Let G be a graph of order n. A path decomposition P of G is a collection of edge-disjoint paths that covers all the edges of G. Let p(G) denote the minimum number of paths needed in a path decomposition of G. Gallai conjectured that if G is connected, then p(G)≤⌈n2⌉. In this paper, we prove that the above conjecture holds for all block graphs.
Citation: Xiaohong Chen, Baoyindureng Wu. Gallai's path decomposition conjecture for block graphs[J]. AIMS Mathematics, 2025, 10(1): 1438-1447. doi: 10.3934/math.2025066
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Let G be a graph of order n. A path decomposition P of G is a collection of edge-disjoint paths that covers all the edges of G. Let p(G) denote the minimum number of paths needed in a path decomposition of G. Gallai conjectured that if G is connected, then p(G)≤⌈n2⌉. In this paper, we prove that the above conjecture holds for all block graphs.
G. Caginalp proposed in [3] and [4] two phase-field system, namely,
∂u∂t−Δu+f(u)=T, | (1.1) |
∂T∂t−ΔT=−∂u∂t, | (1.2) |
called nonconserved system, and
∂u∂t+Δ2u−Δf(u)=−ΔT, | (1.3) |
∂T∂t−ΔT=−∂u∂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=˜T−TE, where ˜T is the absolute temperature and TE is the equilibrium melting temperature) and f is the derivative of a double-well potential F (a typical choice is F(s)=14(s2−1)2, hence the usual cubic nonlinear term f(s)=s3−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
ΨGL=∫Ω(12|∇u|2+F(u)−uT−12T2)dx, | (1.5) |
where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R3, with boundary Γ), 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)
∂u∂u=−DΨGLDu, | (1.7) |
for the nonconserved model, and
∂u∂u=ΔDΨGLDu, | (1.8) |
for the conserved one, where DDu denotes a variational derivative with respect to u, which yields (1.1) and (1.3), respectively. Then, we have the energy equation
∂H∂t=−divq, | (1.9) |
where q is the heat flux. Assuming finally the usual Fourier law for heat conduction,
q=−∇T, | (1.10) |
we obtain (1.2).
In (1.5), the term |∇u|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
ΨHOGL=∫Ω(12∑ki=1∑|β|=iaβ|Dβu|2+F(u)−uT−12T2)dx,k∈N, | (1.11) |
where, for β=(k1,k2,k3)∈(N∪{0})3,
|β|=k1+k2+k3 |
and, for β≠(0,0,0),
Dβ=∂|β|∂xk11∂xk22∂xk33 |
(we agree that 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:
∂u∂t−Δ∑ki=1(−1)i∑|β|=iaβD2βu−Δf(u)=−Δ(∂α∂t−Δ∂α∂t), | (1.12) |
In particular, for k = 1 (anisotropic conserved Caginalp phase-field), we have an equation of the form
∂u∂t+Δ∑3i=1ai∂2u∂x2i−Δf(u)=−Δ(∂α∂t−Δ∂α∂t) |
and, for k = 2 (fourth-order anisotropic conserved Caginalp phase-field system), we have an equation of the form
∂u∂t−Δ∑3i,j=1aij∂4u∂x2i∂x2j+Δ∑3i=1bi∂2u∂x2i−Δf(u)=−Δ(∂α∂t−Δ∂α∂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
∂u∂t−ΔP(−Δ)u−Δf(u)=0, |
where
P(s)=∑ki=1aisi,ak>0,k⩾1, |
endowed with the Dirichlet/Navier boundary conditions
u=Δu=...=Δku=0onΓ. |
Our aim in this paper is to study the model consisting of the higher-order anisotropic equation (1.12) and the temperature equation
∂2α∂t2−Δ∂2α∂t2−Δ∂α∂t−Δα=−∂u∂t. | (1.13) |
In particular, we obtain the existence and uniqueness of solutions.
We consider the following initial and boundary value problem, for k∈N, k⩾2 (the case k = 1 can be treated as in the original conserved system; see, e.g., [23]):
∂u∂t−Δ∑ki=1(−1)i∑|β|=iaβD2βu−Δf(u)=−Δ(∂α∂t−Δ∂α∂t), | (2.1) |
∂2α∂t2−Δ∂2α∂t2−Δ∂α∂t−Δα=−∂u∂t, | (2.2) |
Dβu=α=0onΓ,|β|⩽k, | (2.3) |
u|t=0=u0,α|t=0=α0,∂α∂t|t=0=α1. | (2.4) |
We assume that
aβ>0,|β|=k, | (2.5) |
and we introduce the elliptic operator Ak defined by
⟨Akv,w⟩H−k(Ω),Hk0(Ω)=∑|β|=kaβ((Dβv,Dβw)), | (2.6) |
where H−k(Ω) is the topological dual of Hk0(Ω). Furthermore, ((., .)) denotes the usual L2-scalar product, with associated norm ‖.‖. More generally, we denote by ‖.‖X the norm on the Banach space X; we also set ‖.‖−1=‖(−Δ)−12.‖, where (−Δ)−1 denotes the inverse minus Laplace operator associated with Dirichlet boudary conditions. We can note that
(v,w)∈Hk0(Ω)2↦∑|β|=kaβ((Dβv,Dβw)) |
is bilinear, symmetric, continuous and coercive, so that
Ak:Hk0(Ω)→H−k(Ω) |
is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order 2k (see [1] and [2]) that Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain
D(Ak)=H2k(Ω)∩Hk0(Ω), |
where, for v∈D(Ak),
Akv=(−1)k∑|β|=kaβD2βv. |
We further note that D(A12k)=Hk0(Ω) and, for (v,w)∈D(A12k)2,
((A12kv,A12kw))=∑|β|=kaβ((Dβv,Dβw)). |
We finally note that (see, e.g., [24]) ‖Ak.‖ (resp., ‖A12k.‖) is equivalent to the usual H2k-norm (resp., Hk-norm) on D(Ak) (resp., D(A12k)).
Similarly, we can define the linear operator ¯Ak=−ΔAk
ˉAk:Hk+10(Ω)→H−k−1(Ω) |
which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain
D(ˉAk)=H2k+2(Ω)∩Hk+10(Ω), |
where, for v∈D(ˉAk),
ˉAkv=(−1)k+1Δ∑|β|=kaβD2βv. |
Furthermore, D(ˉA12k)=Hk+10(Ω) and, for (v,w)∈D(ˉA12k),
((ˉA12kv,ˉA12kw))=∑|β|=kaβ((∇Dβv,∇Dβw)). |
Besides ‖ˉAk.‖ (resp., ‖ˉA12k.‖) is equivalent to the usual H2k+2-norm (resp., Hk+1-norm) on D(ˉAk) (resp., D(ˉA12k)).
We finally consider the operator ˜Ak=(−Δ)−1Ak, where
˜Ak:Hk−10(Ω)→H−k+1(Ω); |
note that, as −Δ and Ak commute, then the same holds for (−Δ)−1 and Ak, so that ˜Ak=Ak(−Δ)−1.
We have the (see [17])
Lemme 2.1. The operator ˜Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain
D(˜Ak)=H2k−2(Ω)∩Hk−10(Ω), |
where, for v∈D(˜Ak)
˜Akv=(−1)k∑|β|=kaβD2β(−Δ)−1v. |
Furthermore, D(˜A12k)=Hk−10(Ω) and, for (v,w)∈D(˜A12k),
((˜A12kv,˜A12kw))=∑|β|=kaβ((Dβ(−Δ)−12v,Dβ(−Δ)−12w)). |
Besides ‖˜Ak.‖ (resp., ‖˜A12k.‖) is equivalent to the usual H2k−2-norm (resp., Hk−1-norm) on D(˜Ak) (resp., D(˜A12k)).
Proof. We first note that ˜Ak clearly is linear and unbounded. Then, since (−Δ)−1 and Ak commute, it easily follows that ˜Ak is selfadjoint.
Next, the domain of ˜Ak is defined by
D(˜Ak)={v∈Hk−10(Ω),˜Akv∈L2(Ω)}. |
Noting that ˜Akv=f,f∈L2(Ω),v∈D(˜Ak), is equivalent to Akv=−Δf, where −Δf∈H2(Ω)′, it follows from the elliptic regularity results of [1] and [2] that v∈H2k−2(Ω), so that D(˜Ak)=H2k−2(Ω)∩Hk−10(Ω).
Noting then that ˜A−1k maps L2(Ω) onto H2k−2(Ω) and recalling that k⩾2, we deduce that ˜Ak has compact inverse.
We now note that, considering the spectral properties of −Δ and Ak (see, e.g., [24]) and recalling that these two operators commute, −Δ and Ak have a spectral basis formed of common eigenvectors. This yields that, ∀s1,s2∈R, (−Δ)s1 and As2k commute.
Having this, we see that ˜A12k=(−Δ)−12A12k, so that D(˜A12k)=Hk−10(Ω), and for (v,w)∈D(˜A12k)2,
((˜A12kv,˜A12kw))=∑|β|=kaβ((Dβ(−Δ)−12v,Dβ(−Δ)−12w)). |
Finally, as far as the equivalences of norms are concerned, we can note that, for instance, the norm ‖˜A12k.‖ is equivalent to the norm ‖(−Δ)−12.‖Hk(Ω) and, thus, to the norm ‖(−Δ)k−12.‖.
Having this, we rewrite (2.1) as
∂u∂t−ΔAku−ΔBku−Δf(u)=−Δ(∂α∂t−Δ∂α∂t), | (2.7) |
where
Bkv=∑k−1i=1(−1)i∑|β|=iaβD2βv. |
As far as the nonlinear term f is concerned, we assume that
f∈C2(R),f(0)=0, | (2.8) |
f′⩾−c0,c0⩾0, | (2.9) |
f(s)s⩾c1F(s)−c2⩾−c3,c1>0,c2,c3⩾0,s∈R, | (2.10) |
F(s)⩾c4s4−c5,c4>0,c5⩾0,s∈R, | (2.11) |
where F(s)=∫s0f(τ)dτ. In particular, the usual cubic nonlinear term f(s)=s3−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 (−Δ)−1∂u∂t and (2.2) by ∂α∂t−Δ∂α∂t, sum the two resulting equalities and integrate over Ω 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 ‖∂α∂t‖2+2‖∇∂α∂t‖2+‖Δ∂α∂t‖2=‖∂α∂t−Δ∂α∂t‖2), where
B12k[u]=∑k−1i=1∑|β|=iaβ‖Dβu‖2 | (3.2) |
(note that B12k[u] is not necessarily nonnegative). We can note that, owing to the interpolation inequality
B12k[u]=∑k−1i=1∑|β|=iaβ‖Dβu‖2 | (3.3) |
‖(−Δ)i2v‖⩽c(i)‖(−Δ)m2v‖im‖v‖1−im, |
there holds
v∈Hm(Ω),i∈{1,...,m−1},m∈N,m⩾2, | (3.4) |
This yields, employing (2.11),
|B12k[u]|⩽12‖A12ku‖2+c‖u‖2. |
whence
‖A12ku‖2+B12k[u]+2∫ΩF(u)dx⩾12‖A12ku‖2+∫ΩF(u)dx+c‖u‖4L4(Ω)−c′‖u‖2−c", | (3.5) |
nothing that, owing to Young's inequality,
‖A12ku‖2+B12k[u]+2∫ΩF(u)dx⩾c(‖u‖2Hk(Ω)+∫ΩF(u)dx)−c′,c>0, | (3.6) |
We then multiply (2.7) by (−Δ)−1u and have, owing to (2.10) and the interpolation inequality (3.3),
‖u‖2⩽ϵ‖u‖4L4(Ω)+c(ϵ),∀ϵ>0. |
hence, proceeding as above and employing, in particular, (2.11)
ddt‖u‖2−1+c(‖u‖2Hk(Ω)+∫ΩF(u)dx)⩽c′(‖u‖2+‖∂α∂t‖2+‖Δ∂α∂t‖2)+c", | (3.7) |
Summing (3.1) and δ1 times (3.7), where δ1>0 is small enough, we obtain a differential inegality of the form
ddt‖u‖2−1+c(‖u‖2Hk(Ω)+∫ΩF(u)dx)⩽c′(‖∂α∂t‖2+‖Δ∂α∂t‖2)+c″,c>0. | (3.8) |
where
ddtE1+c(‖u‖2Hk(Ω)+∫ΩF(u)dx+‖∂u∂t‖2−1+‖∂α∂t‖2H2(Ω))⩽c′,c>0, |
satisfies, owing to (3.5)
E1=‖A12ku‖2+B12k[u]+2∫ΩF(u)dx+‖∇α‖2+‖Δα‖2+‖∂α∂t−Δ∂α∂t‖2+δ1‖u‖2−1 | (3.9) |
Multiplying (2.2) by −Δα, we then obtain
E1⩾c(‖u‖2Hk(Ω)+∫ΩF(u)dx+‖α‖2H2(Ω)+‖∂α∂t‖2H2(Ω))−c′,c>0. |
which yields, employing the interpolation inequality
ddt(‖Δα‖2−2((∂α∂t,Δα))+2((Δ∂α∂t,Δα)))+‖Δα‖2⩽‖∂u∂t‖2+‖∇∂α∂t‖2+‖Δ∂α∂t‖2, | (3.10) |
the differential inequality, with 0<ϵ<<1 is small enough
‖v‖2⩽c‖v‖−1‖v‖H1(Ω),v∈H10(Ω), | (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
∂∂t∂u∂t−ΔAk∂u∂t−ΔBk∂u∂t−Δ(f′(u)∂u∂t)=−Δ(Δ∂α∂t+Δα−∂u∂t), | (3.13) |
We multiply (3.11) by (−Δ)−1∂u∂t and obtain, owing to (2.9) and the interpolation inequality (3.3),
Dβ∂u∂t=0onΓ,|β|⩽k. |
hence, owing to (3.10), the differential inequality
ddt‖∂u∂t‖2−1+c‖∂u∂t‖2Hk(Ω)⩽c′(‖∂u∂t‖2+‖Δα‖2+‖Δ∂α∂t‖2),c>0, | (3.14) |
Summing finally (3.8), δ2 times (3.11) and δ3 times (3.14), where δ2,δ3>0 are small enough, we find a differential inequality of the form
ddt‖∂u∂t‖2−1+c‖∂u∂t‖2Hk(Ω)⩽c′(‖∂u∂t‖2−1+‖α‖2H2(Ω)+‖∂α∂t‖2H2(Ω)),c>0. | (3.15) |
where
dE2dt+c(E2+‖∂u∂t‖2Hk(Ω))⩽c′,c>0, |
Owing to the continuous embedding H2k+1(Ω)⊂C(ˉΩ), we deduce that
E2=E1+δ2(‖Δα‖2−2((∂α∂t,Δα))+2((Δ∂α∂t,Δα)))+δ3‖∂u∂t‖2−1. |
and since
|∫ΩF(u0)dx|⩽Q(‖u0‖H2k+1(Ω)) |
we see that (−Δ)−12∂u∂t(0)∈L2(Ω) and
(−Δ)−12∂u∂t(0)=−(−Δ)12Aku0−(−Δ)12Bku0−(−Δ)12f(u0)+(−Δ)12(α1−Δα1), | (3.16) |
Furthermore E2 satisfies
‖∂u∂t(0)‖−1⩽Q(‖u0‖H2k+1(Ω),‖α1‖H3(Ω)). | (3.17) |
It thus follows from (3.15), (3.16), (3.17) and Growall's lemma that
E2⩾c(‖u‖2Hk(Ω)+‖∂u∂t‖2−1+∫ΩF(u)dx+‖α‖2H2(Ω)+‖∂α∂t‖2H2(Ω))−c′,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 ˜Aku, 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),
ddt‖˜A12ku‖2+c‖u‖2H2k(Ω)⩽c′(‖u‖2+‖f(u)‖2+‖∂α∂t‖2+‖Δ∂α∂t‖2),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
E3=E2+‖˜A12ku‖2 | (3.22) |
In particular, it follows from (3.21)-(3.22) that
E3⩾c(‖u‖2Hk(Ω)+‖∂u∂t‖2−1+∫ΩF(u)dx+‖α‖2H2(Ω)+‖∂α∂t‖2H2(Ω))−c′,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)
∫t+rt‖u‖2H2k(Ω)ds⩽e−ctQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c′(r),c>0,t⩾0, |
whence, proceeding as above,
ddt‖u‖2+c‖u‖2Hk+1(Ω)⩽c′(‖u‖2H1(Ω)+‖∂α∂t‖2+‖Δ∂α∂t‖2),c>0, | (3.24) |
We also multiply (2.7) by ∂u∂t and find
ddt‖u‖2+c‖u‖2Hk+1(Ω)⩽e−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H2(Ω),‖α1‖H3(Ω))+c″,c,c′>0. |
where
ddt(‖ˉA12ku‖2+ˉB12k[u])+c‖∂u∂t‖2⩽c′‖Δf(u)‖2−2((Δ∂u∂t,∂α∂t−Δ∂α∂t)), |
Since f is of class C2, it follows from the continuous embedding H2(Ω)⊂C(ˉΩ) that
ˉB12k[u]=∑k−1i=1∑|β|=iaβ‖∇Dβu‖2. |
hence, owing to (3.18),
‖Δf(u)‖2⩽Q(‖u‖H2(Ω)), | (3.25) |
Multiply next (2.2) by −Δ(∂α∂t−Δ∂α∂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 ‖∇∂α∂t‖2+2‖Δ∂α∂t‖2+‖∇Δ∂α∂t‖2=‖∇∂α∂t−∇Δ∂α∂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)
E4=E3+‖u‖2+‖ˉA12ku‖2+ˉB12k[u]+‖Δα‖2+‖∇Δα‖2+‖∇∂α∂t−∇Δ∂α∂t‖2 | (3.29) |
In particular, it follows from (3.28)-(3.29) that
E4⩾c(‖u‖2Hk+1(Ω)+‖∂u∂t‖2−1+∫ΩF(u)dx+‖α‖2H3(Ω)+‖∂α∂t‖2H3(Ω))−c′,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 Aku, we obtain, owing to the interpolation inequality (3.3),
Aku=−(−Δ)−1∂u∂t−Bku−f(u)+∂α∂t−Δ∂α∂t,Dβu=0onΓ,|β|⩽k−1. |
hence, since f is continuous and owing to (3.18)
‖Aku‖2⩽c(‖u‖2+‖f(u)‖2+‖∂u∂t‖2−1+‖∂α∂t‖2+‖Δ∂α∂t‖2), | (3.33) |
We first have the following theorem.
Theorem 4.1. (i) We assume that (u0,α0,α1)∈Hk0(Ω)×(H2(Ω)∩H10(Ω))×(H2(Ω)∩H10(Ω)), with ∫ΩF(u0)dx<+∞. Then, (2.1)−(2.4) possesses at last one solution (u,α,∂α∂t) such that, ∀T>0, u(0)=u0, α(0)=α0, ∂α∂t(0)=α1,
‖u(t)‖2H2k(Ω)⩽ce−c′tQ(‖u0‖H2k+1(Ω),‖α0‖H3(Ω),‖α1‖H3(Ω))+c″,c′>0t⩾0. |
u∈L∞(R+;Hk0(Ω))∩L2(0,T;H2k(Ω)∩Hk0(Ω)), |
∂u∂t∈L∞(R+;H−1(Ω))∩L2(0,T;Hk0(Ω)), |
and
α,∂α∂t∈L∞(R+;H2(Ω)∩H10(Ω)) |
ddt((−Δ)−1u,v))+∑ki=1∑|β|=iai((Dβu,Dβv))+((f(u),v))=ddt(((u,v))+((∇u,∇v))),∀v∈C∞c(Ω), |
in the sense of distributions.
(ii) If we futher assume that (u0,α0,α1)∈(Hk+1(Ω)∩Hk0(Ω))×(H3(Ω)∩H10(Ω))×(H3(Ω)∩H10(Ω)), then, ∀T>0,
ddt(((∂α∂t,w))+((∇∂α∂t,∇w))+((∇α,∇w)))+((∇α,∇w))=−ddt((u,w)),∀w∈C∞c(Ω), |
u∈L∞(R+;Hk+1(Ω)∩Hk0(Ω))∩L2(R+;Hk+1(Ω)∩Hk0(Ω)) |
∂u∂t∈L2(R+;L2(Ω)), |
and
α∈L∞(R+;H3(Ω)∩H10(Ω)) |
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),α(1),∂α(1)∂t) and (u(2),α(2),∂α(2)∂t) be two solutions to (2.1)-(2.3) with initial data (u(1)0,α(1)0,α(1)1) and (u(2)0,α(2)0,α(2)1), respectively. We set
∂α∂t∈L∞(R+;H3(Ω)∩H10(Ω))∩L2(0,T;H3(Ω)∩H10(Ω)) |
and
(u,α,∂α∂t)=(u(1),α(1),∂α(1)∂t)−(u(2),α(2),∂α(2)∂t) |
Then, (u,α) satisfies
(u0,α0,α1)=(u(1)0,α(1)0,α(1)1)−(u(2)0,α(2)0,α(2)1). | (4.1) |
∂u∂t−ΔAku−ΔBku−Δ(f(u(1))−f(u(2)))=−Δ(∂α∂t−Δ∂α∂t), | (4.2) |
∂2α∂t2−Δ∂2α∂t2−Δ∂α∂t−Δα=−∂u∂t, | (4.3) |
Dβu=α=0 on Γ,|β|⩽k, | (4.4) |
Multiplying (4.1) by (−Δ)−1u and integrating over Ω, we obtain
u|t=0=u0,α|t=0=α0,∂α∂t|t=0=α1. |
We note that
ddt‖u‖2−1+c‖u‖2Hk(Ω)⩽c′(‖u‖2+‖∂α∂t−Δ∂α∂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)=∫10f′(su(1)(t)+(1−s)u(2)(t))ds. |
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 (−Δ)−1(u+∂α∂t−Δ∂α∂t), we find
ddt‖u‖2−1+c‖u‖2Hk(Ω)⩽c′(‖u‖2−1+‖∂α∂t−Δ∂α∂t‖2),c>0. | (4.6) |
Summing then δ4 times (4.5) and (4.6), where δ4>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
dE5dt⩽cE5, |
satisfies
E5=δ4‖u‖2−1+‖α‖2+‖∇α‖2+‖u+∂α∂t−Δ∂α∂t‖2−1 | (4.8) |
It follows from (4.7)-(4.8) and Gronwall's lemma that
E5⩾c(‖u‖2−1+‖α‖2H1(Ω)+‖∂α∂t−Δ∂α∂t‖2),c>0. | (4.9) |
hence the uniquess, as well as the continuous dependence with respect to the initial data in H−1×H1×H1-norm.
All authors declare no conflicts of interest in this paper.
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