\vert=i}a_{\beta}\mathcal{D}^{2\beta}u-\Delta f(u)=-\Delta(\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}), $

(2.1)

$\dfrac{\partial^{2} \alpha}{\partial t^{2}}-\Delta\dfrac{\partial^{2} \alpha}{\partial t^{2}}-\Delta\dfrac{\partial \alpha}{\partial t}-\Delta \alpha=-\dfrac{\partial u}{\partial t}, $

(2.2)

$\mathcal{D^{\beta}}u=\alpha=0\quad on \quad\Gamma, \quad \vert\beta\vert\leqslant k, $

(2.3)

$u|_{t=0}=u_{0}, \quad \alpha|_{t=0}=\alpha_{0}, \quad \dfrac{\partial \alpha}{\partial t}|_{t=0}=\alpha_{1}.$

(2.4)

We assume that

$a_{\beta} > 0, \quad \vert\beta\vert=k, $

(2.5)

and we introduce the elliptic operator $A_{k}$ defined by

$\langle A_{k}v, w\rangle_{H^{-k}(\Omega), H_{0}^{k}(\Omega)}=\sum_{\vert\beta\vert=k}a_{\beta}((\mathcal{D^{\beta}}v, \mathcal{D^{\beta}}w)), $

(2.6)

where $ H^{-k}(\Omega)$ is the topological dual of $ H_{0}^{k}(\Omega)$. Furthermore, ((., .)) denotes the usual $L^{2}$-scalar product, with associated norm $\|.\|$. More generally, we denote by $\|.\|_{X}$ the norm on the Banach space X; we also set $\|.\|_{-1}=\|(-\Delta)^{-\frac{1}{2}}.\|$, where $(-\Delta)^{-1}$ denotes the inverse minus Laplace operator associated with Dirichlet boudary conditions. We can note that

$ (v, w)\in H_{0}^{k}(\Omega)^{2}\mapsto\sum_{\vert\beta\vert=k}a_{\beta}((\mathcal{D^{\beta}}v, \mathcal{D^{\beta}}w)) $

is bilinear, symmetric, continuous and coercive, so that

$ A_{k}:H_{0}^{k}(\Omega)\rightarrow H^{-k}(\Omega) $

is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order 2k (see ^[1] and ^[2]) that $A_{k}$ is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

$ D(A_{k})=H^{2k}(\Omega)\cap H_{0}^{k}(\Omega), $

where, for $v\in D (A_{k})$,

$ A_{k}v=(-1)^{k}\sum_{\vert\beta\vert=k}a_{\beta}\mathcal{D}^{2\beta}v. $

We further note that $D (A_{k}^{\frac{1}{2}})=H_{0}^{k}(\Omega)$ and, for $(v, w)\in D (A_{k}^{\frac{1}{2}})^{2}$,

$ ((A_{k}^{\frac{1}{2}}v, A_{k}^{\frac{1}{2}}w))=\sum_{\vert\beta\vert=k}a_{\beta}((\mathcal{D^{\beta}}v, \mathcal{D^{\beta}}w)). $

We finally note that (see, e.g., ^[24]) $\|A_{k}.\|$ (resp., $\|A_{k}^{\frac{1}{2}}.\|$) is equivalent to the usual $H^{2k}$-norm (resp., $H^{k}$-norm) on $D (A_{k})$ (resp., $D (A_{k}^{\frac{1}{2}})$).

Similarly, we can define the linear operator $\bar{A_{k}}=-\Delta A_{k}$

$ \bar{A}_{k}:H_{0}^{k+1}(\Omega)\rightarrow H^{-k-1}(\Omega) $

which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

$ D(\bar{A}_{k})=H^{2k+2}(\Omega)\cap H_{0}^{k+1}(\Omega), $

where, for $v\in D (\bar{A}_{k})$,

$ \bar{A}_{k}v=(-1)^{k+1}\Delta\sum_{\vert\beta\vert=k}a_{\beta}\mathcal{D}^{2\beta}v. $

Furthermore, $D (\bar{A}_{k}^{\frac{1}{2}})=H_{0}^{k+1}(\Omega)$ and, for $(v, w)\in D (\bar{A}_{k}^{\frac{1}{2}})$,

$ ((\bar{A}_{k}^{\frac{1}{2}}v, \bar{A}_{k}^{\frac{1}{2}}w))=\sum_{\vert\beta\vert=k}a_{\beta}((\nabla\mathcal{D^{\beta}}v, \nabla\mathcal{D^{\beta}}w)). $

Besides $\|\bar{A}_{k}.\|$ (resp., $\|\bar{A}_{k}^{\frac{1}{2}}.\|$) is equivalent to the usual $H^{2k+2}$-norm (resp., $H^{k+1}$-norm) on $D (\bar{A}_{k})$ (resp., $D (\bar{A}_{k}^{\frac{1}{2}})$).

We finally consider the operator $\tilde{A}_{k}=(-\Delta)^{-1}A_{k}$, where

$ \tilde{A}_{k}:H_{0}^{k-1}(\Omega)\rightarrow H^{-k+1}(\Omega); $

note that, as $-\Delta$ and $A_{k}$ commute, then the same holds for $(-\Delta)^{-1}$ and $A_{k}$, so that $\tilde{A}_{k}=A_{k}(-\Delta)^{-1}.$

We have the (see ^[17])

Lemme 2.1. The operator $\tilde{A}_{k}$ is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain

$ D(\tilde{A}_{k})=H^{2k-2}(\Omega)\cap H_{0}^{k-1}(\Omega), $

where, for $v\in D (\tilde{A}_{k})$

$\tilde{A}_{k}v=(-1)^{k}\sum_{\vert\beta\vert=k}a_{\beta}\mathcal{D}^{2\beta}(-\Delta)^{-1}v. $

Furthermore, $D (\tilde{A}_{k}^{\frac{1}{2}})=H_{0}^{k-1}(\Omega)$ and, for $(v, w)\in D (\tilde{A}_{k}^{\frac{1}{2}})$,

$ ((\tilde{A}_{k}^{\frac{1}{2}}v, \tilde{A}_{k}^{\frac{1}{2}}w))=\sum_{\vert\beta\vert=k}a_{\beta}((\mathcal{D^{\beta}}(-\Delta)^{-\frac{1}{2}}v, \mathcal{D^{\beta}}(-\Delta)^{-\frac{1}{2}}w)). $

Besides $\|\tilde{A}_{k}.\|$ (resp., $\|\tilde{A}_{k}^{\frac{1}{2}}.\|$) is equivalent to the usual $H^{2k-2}$-norm (resp., $H^{k-1}$-norm) on $D (\tilde{A}_{k})$ (resp., $D (\tilde{A}_{k}^{\frac{1}{2}})$).

Proof. We first note that $\tilde{A}_{k}$ clearly is linear and unbounded. Then, since $(-\Delta)^{-1}$ and $A_{k}$ commute, it easily follows that $\tilde{A}_{k}$ is selfadjoint.

Next, the domain of $\tilde{A}_{k}$ is defined by

$ D(\tilde{A}_{k})=\lbrace v\in H_{0}^{k-1}(\Omega), \tilde{A}_{k}v\in L^{2}(\Omega)\rbrace. $

Noting that $\tilde{A}_{k}v=f, f\in L^{2}(\Omega), v\in D (\tilde{A}_{k})$, is equivalent to $A_{k}v=-\Delta f$, where $-\Delta f\in H^{2}(\Omega)'$, it follows from the elliptic regularity results of ^[1] and ^[2] that $v\in H^{2k-2}(\Omega)$, so that $D (\tilde{A}_{k})=H^{2k-2}(\Omega)\cap H_{0}^{k-1}(\Omega)$.

Noting then that $\tilde{A}_{k}^{-1}$ maps $L^{2}(\Omega)$ onto $H^{2k-2}(\Omega)$ and recalling that $k\geqslant 2$, we deduce that $\tilde{A}_{k}$ has compact inverse.

We now note that, considering the spectral properties of $-\Delta$ and $A_{k}$ (see, e.g., ^[24]) and recalling that these two operators commute, $-\Delta$ and $A_{k}$ have a spectral basis formed of common eigenvectors. This yields that, $\forall s_{1}, s_{2}\in \mathbb{R}$, $(-\Delta)^{s_{1}}$ and $A_{k}^{s_{2}}$ commute.

Having this, we see that $\tilde{A}_{k}^{\frac{1}{2}}=(-\Delta)^{-\frac{1}{2}}A_{k}^{\frac{1}{2}}$, so that $D (\tilde{A}_{k}^{\frac{1}{2}})=H_{0}^{k-1}(\Omega)$, and for $(v, w)\in D (\tilde{A}_{k}^{\frac{1}{2}})^{2}$,

$ ((\tilde{A}_{k}^{\frac{1}{2}}v, \tilde{A}_{k}^{\frac{1}{2}}w))=\sum_{\vert\beta\vert=k}a_{\beta}((\mathcal{D^{\beta}}(-\Delta)^{-\frac{1}{2}}v, \mathcal{D^{\beta}}(-\Delta)^{-\frac{1}{2}}w)). $

Finally, as far as the equivalences of norms are concerned, we can note that, for instance, the norm $\|\tilde{A}_{k}^{\frac{1}{2}}.\|$ is equivalent to the norm $\|(-\Delta)^{-\frac{1}{2}}.\|_{H^{k}(\Omega)}$ and, thus, to the norm $\|(-\Delta)^{\frac{k-1}{2}}.\|_{.}$

Having this, we rewrite (2.1) as

$\dfrac{\partial u}{\partial t}-\Delta A_{k}u-\Delta B_{k}u-\Delta f(u)=-\Delta(\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}), $

(2.7)

where

$ B_{k}v=\sum_{i=1}^{k-1}(-1)^{i}\sum_{\vert\beta\vert=i}a_{\beta}\mathcal{D}^{2\beta}v. $

As far as the nonlinear term f is concerned, we assume that

$f\in C^{2}(\mathbb{R}), \quad f(0)=0, $

(2.8)

$f'\geqslant -c_{0}, \quad c_{0}\geqslant 0, $

(2.9)

$f(s)s\geqslant c_{1}F(s)-c_{2}\geqslant-c_{3}, \quad c_{1} >0, \quad c_{2}, \quad c_{3}\geqslant 0, \quad s\in \mathbb{R}, $

(2.10)

$F(s)\geqslant c_{4}s^{4}-c_{5}, \quad c_{4} >0, \quad c_{5}\geqslant 0, \quad s\in \mathbb{R}, $

(2.11)

where $F (s)=\int_{0}^{s}f (\tau)\mathrm{d}\tau$. In particular, the usual cubic nonlinear term $f (s)= s^{3}-s$ satisfies these assumptions.

Throughout the paper, the same letters c, c' and c" denote (generally positive) constants which may vary from line to line. Similary, the same letter Q denotes (positive) monotone increasing (with respect to each argument) and continuous functions which may vary from line to line.

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3. A priori estimates

We multiply (2.7) by $(-\Delta)^{-1}\dfrac{\partial u}{\partial t}$ and (2.2) by $\dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t}$, sum the two resulting equalities and integrate over $\Omega$ and by parts. This gives

$ $$\begin{align} & \frac{d}{dt}(\|A_{k}^{\frac{1}{2}}u{{\|}^{2}}+B_{k}^{\frac{1}{2}}\left[u \right]+2\int_{\Omega }{F}(u)\text{d}x+\|\nabla \alpha {{\|}^{2}}+\|\Delta \alpha {{\|}^{2}}+\|\frac{\partial \alpha }{\partial t}-\Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}) \\ & +2\|\frac{\partial u}{\partial t}\|_{-1}^{2}+2\|\nabla \frac{\partial \alpha }{\partial t}{{\|}^{2}}+2\|\Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}=0 \\ \end{align}$$ $

(3.1)

(note indeed that $\|\dfrac{\partial \alpha}{\partial t}\|^{2}+2\|\nabla\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}=\|\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}$), where

$B_{k}^{\frac{1}{2}}\left[u\right]=\sum_{i=1}^{k-1}\sum_{\vert\beta\vert=i}a_{\beta}\|\mathcal{D^{\beta}}u\|^{2}$

(3.2)

(note that $B_{k}^{\frac{1}{2}}\left[u\right]$ is not necessarily nonnegative). We can note that, owing to the interpolation inequality

$B_{k}^{\frac{1}{2}}\left[u\right]=\sum_{i=1}^{k-1}\sum_{\vert\beta\vert=i}a_{\beta}\|\mathcal{D^{\beta}}u\|^{2}$

(3.3)

$\|(-\Delta)^{\frac{i}{2}}v\|\leqslant c(i)\|(-\Delta)^{\frac{m}{2}}v\|^{\frac{i}{m}}\|v\|^{1-\frac{i}{m}}, $

there holds

$ v\in H^{m}(\Omega), \quad i\in\lbrace1, ..., {m-1}\rbrace, \quad m\in\mathbb{N}, \quad m\geqslant 2, $

(3.4)

This yields, employing (2.11),

$\vert B_{k}^{\frac{1}{2}}\left[u\right]\vert\leqslant \dfrac{1}{2}\|A_{k}^{\frac{1}{2}}u\|^{2}+c\|u\|^{2}.$

whence

$ \|A_{k}^{\frac{1}{2}}u\|^{2}+B_{k}^{\frac{1}{2}}\left[u\right]+2\int_{\Omega}F(u)\mathrm{d}x\geqslant\frac{1}{2}\|A_{k}^{\frac{1}{2}}u\|^{2}+\int_{\Omega}F(u)\mathrm{d}x+c\|u\|_{L^{4}(\Omega)}^{4}-c'\|u\|^{2}-c", $

(3.5)

nothing that, owing to Young's inequality,

$\|A_{k}^{\frac{1}{2}}u\|^{2}+B_{k}^{\frac{1}{2}}\left[u\right]+2\int_{\Omega}F(u)\mathrm{d}x\geqslant c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x)-c', \quad c > 0, $

(3.6)

We then multiply (2.7) by $(-\Delta)^{-1}u$ and have, owing to (2.10) and the interpolation inequality (3.3),

$\|u\|^{2}\leqslant\epsilon\|u\|_{L^{4}(\Omega)}^{4}+c(\epsilon), \quad\forall\epsilon > 0. $

hence, proceeding as above and employing, in particular, (2.11)

$ \dfrac{d}{dt}\|u\|_{-1}^{2}+c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x)\leqslant c'(\|u\|^{2}+\|\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\Delta\dfrac{\partial \alpha}{\partial t}\|^{2})+c", $

(3.7)

Summing (3.1) and $\delta_1$ times (3.7), where $\delta_1>0$ is small enough, we obtain a differential inegality of the form

$\dfrac{d}{dt}\|u\|_{-1}^{2}+c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x)\leqslant c'(\|\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\Delta\dfrac{\partial \alpha}{\partial t}\|^{2})+c'', \quad c>0.$

(3.8)

where

$\dfrac{d}{dt}E_1+c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\|\dfrac{\partial \alpha}{\partial t }\|_{H^{2}(\Omega)}^{2})\leqslant c', \quad c>0, $

satisfies, owing to (3.5)

$ E_1=\|A_{k}^{\frac{1}{2}}u\|^{2}+B_{k}^{\frac{1}{2}}\left[u\right]+2\int_{\Omega}F(u)\mathrm{d}x+\|\nabla \alpha\|^{2}+\|\Delta \alpha\|^{2}+\|\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}+\delta_1\|u\|_{-1}^{2} $

(3.9)

Multiplying $(2.2)$ by $-\Delta\alpha$, we then obtain

$E_{1}\geqslant c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\alpha\|_{H^{2}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{2}(\Omega)}^{2})-c', \quad c > 0.$

which yields, employing the interpolation inequality

$ \dfrac{d}{dt}(\|\Delta\alpha\|^{2}-2((\dfrac{\partial\alpha} {\partial t}, \Delta\alpha))+2((\Delta\dfrac{\partial\alpha} {\partial t}, \Delta\alpha)))+\|\Delta\alpha\|^{2}\leqslant \|\dfrac{\partial u}{\partial t}\|^{2}+\|\nabla\dfrac{\partial\alpha}{\partial t}\|^{2}+\|\Delta\dfrac{\partial\alpha}{\partial t}\|^{2}, $

(3.10)

the differential inequality, with $0 < \epsilon < < 1$ is small enough

$\|v\|^{2}\leqslant c\|v\|_{-1}\|v\|_{H^{1}(\Omega)}, \quad v\in H_{0}^{1}(\Omega), $

(3.11)

We now differentiate (2.7) with respect to time to find, owing to (2.2),

$ $$\begin{align} & \frac{d}{dt}(\|\Delta \alpha {{\|}^{2}}-2((\frac{\partial \alpha }{\partial t}, \Delta \alpha ))+2((\Delta \frac{\partial \alpha }{\partial t}, \Delta \alpha )))+c\|\alpha \|_{{{H}^{2}}(\Omega )}^{2} \\ & \le {c}'(\|\frac{\partial u}{\partial t}\|_{-1}^{2}+\epsilon \|\frac{\partial u}{\partial t}\|_{{{H}^{1}}(\Omega )}^{2}+\|\frac{\partial \alpha }{\partial t}\|_{{{H}^{2}}(\Omega )}^{2}), \quad c>0. \\ \end{align}$$ $

(3.12)

together with the boundary condition

$\dfrac{\partial }{\partial t}\dfrac{\partial u}{\partial t}-\Delta A_{k}\dfrac{\partial u}{\partial t}-\Delta B_{k}\dfrac{\partial u}{\partial t}-\Delta (f'(u)\dfrac{\partial u}{\partial t})=-\Delta(\Delta\dfrac{\partial \alpha}{\partial t}+\Delta\alpha-\dfrac{\partial u}{\partial t}), $

(3.13)

We multiply (3.11) by $(-\Delta)^{-1}\dfrac{\partial u}{\partial t}$ and obtain, owing to (2.9) and the interpolation inequality (3.3),

$\mathcal{D^{\beta}}\dfrac{\partial u}{\partial t}=0\quad on\quad \Gamma, \quad \vert\beta\vert\leqslant k.$

hence, owing to (3.10), the differential inequality

$ \dfrac{d}{dt}\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+c\|\dfrac{\partial u}{\partial t}\|_{H^{k}(\Omega)}^{2}\leqslant c'(\|\dfrac{\partial u}{\partial t}\|^{2}+\|\Delta\alpha\|^{2}+\|\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}), \quad c>0, $

(3.14)

Summing finally (3.8), $\delta_2$ times (3.11) and $\delta_3$ times (3.14), where $\delta_2, \delta_3>0$ are small enough, we find a differential inequality of the form

$\dfrac{d}{dt}\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+c\|\dfrac{\partial u}{\partial t}\|_{H^{k}(\Omega)}^{2}\leqslant c'(\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\|\alpha\|_{H^{2}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{2}(\Omega)}^{2}), \quad c>0.$

(3.15)

where

$\dfrac{dE_{2}}{dt}+c(E_2+\|\dfrac{\partial u}{\partial t}\|_{H^{k}(\Omega)}^{2})\leqslant c', \quad c>0, $

Owing to the continuous embedding $H^{2k+1}(\Omega)\subset C (\bar{\Omega})$, we deduce that

$ E_2=E_1+\delta_2(\|\Delta\alpha\|^{2}-2((\dfrac{\partial\alpha} {\partial t}, \Delta\alpha))+2((\Delta\dfrac{\partial\alpha} {\partial t}, \Delta\alpha)))+\delta_3\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}. $

and since

$ \vert\int_{\Omega}F(u_{0})\mathrm{d}x\vert\leqslant Q(\|u_{0}\|_{H^{2k+1}(\Omega)}) $

we see that $(-\Delta)^{-\frac{1}{2}}\dfrac{\partial u}{\partial t}(0)\in L^{2}(\Omega)$ and

$ (-\Delta)^{-\frac{1}{2}}\dfrac{\partial u}{\partial t}(0)=-(-\Delta)^{\frac{1}{2}} A_{k}u_{0}-(-\Delta)^{\frac{1}{2}} B_{k}u_{0}-(-\Delta)^{\frac{1}{2}} f(u_{0})+(-\Delta)^{\frac{1}{2}}(\alpha_{1}-\Delta\alpha_{1}), $

(3.16)

Furthermore $E_2$ satisfies

$\|\dfrac{\partial u}{\partial t}(0)\|_{-1}\leqslant Q(\|u_{0}\|_{H^{2k+1}(\Omega)}, \|\alpha_{1}\|_{H^{3}(\Omega)}).$

(3.17)

It thus follows from (3.15), (3.16), (3.17) and Growall's lemma that

$E_{2}\geqslant c(\|u\|_{H^{k}(\Omega)}^{2}+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\alpha\|_{H^{2}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{2}(\Omega)}^{2})-c', \quad c > 0.$

(3.18)

and

$ $$\begin{align} & \|u(t)\|_{{{H}^{k}}(\Omega )}^{2}+\|\frac{\partial u}{\partial t}(t)\|_{-1}^{2}+\|\alpha (t)\|_{{{H}^{2}}(\Omega )}^{2}+\|\frac{\partial \alpha }{\partial t}(t)\|_{{{H}^{2}}(\Omega )}^{2} \\ & \le {{e}^{-ct}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}', \quad c>0, \quad t0, \\ \end{align}$$ $

(3.19)

$r>0$ given.

Multiplying next (2.7) by $\tilde{A}_{k}u$, we find, owing to the interpolation inequality (3.3),

$ $$\begin{align} & \int_{t}^{t+r}{\|\frac{\partial u}{\partial t}\|_{{{H}^{k}}(\Omega )}^{2}}\text{d}s \\ & \le {{e}^{-ct}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}'(r), \quad c>0, \quad t\ge 0, \\ \end{align}$$ $

hence, since f and F are continuous and owing to (3.18),

$ \dfrac{d}{dt}\|\tilde{A}_{k}^{\frac{1}{2}}u\|^{2}+c\|u\|_{H^{2k}(\Omega)}^{2}\leqslant c'(\|u\|^{2}+\|f(u)\|^{2}+\|\dfrac{\partial \alpha}{\partial t }\|^{2}+\|\Delta \dfrac{\partial \alpha}{\partial t }\|^{2}), \quad c>0, $

(3.20)

Summing (3.15) and (3.22), we have a differential inequality of the form

$ $$\begin{align} & \frac{d}{dt}\|\tilde{A}_{k}^{\frac{1}{2}}u{{\|}^{2}}+c\|u\|_{{{H}^{2k}}(\Omega )}^{2} \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}", \quad c, {c}'>0, \quad t\ge 0. \\ \end{align}$$ $

(3.21)

where

$ $$\begin{align} & \frac{d{{E}_{3}}}{dt}+c({{E}_{3}}+\|u\|_{{{H}^{2k}}(\Omega )}^{2}+\|\frac{\partial u}{\partial t}\|_{{{H}^{k}}(\Omega )}^{2}) \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}", \quad c, {c}'>0, \quad t\ge 0, \\ \end{align}$$ $

satisfies

$ E_{3}=E_{2}+\|\tilde{A}_{k}^{\frac{1}{2}}u\|^{2} $

(3.22)

In particular, it follows from (3.21)-(3.22) that

$E_{3}\geqslant c(\|u\|_{H^{k}(\Omega)}^{2}+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\alpha\|_{H^{2}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{2}(\Omega)}^{2})-c', \quad c > 0.$

(3.23)

$r >0$ given.

We now multiply (2.7) by u and obtain, employing (2.9) and the interpolation inequality (3.3)

$\int_t^{t+r}\|u\|_{H^{2k}(\Omega)}^{2}\mathrm{d}s \leqslant e^{-ct}Q(\|u_{0}\|_{H^{2k+1}(\Omega)}, \|\alpha_0\|_{H^{2}(\Omega)}, \|\alpha_1\|_{H^{3}(\Omega)})+c'(r), \quad c>0, \quad t\geqslant 0, $

whence, proceeding as above,

$ \dfrac{d}{dt}\|u\|^{2}+c\|u\|_{H^{k+1}(\Omega)}^{2}\leqslant c'(\|u\|_{H^{1}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\Delta \dfrac{\partial \alpha}{\partial t}\|^{2}), \quad c > 0, $

(3.24)

We also multiply (2.7) by $\dfrac{\partial u}{\partial t}$ and find

$\dfrac{d}{dt}\|u\|^{2}+c\|u\|_{H^{k+1}(\Omega)}^{2}\leqslant e^{-c't}Q(\|u_{0}\|_{H^{2k+1}(\Omega)}, \|\alpha_0\|_{H^{2}(\Omega)}, \|\alpha_1\|_{H^{3}(\Omega)})+c'', \quad c, c'>0.$

where

$ \dfrac{d}{dt}(\|\bar{A}_{k}^{\frac{1}{2}}u\|^{2}+\bar{B}_{k}^{\frac{1}{2}}\left[u\right])+c\|\dfrac{\partial u}{\partial t}\|^{2}\leqslant c'\|\Delta f(u)\|^{2}-2((\Delta\dfrac{\partial u} {\partial t}, \dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t})), $

Since f is of class $C^{2}$, it follows from the continuous embedding $H^{2}(\Omega)\subset C (\bar{\Omega})$ that

$ \bar{B}_{k}^{\frac{1}{2}}\left[u\right]=\sum_{i=1}^{k-1}\sum_{\vert\beta\vert=i}a_{\beta}\|\nabla\mathcal{D^{\beta}}u\|^{2}. $

hence, owing to (3.18),

$ \|\Delta f(u)\|^{2}\leqslant Q(\|u\|_{H^{2}(\Omega)}), $

(3.25)

Multiply next (2.2) by $-\Delta (\dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t})$, we have

$ $$\begin{align} & \frac{d}{dt}(\|\bar{A}_{k}^{\frac{1}{2}}u{{\|}^{2}}+\bar{B}_{k}^{\frac{1}{2}}\left[u \right])+c\|\frac{\partial u}{\partial t}{{\|}^{2}} \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})-2((\Delta \frac{\partial u}{\partial t}, \frac{\partial \alpha }{\partial t}-\Delta \frac{\partial \alpha }{\partial t}))+{c}'', \quad c, {c}'>0. \\ \end{align}$$ $

(3.26)

(note indeed that $\|\nabla\dfrac{\partial \alpha}{\partial t}\|^{2}+2\|\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\nabla\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}=\|\nabla\dfrac{\partial \alpha}{\partial t}-\nabla\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}$).

Summing (3.25) and (3.26), we obtain

$ $$\begin{align} & \frac{d}{dt}(\|\Delta \alpha {{\|}^{2}}+\|\nabla \Delta \alpha {{\|}^{2}}+\|\nabla \frac{\partial \alpha }{\partial t}-\nabla \Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}})+c(\|\Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}+\|\nabla \Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}) \\ & \le 2((\Delta \frac{\partial u}{\partial t}, \frac{\partial \alpha }{\partial t}-\Delta \frac{\partial \alpha }{\partial t})), \quad c>0 \\ \end{align}$$ $

(3.27)

Summing finally (3.21), (3.24) and (3.27), we find a differential inegality of the form

$ $$\begin{align} & \frac{d}{dt}(\|\bar{A}_{k}^{\frac{1}{2}}u{{\|}^{2}}+\bar{B}_{k}^{\frac{1}{2}}\left[u \right]+\|\Delta \alpha {{\|}^{2}}+\|\nabla \Delta \alpha {{\|}^{2}}+\|\nabla \frac{\partial \alpha }{\partial t}-\nabla \Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}) \\ & +c(\|\frac{\partial u}{\partial t}{{\|}^{2}}+\|\Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}+\|\nabla \Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}) \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}'', \quad c, {c}'>0. \\ \end{align}$$ $

(3.28)

where

$ $$\begin{align} & \frac{d{{E}_{4}}}{dt}+c({{E}_{3}}+\|u\|_{{{H}^{k+1}}(\Omega )}^{2}+\|u\|_{{{H}^{2k}}(\Omega )}^{2}+\|\frac{\partial u}{\partial t}{{\|}^{2}}+\|\frac{\partial u}{\partial t}\|_{{{H}^{k}}(\Omega )}^{2}+\|\frac{\partial \alpha }{\partial t}\|_{{{H}^{3}}(\Omega )}^{2}) \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}'', \quad c, {c}'>0, \quad t\ge 0 \\ \end{align}$$ $

satisfies, owing to (2.11) and the interpolation inegality (3.3)

$ E_4=E_3+\|u\|^{2}+\|\bar{A}_{k}^{\frac{1}{2}}u\|^{2}+\bar{B}_{k}^{\frac{1}{2}}\left[u\right]+\|\Delta \alpha\|^{2}+\|\nabla\Delta \alpha\|^{2}+\|\nabla\dfrac{\partial \alpha}{\partial t }-\nabla\Delta \dfrac{\partial \alpha}{\partial t }\|^{2} $

(3.29)

In particular, it follows from (3.28)-(3.29) that

$E_{4}\geqslant c(\|u\|_{H^{k+1}(\Omega)}^{2}+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\alpha\|_{H^{3}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{3}(\Omega)}^{2})-c', \quad c > 0.$

(3.30)

and

$ $$\begin{align} & \|u(t){{\|}_{{{H}^{k+1}}(\Omega )}}+\|\alpha (t){{\|}_{{{H}^{3}}(\Omega )}}+\|\frac{\partial \alpha }{\partial t}(t){{\|}_{{{H}^{3}}(\Omega )}} \\ & \le {{e}^{-ct}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{3}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}', \quad c>0, \quad t\ge 0, \\ \end{align}$$ $

(3.31)

$r$ given.

We finally rewrite (2.7) as an elliptic equation, for t > 0 fixed,

$ $$\begin{align} & \int_{t}^{t+r}{(\|}\frac{\partial u}{\partial t}{{\|}^{2}}+\|\frac{\partial \alpha }{\partial t}\|_{{{H}^{3}}(\Omega )}^{2})\text{d}s \\ & \le {{e}^{-ct}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{3}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}'(r), \quad c>0, \quad t\ge 0, \\ \end{align}$$ $

(3.32)

Multiplying (3.32) by $A_{k}u$, we obtain, owing to the interpolation inequality (3.3),

$A_{k}u=-(-\Delta)^{-1}\dfrac{\partial u}{\partial t}-B_{k}u-f(u)+\dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t}, \quad\mathcal{D^{\beta}}u=0\quad on\quad \Gamma, \quad \vert\beta\vert\leqslant k-1.$

hence, since f is continuous and owing to (3.18)

$ \|A_{k}u\|^{2}\leqslant c(\|u\|^{2}+\|f(u)\|^{2}+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\Delta \dfrac{\partial \alpha}{\partial t}\|^{2}), $

(3.33)

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4. Existence and uniqueness of solutions

We first have the following theorem.

Theorem 4.1. (i) We assume that $(u_{0}, \alpha_{0}, \alpha_{1})\in H_{0}^{k}(\Omega)\times (H^{2}(\Omega)\cap H_{0}^{1}(\Omega))\times (H^{2}(\Omega)\cap H_{0}^{1}(\Omega))$, with $\int_{\Omega}F (u_{0})\mathrm{d}x < +\infty$. Then, $(2.1)-(2.4)$ possesses at last one solution $(u, \alpha, \dfrac{\partial \alpha}{\partial t})$ such that, $\forall T > 0$, $u (0)=u_{0}$, $\alpha (0)=\alpha_{0}$, $\dfrac{\partial \alpha}{\partial t}(0)=\alpha_{1}$,

$\|u(t)\|_{H^{2k}(\Omega)}^{2}\leqslant ce^{-c't}Q(\|u_{0}\|_{H^{2k+1}(\Omega)}, \|\alpha_{0}\|_{H^{3}(\Omega)}, \|\alpha_{1}\|_{H^{3}(\Omega)})+c'', \quad c'>0\quad t\geqslant 0.$

$ u \in L^{\infty} (\mathbb{R^{+}}; H_{0}^{k}(\Omega))\cap L^{2}(0, T;H^{2k}(\Omega)\cap H_{0}^{k}(\Omega)), $

$ \dfrac{\partial u}{\partial t}\in L^{\infty} (\mathbb{R^{+}}; H^{-1}(\Omega ))\cap L^{2} (0, T; H_{0}^{k}(\Omega)), $

and

$ \alpha, \dfrac{\partial \alpha}{\partial t}\in L^{\infty} (\mathbb{R^{+}}; H^{2}(\Omega)\cap H_{0}^{1}(\Omega)) $

$ \dfrac{d}{dt}((-\Delta)^{-1}u, v))+\sum_{i=1}^{k}\sum_{\vert\beta\vert=i} a_{i}((\mathcal{D^{\beta}}u, \mathcal{D^{\beta}}v))+((f(u), v))=\dfrac{d}{dt}(((u, v)) \\ +((\nabla u, \nabla v))), \forall v\in C_c^{\infty}(\Omega), $

in the sense of distributions.

(ii) If we futher assume that $(u_{0}, \alpha_{0}, \alpha_{1})\in (H^{k+1}(\Omega)\cap H_{0}^{k}(\Omega))\times (H^{3}(\Omega)\cap H_{0}^{1}(\Omega))\times (H^{3}(\Omega)\cap H_{0}^{1}(\Omega))$, then, $\forall T > 0$,

$ \dfrac{d}{dt}(((\dfrac{\partial \alpha}{\partial t}, w))+((\nabla\dfrac{\partial \alpha}{\partial t}, \nabla w))+((\nabla\alpha, \nabla w)))+((\nabla\alpha, \nabla w))=-\dfrac{d}{dt}((u, w)), \forall w\in C_c^{\infty}(\Omega), $

$ u \in L^{\infty} (\mathbb{R^{+}}; H^{k+1}(\Omega)\cap H_{0}^{k}(\Omega))\cap L^{2} (\mathbb{R^{+}}; H^{k+1}(\Omega)\cap H_{0}^{k}(\Omega)) $

$ \dfrac{\partial u}{\partial t}\in L^{2} (\mathbb{R^{+}}; L^{2}(\Omega )), $

and

$ \alpha\in L^{\infty} (\mathbb{R^{+}}; H^{3}(\Omega)\cap H_{0}^{1}(\Omega)) $

The proofs of existence and regularity in (i) and (ii) follow from the a priori estimates derived in the previous section and, e.g., a standard Galerkin scheme.

We then have the following theorem.

Theorem 4.2. The system (1.1)-(1.4) possesses a unique solution with the above regularity.

proof. Let $(u^{(1)}, \alpha^{(1)}, \dfrac{\partial\alpha^{(1)}}{\partial t})$ and $(u^{(2)}, \alpha^{(2)}, \dfrac{\partial\alpha^{(2)}}{\partial t})$ be two solutions to (2.1)-(2.3) with initial data $(u_{0}^{(1)}, \alpha_{0}^{(1)}, \alpha_{1}^{(1)})$ and $(u_{0}^{(2)}, \alpha_{0}^{(2)}, \alpha_{1}^{(2)})$, respectively. We set

$ \dfrac{\partial \alpha}{\partial t}\in L^{\infty} (\mathbb{R^{+}}; H^{3}(\Omega)\cap H_{0}^{1}(\Omega))\cap L^{2} (0, T; H^{3}(\Omega)\cap H_{0}^{1}(\Omega)) $

and

$ (u, \alpha, \dfrac{\partial\alpha}{\partial t})=( u^{(1)}, \alpha^{(1)}, \dfrac{\partial\alpha^{(1)}}{\partial t})-( u^{(2)}, \alpha^{(2)}, \dfrac{\partial\alpha^{(2)}}{\partial t}) $

Then, $(u, \alpha)$ satisfies

$ ( u_{0}, \alpha_{0}, \alpha_{1})=( u_{0}^{(1)}, \alpha_{0}^{(1)}, \alpha_{1}^{(1)})-( u_{0}^{(2)}, \alpha_{0}^{(2)}, \alpha_{1}^{(2)}). $

(4.1)

$\dfrac{\partial u}{\partial t}-\Delta A_{k}u-\Delta B_{k}u-\Delta (f(u^{(1)})-f(u^{(2)}))=-\Delta(\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}), $

(4.2)

$\dfrac{\partial^{2} \alpha}{\partial t^{2}}-\Delta\dfrac{\partial^{2} \alpha}{\partial t^{2}}-\Delta\dfrac{\partial \alpha}{\partial t}-\Delta \alpha=-\dfrac{\partial u}{\partial t}, $

(4.3)

$\mathcal{D^{\beta}} u=\alpha=0\quad\ on \quad\ \Gamma, \quad \vert\beta\vert\leqslant k, $

(4.4)

Multiplying (4.1) by $(-\Delta)^{-1}u$ and integrating over $\Omega$, we obtain

$u|_{t=0}=u_{0}, \alpha|_{t=0}=\alpha_{0}, \dfrac{\partial \alpha}{\partial t}|_{t=0}=\alpha_{1}.$

We note that

$ \dfrac{d}{dt}\|u\|_{-1}^{2}+c\|u\|_{H^{k}(\Omega)}^{2}\leqslant c'(\|u\|^{2}+\|\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}\|^{2})-2((f(u^{(1)})-f(u^{(2)}, u)). $

with l defined as

$ f(u^{(1)})-f(u^{(2)})=l(t)u, $

Owing to (2.9), we have

$ l(t)=\int_0^{1}f'(su^{(1)}(t)+(1-s)u^{(2)}(t))\mathrm{d}s. $

and we obtain owing to the intepolation inequalities (3.3) and (3.10),

$ $$\begin{align} & -2((f({{u}^{(1)}})-f({{u}^{(2)}}, u))\le 2{{c}_{0}}\|u{{\|}^{2}} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le c\|u{{\|}^{2}} \\ \end{align}$$ $

(4.5)

Next, multiplying (4.2) by $(-\Delta)^{-1}(u+\dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t})$, we find

$\dfrac{d}{dt}\|u\|_{-1}^{2}+c\|u\|_{H^{k}(\Omega)}^{2}\leqslant c'(\|u\|_{-1}^{2}+\|\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}), \quad c > 0.$

(4.6)

Summing then $\delta_4$ times (4.5) and (4.6), where $\delta_4>0$ is small enough, we have, employing once more the interpolation inequality (3.10), a differential inequality of the form

$ $$\begin{align} & \frac{d}{dt}(\|\alpha {{\|}^{2}}+\|\nabla \alpha {{\|}^{2}}+\|u+\frac{\partial \alpha }{\partial t}-\Delta \frac{\partial \alpha }{\partial t}\|_{-1}^{2})+c(\|\frac{\partial \alpha }{\partial t}{{\|}^{2}}+\|\frac{\partial \alpha }{\partial t}\|_{{{H}^{1}}(\Omega )}^{2}) \\ & \le {c}'(\|u{{\|}^{2}}+\|\alpha {{\|}^{2}}). \\ \end{align}$$ $

(4.7)

where

$\dfrac{dE_{5}}{dt}\leqslant cE_{5}, $

satisfies

$ E_{5}=\delta_4\|u\|_{-1}^{2}+\|\alpha\|^{2}+\|\nabla \alpha\|^{2}+\|u+\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}\|_{-1}^{2} $

(4.8)

It follows from (4.7)-(4.8) and Gronwall's lemma that

$E_{5}\geqslant c(\|u\|_{-1}^{2}+\|\alpha\|_{H^{1}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t}\| ^{2}), c>0.$

(4.9)

hence the uniquess, as well as the continuous dependence with respect to the initial data in $H^{-1}\times H^{1}\times H^{1}$-norm.

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Conflict of Interest

All authors declare no conflicts of interest in this paper.

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