The methodology named LIFE (Linear-in-Flux-Expressions) was developed with the purpose of simulating and analyzing large metabolic systems. With LIFE, the number of model parameters is reduced by accounting for correlations among the parameters of the system. Perturbation analysis on LIFE systems results in less overall variability of the system, leading to results that more closely resemble empirical data. These systems can be associated to graphs, and characteristics of the graph give insight into the dynamics of the system.
This work addresses two main problems: 1. for fixed metabolite levels, find all fluxes for which the metabolite levels are an equilibrium, and 2. for fixed fluxes, find all metabolite levels which are equilibria for the system. We characterize the set of solutions for both problems, and show general results relating stability of systems to the structure of the associated graph. We show that there is a structure of the graph necessary for stable dynamics. Along with these general results, we show how stability analysis from the fields of network flows, compartmental systems, control theory and Markov chains apply to LIFE systems.
Citation: Nathaniel J. Merrill, Zheming An, Sean T. McQuade, Federica Garin, Karim Azer, Ruth E. Abrams, Benedetto Piccoli. Stability of metabolic networks via Linear-in-Flux-Expressions[J]. Networks and Heterogeneous Media, 2019, 14(1): 101-130. doi: 10.3934/nhm.2019006
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The methodology named LIFE (Linear-in-Flux-Expressions) was developed with the purpose of simulating and analyzing large metabolic systems. With LIFE, the number of model parameters is reduced by accounting for correlations among the parameters of the system. Perturbation analysis on LIFE systems results in less overall variability of the system, leading to results that more closely resemble empirical data. These systems can be associated to graphs, and characteristics of the graph give insight into the dynamics of the system.
This work addresses two main problems: 1. for fixed metabolite levels, find all fluxes for which the metabolite levels are an equilibrium, and 2. for fixed fluxes, find all metabolite levels which are equilibria for the system. We characterize the set of solutions for both problems, and show general results relating stability of systems to the structure of the associated graph. We show that there is a structure of the graph necessary for stable dynamics. Along with these general results, we show how stability analysis from the fields of network flows, compartmental systems, control theory and Markov chains apply to LIFE systems.
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.
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.
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
$ddt(‖A12ku‖2+B12k[u]+2∫ΩF(u)dx+‖∇α‖2+‖Δα‖2+‖∂α∂t−Δ∂α∂t‖2)+2‖∂u∂t‖2−1+2‖∇∂α∂t‖2+2‖Δ∂α∂t‖2=0$ | (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),
$ddt(‖Δα‖2−2((∂α∂t,Δα))+2((Δ∂α∂t,Δα)))+c‖α‖2H2(Ω)≤c′(‖∂u∂t‖2−1+ϵ‖∂u∂t‖2H1(Ω)+‖∂α∂t‖2H2(Ω)),c>0.$ | (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
$‖u(t)‖2Hk(Ω)+‖∂u∂t(t)‖2−1+‖α(t)‖2H2(Ω)+‖∂α∂t(t)‖2H2(Ω)≤e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c′,c>0,t0,$ | (3.19) |
$r>0$ given.
Multiplying next (2.7) by $\tilde{A}_{k}u$, we find, owing to the interpolation inequality (3.3),
$∫t+rt‖∂u∂t‖2Hk(Ω)ds≤e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c′(r),c>0,t≥0,$ |
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
$ddt‖˜A12ku‖2+c‖u‖2H2k(Ω)≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c",c,c′>0,t≥0.$ | (3.21) |
where
$dE3dt+c(E3+‖u‖2H2k(Ω)+‖∂u∂t‖2Hk(Ω))≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c",c,c′>0,t≥0,$ |
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
$ddt(‖ˉA12ku‖2+ˉB12k[u])+c‖∂u∂t‖2≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))−2((Δ∂u∂t,∂α∂t−Δ∂α∂t))+c″,c,c′>0.$ | (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
$ddt(‖Δα‖2+‖∇Δα‖2+‖∇∂α∂t−∇Δ∂α∂t‖2)+c(‖Δ∂α∂t‖2+‖∇Δ∂α∂t‖2)≤2((Δ∂u∂t,∂α∂t−Δ∂α∂t)),c>0$ | (3.27) |
Summing finally (3.21), (3.24) and (3.27), we find a differential inegality of the form
$ddt(‖ˉA12ku‖2+ˉB12k[u]+‖Δα‖2+‖∇Δα‖2+‖∇∂α∂t−∇Δ∂α∂t‖2)+c(‖∂u∂t‖2+‖Δ∂α∂t‖2+‖∇Δ∂α∂t‖2)≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c″,c,c′>0.$ | (3.28) |
where
$dE4dt+c(E3+‖u‖2Hk+1(Ω)+‖u‖2H2k(Ω)+‖∂u∂t‖2+‖∂u∂t‖2Hk(Ω)+‖∂α∂t‖2H3(Ω))≤e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c″,c,c′>0,t≥0$ |
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
$‖u(t)‖Hk+1(Ω)+‖α(t)‖H3(Ω)+‖∂α∂t(t)‖H3(Ω)≤e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H3(Ω),‖α1‖H3(Ω))+c′,c>0,t≥0,$ | (3.31) |
$r$ given.
We finally rewrite (2.7) as an elliptic equation, for t > 0 fixed,
$∫t+rt(‖∂u∂t‖2+‖∂α∂t‖2H3(Ω))ds≤e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H3(Ω),‖α1‖H3(Ω))+c′(r),c>0,t≥0,$ | (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) |
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),
$ −2((f(u(1))−f(u(2),u))≤2c0‖u‖2 ≤c‖u‖2 $ | (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
$ddt(‖α‖2+‖∇α‖2+‖u+∂α∂t−Δ∂α∂t‖2−1)+c(‖∂α∂t‖2+‖∂α∂t‖2H1(Ω))≤c′(‖u‖2+‖α‖2).$ | (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.
All authors declare no conflicts of interest in this paper.
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A directed graph
A directed graph
A directed graph where vertices
A directed cycle graph
Reverse Cholesterol Transport Network from [19]. This network contains 6 vertices which represent metabolites, 10 edges which represent fluxes and 2 virtual vertices
The trajectories of the values of metabolites over 25 hours