Case report Special Issues

Percutaneous Coronary Intervention in a Patient with Acute Thrombosis of Saphenous Vein Graft and Patent Native Coronary Artery: Which is the Vessel to Approach?

  • Received: 12 June 2015 Accepted: 15 September 2015 Published: 22 September 2015
  • We describe a case of a patient with a clinical history of coronary artery disease, previously treated by coronary surgery and, one year later, by percutaneous coronary intervention plus stenting for sub-occlusive disease of the saphenous vein graft to first obtuse marginal (OM) branch. The patient, admitted to our emergency room with chest pain, nausea, hypotension and diaphoresis, had elevated blood levels of cardiac troponin T and EKG showed elevation of the ST segment in the in lateral leads, suggesting a diagnosis of ST elevated myocardial infarction (STEMI). Thus, coronary angiography was immediately performed, showing the massive thrombosis of the saphenous vein graft previously treated by stenting and the slight patency of the native vessel. We decided to approach the native vessel instead of clashing to the massive thrombus of the saphenous vein graft, overcoming the actual guidelines indications.

    Citation: Marco Ferrone, Anna Franzone, Bruno Trimarco, Giovanni Esposito, Plinio Cirillo. Percutaneous Coronary Intervention in a Patient with Acute Thrombosis of Saphenous Vein Graft and Patent Native Coronary Artery: Which is the Vessel to Approach?[J]. AIMS Medical Science, 2015, 2(4): 310-315. doi: 10.3934/medsci.2015.4.310

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  • We describe a case of a patient with a clinical history of coronary artery disease, previously treated by coronary surgery and, one year later, by percutaneous coronary intervention plus stenting for sub-occlusive disease of the saphenous vein graft to first obtuse marginal (OM) branch. The patient, admitted to our emergency room with chest pain, nausea, hypotension and diaphoresis, had elevated blood levels of cardiac troponin T and EKG showed elevation of the ST segment in the in lateral leads, suggesting a diagnosis of ST elevated myocardial infarction (STEMI). Thus, coronary angiography was immediately performed, showing the massive thrombosis of the saphenous vein graft previously treated by stenting and the slight patency of the native vessel. We decided to approach the native vessel instead of clashing to the massive thrombus of the saphenous vein graft, overcoming the actual guidelines indications.


    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:

    $ ψ=Ω(12|u|2+F(u)uθ12θ2)dx,
    $
    (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

    $ H=u+θ.
    $
    (1.4)

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

    $ ut=δuψ,
    $
    (1.5)

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

    $ Ht=divq,
    $
    (1.6)

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

    $ q=θ,
    $
    (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]):

    $ θ=φΔφ.
    $
    (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 $\|.\|_2$.


    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 $\theta$,

    $ ψ(u,θ)=Ω(12|u|2+F(u)G(u)θ12θ2)dx,
    $
    (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

    $ H=G(u)+θ=G(u)+φΔφ,
    $
    (2.2)

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

    $ φtΔφt+divq=g(u)ut.
    $
    (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.


    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)
    $ |g(s)s|c3(|G(s)|2+1),c30,
    $
    (3.2)
    $ c4sk+2c5F(s)f(s)s+c0c6sk+2c7,c4,c6>0,c5,c70,
    $
    (3.3)
    $ |g(s)|c8(|s|+1),|g(s)|c9c8,c90,
    $
    (3.4)
    $ |f(s)|c10(|s|k+1),c100,
    $
    (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

    $ 12ddt(u2+2ΩF(u)dx)+ut2=Ωg(u)ut(φΔφ)dx.
    $
    (3.6)

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

    $ 12ddt(φ2+2φ2+Δφ2)+φ2+Δφ2=Ωg(u)ut(φΔφ)dx.
    $
    (3.7)

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

    $ ddt(u2+2ΩF(u)dx+φ2+2φ2+Δφ2)+2(ut2+φ2+Δφ2)=0.
    $
    (3.8)

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

    $ 12ddtu2+u2+Ωf(u)udx=Ωg(u)u(φΔφ)dx.
    $
    (3.9)

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

    $ 12ddtu2+u2+cΩF(u)dxcΩ|G(u)|2dx+12(φ2+Δφ2)+c.
    $
    (3.10)

    Adding (3.8) and (3.10), one has

    $ dE1dt+2(u2+cΩF(u)dx+ut2+φ2)+Δφ2cΩ|G(u)|2dx+φ2+c,
    $
    (3.11)

    where

    $ E1=u2+u2+2ΩF(u)dx+φ2+2φ2+Δφ2
    $
    (3.12)

    enjoys

    $ E1c(u2H1(Ω)+uk+2k+2+φ2H2(Ω))c
    $
    (3.13)

    and

    $ E1c(u2H1(Ω)+uk+2k+2+φ2H2(Ω))c.
    $
    (3.14)

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

    $ 12ddtφ2+φt2+φt2=Ωg(u)utφtdx.
    $
    (3.15)

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

    $ 12ddtφ2+12φt2+φt2c(u2+1)ut2
    $
    (3.16)

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

    $ dE2dt+2(u2+cΩF(u)dx+ut2+φ2+12Δφ2+12φt2+φt2)cΩ|G(u)|2dx+φ2+c(u2+1)ut2+c,
    $
    (3.17)

    where

    $ E2=E1+φ2
    $
    (3.18)

    satisfies similar estimates as $E_1$.

    We deduce from (3.1) and (3.17)

    $ dE2dt+c(φt2+φt2)cE2+c,
    $
    (3.19)

    which achieve the proof.

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

    $ f(0)=0andf(s)c,c0.
    $
    (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 $\Omega$,

    $ 12ddtφ2+φt2+Δφt2=Ωg(u)utΔφtdx.
    $
    (3.21)

    Thanks to (3.4) and Hölder's inequality:

    $ Ωg(u)utΔφtdxcΩ(|u|+1)|ut||Δφt|dxc(u2+1)ut2+12Δφt2
    $
    (3.22)

    and then,

    $ 12ddtφ2+φt2+12Δφt2c(u2+1)ut2.
    $
    (3.23)

    Differentiating (1.9) with respect to time, we get

    $ 2ut2Δut+f(u)ut=g(u)ut(φΔφ)+g(u)(φtΔφt).
    $
    (3.24)

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

    $ 12ddtut2+ut2+Ωf(u)|ut|2dx=Ωg(u)|ut|2(φΔφ)dx+Ωg(u)ut(φtΔφt)dx.
    $
    (3.25)

    Using (1.10), we write,

    $ Ωg(u)ut(φtΔφt)dx=Ωg(u)ut(g(u)ut+Δφ)dx=Ω|g(u)ut|2dx+Ωg(u)utΔφdx.
    $
    (3.26)

    Owing to (3.26), (3.25) reads

    $ 12ddtut2+ut2+Ωf(u)|ut|2dx=Ωg(u)|ut|2(φΔφ)dx+Ωg(u)utΔφdxΩ|g(u)ut|2dx,
    $
    (3.27)

    since

    $ Ωg(u)|ut|2(φΔφ)dxcΩ|ut|2(|φ|+|Δφ|)dx12ut2+c(φ2+Δφ2),
    $
    (3.28)
    $ Ωg(u)utΔφdx=Ωg(u)uutφdxΩg(u)utφdx
    $
    (3.29)

    and then,

    $ |Ωg(u)uutφdx|cΩ|u||ut||φ|dx16ut2+cu2Δφ2
    $
    (3.30)

    and

    $ |Ωg(u)utφdx|cΩ(|u|+1)|ut||φ|dx16ut2+c(u2+1)φ2.
    $
    (3.31)

    Furthemore,

    $ Ω|g(u)ut|2dxcΩ(|u|+1)2|ut|2dxc(u2+u2+1)ut2.
    $
    (3.32)

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

    $ ddtut2+cut2c(u2H1(Ω)+1)(ut2+φ2H2(Ω)).
    $
    (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

    $ dE3dt+c(ut2H1(Ω)+φt2H2(Ω))cE3+c,
    $
    (3.34)

    where

    $ E3=E2+ε1φ2+ε2ut2
    $
    (3.35)

    enjoys

    $ E3c(u2H(Ω)+uk+2k+2+φ2H2(Ω))c
    $
    (3.36)

    and

    $ E3c(u2H(Ω)+uk+2k+2+φ2H2(Ω))c.
    $
    (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

    $ 12ddtu2+ut2+Ω(f(u1f(u2)))utdx=Ωg(u1)(φΔφ)utdx+Ω(g(u1)g(u2))(φ2Δφ2)utdx.
    $
    (3.42)

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

    $ 12ddt(φ2+φ2)+φ2=Ωg(u1)utφdxΩ(g(u1)g(u2))u2tφdx.
    $
    (3.43)

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

    $ 12ddt(φ2+Δφ2)+Δφ2=Ωg(u1)utΔφdx+Ω(g(u1)g(u2))u2tΔφdx.
    $
    (3.44)

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

    $ dE4dt+ut2+φ2+Δφ2+Ω(f(u1)f(u2))utdx=Ω(g(u1)g(u2))(φ2Δφ2)utdxΩ(g(u1)g(u2))(φΔφ)u2tdx,
    $
    (3.45)

    where

    $ E4=u2+φ2+2φ2+Δφ2.
    $
    (3.46)

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

    $ Ω(f(u1)f(u2))utdxcΩ(|u2|k+1)|u||ut|dxc(u22k+1)u2+ut2,
    $
    (3.47)

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

    $ Ω(g(u1)g(u2))(φ2Δφ2)utdxcΩ|u||φ2Δφ2||ut|dxcu2(φ22+Δφ22)+ut2
    $
    (3.48)

    and

    $ Ω(g(u1)g(u2))(φΔφ)u2tdxcΩ|u||ut||φΔφ|dxcu2t2(φ2+Δφ2)+u2.
    $
    (3.49)

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

    $ dE4dt+cut2c(u22k+u2t2+φ22+Δφ22+1)E4.
    $
    (3.50)

    In particular,

    $ dE4dtcE4
    $
    (3.51)

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


    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,

    $ ϵ>0,|G(u)|2ϵF(s)+cϵ,sR.
    $
    (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,

    $ dE5dt+2(cu2+δΩF(u)dx+ut2+φ2+Δφ2)2cδΩ|G(u)|2dx+δ(φ2+Δφ2)+c2cδΩ|G(u)|2dx+δ(cφ2+Δφ2)+c,
    $
    (4.2)

    where

    $ E5=δu2+u2+2ΩF(u)dx+φ2+2φ2+Δφ2
    $
    (4.3)

    satisfies

    $ E5c(u2H1(Ω)+uk+2k+2+φ2H2(Ω))c
    $
    (4.4)

    and

    $ E5c(u2H1(Ω)+uk+2k+2+φ2H2(Ω))c.
    $
    (4.5)

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

    $ dE5dt+2(cu2+δΩF(u)dx+ut2+φ2+Δφ2)CϵΩF(u)dx+δ(cφ2+Δφ2)+Cϵ,
    $
    (4.6)

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

    $ 2δCϵand2>cδ,
    $
    (4.7)

    we then deduce from (4.6),

    $ dE5dt+c(E5+ut2)c,
    $
    (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:

    $ S(t):ΦΦ,(u0,φ0)(u(t),φ(t)),t0,
    $
    (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).


    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:

    $ u=φ=0on(0,+)×D×(0,T)
    $
    (5.1)

    and

    $ u(0,x2,x3;t)=h(x2,x3;t),φ(0,x2,x3;t)=l(x2,x3;t)on{0}×D×(0,T),
    $
    (5.2)

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

    We also consider following initial data

    $ u|t=0=φ|t=0=0onR.
    $
    (5.3)

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

    $ Fw(z,t)=t0D(z)ews(usu,1+φ(φ,1+φ,1s)+φsφ,1)dads,
    $
    (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

    $ Fw(z+h,t)Fw(z,t)=ewt2R(z,z+h)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dx+t0R(z,z+h)ews(|us|2+|φ|2+|Δφ|2)dxds+w2t0R(z,z+h)ews(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dxds,
    $
    (5.5)

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

    Hence,

    $ Fwt(z,t)=ewt2D(z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)da+t0D(z)ews(|us|2+|φ|2+|Δφ|2)dads+w2t0D(z)ews(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dads.
    $
    (5.6)

    We consider a second function, namely,

    $ Gw(z,t)=t0D(z)ews(usu,1+φ(θ,1+φ,1s))dads,
    $
    (5.7)

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

    Similarly, we have

    $ Gw(z+h,t)Gw(z,t)=ewt2R(z,z+h)(|u|2+|θ|2)dx+t0R(z,z+h)ews(|u|2+f(u)u+uΔφ+|φ|2+|φ|2)dxds+w2t0R(z,z+h)ews(|u|2+|θ|2)dxds+t0R(z,z+h)ews(G(u)g(u)u)φdxds
    $
    (5.8)

    and then

    $ Gwt(z,t)=ewt2D(z)(|u|2+|θ|2)da+t0D(z)ews(|u|2+f(u)u+uΔφ+|φ|2+|φ|2)dads+w2t0D(z)ews(|u|2+|θ|2)dads+t0D(z)ews(G(u)g(u)u)φdads.
    $
    (5.9)

    We choose $\tau$ large enough such as

    $ 2F(u)+τu2C1u2,C1>0.
    $
    (5.10)

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

    $ w(F(u)+τ2|u|2)+τf(u)u+τ(G(u)g(u)u)φ+w2|φ|2.
    $
    (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

    $ w(F(u)+τ2|u|2)+τf(u)u+τ(G(u)g(u)u)φ+w2|φ|2C3(|u|2+|φ|2+|Δφ|2).
    $
    (5.12)

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

    $ Hw=Fw+τGw
    $
    (5.13)

    satisfies

    $ Hwt(z,t)C4t0D(z)ews(|u|2+|u|2+|us|2+|φ|2+|φ|2+|Δφ|2+|θ|2)dads.
    $
    (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

    $ |Fw|(t0D(z)ewsu2sdads)1/2(ewsu2,1)1/2+(t0D(z)ewsφ2dads)1/2(ewsφ2,1)1/2+(t0D(z)ewsφ2dads)1/2(ewsφ2,1s)1/2+(t0D(z)ewsφ2sdads)1/2(ewsφ2,1)1/2C5t0D(z)ews(|u|2+|us|2+|φ|2+|φ|2+|φs|2+|φs|2)dads,C5>0.
    $
    (5.15)

    Similarly,

    $ |Gw|(t0D(z)ewsu2dads)1/2(t0D(z)ewsu2,1dads)1/2+(t0D(z)ewsφ2dads)1/2(t0D(z)ewsθ2,1dads)1/2+(t0D(z)ewsφ2sdads)1/2(t0D(z)ewsφ2,1dads)1/2C6t0D(z)ews(|u|2+|u|2+|φ|2+|φ|2+|θ|2)dads,C6>0.
    $
    (5.16)

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

    $ |Hw|C7Hwz.
    $
    (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

    $ Hw(z,t)Hw(z0,t)eC17(zz0),zz0.
    $
    (5.18)

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

    $ ewt2R(0,z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dx+τewt2R(0,z)(|u|2+|θ|2)dx+t0R(0,z)ews(|us|2+|φ|2+|Δφ|2)dxds+τt0R(0,z)ews(|u|2+f(u)u+g(u)uΔφ+|φ|2+2|φ|2)dxds+w2t0R(0,z)ews(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dxds+τw2t0R(0,z)ews(|u|2+|θ|2)dx+τt0R(0,z)ews(G(u)g(u)u)φdxds
    $
    (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

    $ Hw(z,t)Hw(0,t)eC17z,z0,
    $
    (5.20)

    where

    $ Ew(z,t)=ewt2R(z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dx+τewt2R(z)(|u|2+|θ|2)dx+t0R(z)ews(|us|2+|φ|2+|Δφ|2)dxds+τt0R(z)ews(|u|2+f(u)u+g(u)uΔφ+|φ|2+2|φ|2)dxds+w2t0R(z)ews(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dxds+τw2t0R(z)ews(|u|2+|θ|2)dx+τt0R(z)ews(G(u)g(u)u)φdxds
    $
    (5.21)

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

    Finally, setting

    $ Ew(z,t)=12R(z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dx+τ12R(z)(|u|2+|θ|2)dx+t0R(z)(|us|2+|φ|2+|Δφ|2)dxds+τt0R(z)(|u|2+f(u)u+g(u)uΔφ+|φ|2+2|φ|2)dxds+w2t0R(z)(|u|2+2F(u)+|φ|2+2|φ|2+|Δφ|2)dxds+τw2t0R(z)(|u|2+|θ|2)dx+τt0R(z)(G(u)g(u)u)φdxds.
    $
    (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

    $ Ew(z,t)Ew(0,t)ewtC17z,z0,
    $
    (5.23)

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


    Acknowledgments

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


    Conflict of interest

    The author declares no conflicts of interest in this paper.


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