The subject of this paper is the solvability of the
quasilinear parabolic system
∂
∂tu =X
i,j=1,...,l ∇i(aij (x, t, u)∇ju) +
b(x, t, u, ∇u)
defined on Rl×[0, T ], l > 2 with elliptic matrix
a(x, t, u) and N-dimension singular vector
b.
The existence of solutions to boundary problems
for quasilinear parabolic equations has been intensively
studied during the last decades, see list of references [1
- 37]. These studies produce several universal meth-
ods such as methods of the fixed point introduced in
the works of Leray and Schauder [6], the perturbation
method, and the method of a priory estimation and
their combinations. In the linear case, fundamental re-
sults were obtained in the works of Nash, Degiorgi [8, 9],
Moser, and Aronson with further development produced
by Zhang, Qian, Xi [34 - 37], and many others. The
quasilinear case is less explored and presents great inter-
est due to the plethora of applications in signal process-
ing and quantum physics. Some foundational questions
were explored in [6] by Ladyzenskaja and Solonnikov.
The main progress in linear parabolic theory is the
extension to general linear parabolic equation contain-
ing lower order term Nash-Degiorgi results, so the con-
ditions on the coefficient, which guarantee certain reg-
ularity of solutions were formulated in terms of form-
boundary functions and Kato functional classes. In col-
loquial terms, a function f:Rl→R,f∈L2loc is
said to be form-bounded if there are positive constants
β, c (β) such that inequality
∥fφ∥2
2≤β∥∇φ∥2
2+c(β)∥φ∥2
2
holds for all φ∈C∞
0; the Kato class Kl
ν,ν > 0 consists
of all functions fsuch that
(λ−∆)−1|f|
∞≤ν. We
consider a simple linear equation
∂
∂t − ∇a∇+f· ∇u(x, t) = 0,(x, t)∈Rl×[0,∞)
then the heat kernel of this simple linear equation sat-
isfies two Gaussian bounds if the coefficient fis form-
bounded and diverges of fbelongs to some Kato class.
In the present article, we consider a more complex
case of the quasilinear parabolic system with singular co-
efficients given in the specific form (1), we establish suf-
ficient conditions for the initial-boundary value problem
u|Γ=ϕ|Γ,ϕ∈C2,1({(x, t) : x∈∂Ω, t ∈[0, T ]})
for quasilinear parabolic system (1) has a unique solu-
tion u ∈Hα, α
2(clos (DT)). The Heinz example
∂tu1−∂xxu1=u1∂xu12+∂xu22
∂tu2−∂xxu2=u2∂xu12+∂xu22,
with the solution u1= cos (mx) and u2= sin (mx) that
does not satisfy the condition max
[0,2π]|∇u|, this shows the
necessity of some additional growth conditions for a non-
linear system with an unknown vector in contrast to the
case of a single equation, in this work such conditions
are
∂aij
∂uk∇iuk+∂aij
∂xi
≤µ(|u|) (1 + |∇u|),
b
≤(ε(|u|) + θ(|∇u|,|u|)) (1 + |∇u|)2
with lim
|∇u|→∞θ(|∇u|,|u|) = 0 and ε(M1) is small
enough.
Aspects of Solvability Theory for Quasilinear Parabolic Systems in
Specific Form with the Singular Coefficients
MYKOLA YAREMENKO
National Technical University of Ukraine, Pryrichna 17G, 117, 04123, Kiev, UKRAINE
Abstract: In this paper, we study a quasilinear parabolic system in the form ∂tu = ∇i (αij (x, t, u) ∇ju) +
b (x, t, u, ∇u), where u (x, t) is an unknown N-dimensional vector over a domain DT = Ω × [0, T], we assume
the weak general conditions on the structural coefficients, demanding that the singular term satisfy the
formboundary conditions.
Keywords: Quasilinear Partial Differential Equation, Holder solution, regularity theory, form-bounded,
Parabolic equation, Weak Solution, a Priori Estimation.
Received: September 8, 2022. Revised: May 12, 2023. Accepted: Jane 7, 2023. Published: July 7, 2023.
1. Introduction
PROOF
DOI: 10.37394/232020.2023.3.2
Mykola Yaremenko
E-ISSN: 2732-9941
8
Volume 3, 2023
Consider a quasilinear parabolic system in the specific
divergent form
∂
∂tu =X
i,j=1,...,l ∇i(aij (x, t, u)∇ju) +
b(x, t, u, ∇u),
(1)
in the domain (x, t)∈DT= Ω ×[0, T ], u (x, t) =
u1(x, t), ..., uN(x, t)is an unknown N-dimensional
vector in clos (DT), l≥3;
b: Ω ×[0, T ]×RN×
Rl×RN→RNis a known vector-function. Functions
aij comprise a symmetric l×l-matrix uniformly elliptic,
namely,
ν(u)ξ2≤aij ξiξj≤µ(u)ξ2(2)
for all (x, t)∈Rl×[0, T ] and all ξ∈Rl.
We formulate the restrictions on the measurable
structural coefficients of the system (1) as
aij (x, t, u)
ki
kj≥ν(|u|)
k
2−γ0(x, t) (3)
aij (x, t, u)
kj
≤µ(|u|)
k
+γ1(x, t) (4)
bx, t, u,
k
≤˜µ(|u|)
k
2+γ2(x, t) (5)
where ν(τ), µ(τ) and ˜µ(τ) are given positive continuous
functions.
Definition 1. A function
f:DT→RNis
said to be form-boundary or belongs to the class
P K (β)if there exist some positive constants β
and c(β)such that the inequality
R[0, T ]RΩ
f φ
2
dxdt ≤
≤βR[0, T ]RΩ|∇φ|2dxdt +c(β)R[0, T ]RΩ|φ|2dxdt
(6)
holds all functions φ :Rl×[0, T ]→RN, φ ∈C∞
0.
Definition 2. We call real-valued vector-
function u (x, t)a weak bounded solution to sys-
tem (1) if u ∈V2
1,0(DT),ess max
(x, t)∈DT|u (x, t)|<∞and
RΩu (x, t)φ (x, t)dx
T
0=
=R[0, T ]RΩu∂tφdxdt−
−R[0, T ]RΩaij (x, t, u)∇ju∇iφdxdt+
+R[0, T ]RΩ
bφdxdt
(7)
for all φ ∈C∞
0.
We assume the function u is a weak bounded solution
to system (1) so that from (7), we obtain an integral
inequality
RΩu (x, t)φ (x, t)dx
T
0+
+R[0, T ]RΩaij (x, t, u)∇ju∇iφdxdt ≤
≤R[0, T ]RΩu∂tφdxdt
+R[0, T ]RΩ˜µ|∇u|2+γ2(x, t)|φ|dxdt,
(8)
where φ ∈C∞
0(DT)∩W2
1,2(DT).
We construct N1functions ϖm(x, t) =
ϕmu1(x, t), ..., uN(x, t), m = 1, ..., N1, where
ϕ1u1, ..., uN, . . . ., ϕN1u1, ..., uNare continuously
differentiable over their domains, and such that func-
tions ϖm(x, t) = ϕmu1(x, t), ..., uN(x, t), m =
1, ..., N1satisfy the following conditions:
1)ess max
(x, t)∈DT|ϖm(x, t)|< M1, ϖm∈V2
1,0(DT) ;
2) for an arbitrary cylinder D2ρ=B(2ρ)×
˜
t, ˜
t+τ⊂DTand a point t1∈˜
t, ˜
t+τthere is a
number ˜msuch that
osc {ϖm(x, t), D2ρ} ≥
≥δ1max
k=1,..., Nosc uk(x, t), D2ρ(9)
and
µ
x∈B(ρ) : ϖm(x, t1)≤
≤ess max
D2ρ{ϖm(x, t)}−
−δ2osc {ϖm(x, t), D2ρ}
≥
≥(1 −δ3)c(l)ρl,
(10)
where B(ρ) is a ball, of radius ρ, concentric with B(2ρ);
δ1, δ2, δ3are positive constants;
3) we denote the Lebesgue measure by µ; for each
function ϖm, m = 1, ..., N1, we have
max
t∈[˜
t, ˜
t+τ]∥ϖmn(·, t)∥2
2,B(ρ−ϑ1ρ)≤
≤max
t∈[˜
t, ˜
t+τ]
ϖmn·,˜
t
2
2,B(ρ)+
+˘c1
(ϑ1ρ)2
ϖmn·,˜
t
2
2,B(ρ)×[˜
t, ˜
t+τ]+ ˆc(µ(Λn,ρ)),
(11)
and
∥ϖmn(·, t)∥H, 2
2, B(ρ−ϑ1ρ)×[˜
t, ˜
t+ϑ2τ]≤
≤˘c1
(ϑ1ρ)2+1
(ϑ2τ)2
ωϖmn·,˜
t
2
2,B(ρ)×[˜
t, ˜
t+τ]+
+ˆc(µ(Λn,ρ)) ,
(12)
∥·∥His the Holder norm, we denote ϖmn(x, t) =
max {ϖm(x, t)−n, 0}and Λn,ρ is a set of all x∈
B(ρ) such that min
m=1,...,N1
ϖm(x)> n for natural number
n.
Since γ0, γ2
1, γ2∈P K (β) we obtain the estimation
ϖnx, ˜
t+τξx, ˜
t+τ
2
2,B(ρ)+
+ν∥ξ∇ϖn∥2
2, Dρ≤
≤
ϖnx, ˜
tξx, ˜
t
2
2,B(ρ)+
+˜cRDρ|ϖn|2|∇ξ|2+ξ|∂tξ|dxdt
+ˆc(µ(Λn,ρ)) ,
(13)
where we used
Z[0, T ]ZΩ|√γ2ξ|2dxdt ≤
≤βZ[0, T ]ZΩ|∇ξ|2dxdt +c(β)Z[0, T ]ZΩ|ξ|2dxdt,
where ξ∈C∞
0(DT).
2. Problem formalization
PROOF
DOI: 10.37394/232020.2023.3.2
Mykola Yaremenko
E-ISSN: 2732-9941
9
Volume 3, 2023
We denote
uh(x, t) = 1
hZ[t−h, t]
u (x, τ)dτ (14)
and
uh(x, t) = 1
hZ[t, t+h]
u (x, τ)dτ. (15)
By taking φ =u exp λ|uh|2ξ2(x)hin (8), then
taking the limit h→0, we obtain
1
2λRB(2ρ)exp λ|u|2ξ2(x)dx
t2
t1
+
+λν
2R[t1, t2]RB(2ρ)exp λ|u|2ξ2∇|u|22
dxdt+
+νR[t1, t2]RB(2ρ)exp λ|u|2ξ2|∇u|2dxdt ≤
≤λ
2R[t1, t2]RB(2ρ)exp λ|u|2ξ2γ0dxdt+
+R[t1, t2]RB(2ρ)exp λ|u|2ξ2γ0dxdt+
+2 R[t1, t2]RB(2ρ)exp λ|u|2
(µ|∇u|+γ1)|u|ξ|∇ξ|dxdt+
+R[t1, t2]RB(2ρ)˜µ|∇u|2+γ2|u|exp λ|u|2ξ2dxdt
(16)
and further we have
1
2λRB(2ρ)exp λ|u|2ξ2(x)dx
t2
t1
+
+λν
2R[t1, t2]RB(2ρ)exp λ|u|2ξ2∇|u|22
dxdt+
+νR[t1, t2]RB(2ρ)exp λ|u|2ξ2|∇u|2dxdt ≤
≤R[t1, t2]RB(2ρ)exp λM12
λ
2+ 1γ0+γ2
1+M1γ2ξ2dxdt+
+2µM1exp λM12R[t1, t2]RB(2ρ)|∇u|ξ|∇ξ|dxdt+
+˜µM1exp λM12R[t1, t2]RB(2ρ)|∇u|2ξ2dxdt ≤
≤µM1exp λM12R[t1, t2]RB(2ρ)|∇ξ|2dxdt+
+ (µ+ ˜µ)M1exp λM12R[t1, t2]RB(2ρ)|∇u|2ξ2dxdt+
+c1exp λM12βR[0, T ]RΩ|∇ξ|2dxdt+
+c(β)R[0, T ]RΩ|ξ|2dxdt.
Thus, we obtain
R[t1, t2]RB(2ρ)ξ2|∇u|2dxdt ≤c1ρl+
+c2(t2−t1)ρlmax |∇ξ|2+ max ξ2.
If η(x, t)≤1 and equals zero on the boundary then
max
t∈[˜
t−τ, ˜
t]RB(ρ)|ϖmnη|2dx+
+R[˜
t−τ, ˜
t]RB(2ρ)|∇ϖmn|2η2dxdt ≤
≤c1R[˜
t−τ, ˜
t]RB(2ρ)|ϖmn|2|∇η|2+η|∂tη|dxdt+
+c1R[0, T ]RΩ|∇ξ|2dxdt +R[0, T ]RΩ|ξ|2dxdt.
(17)
The Holder estimation of u follows from
osc {ϖm, Dρ} ≤
≤(1 −ϑ)osc {ϖm, D2ρ}+ϑ2ρϑ1,(18)
where constants ϑ, ϑ1, ϑ2are depending on the struc-
tural coefficients.
Proposition 1. Let function u ∈C2,1(DT)be
a solution to the system (1) and let function u
equals zero on the boundary. Let structural coef-
ficients of the system (1) satisfy conditions (2)-(5)
and
∂aij
∂uk∇iuk∇ju +∂aij
∂xi∇ju +
b
≤µ(|u|) (1 + |∇u|)2,
(19)
∂aij
∂uk∇iuk+∂aij
∂xi
≤µ(|u|) (1 + |∇u|),(20)
b
≤(ε(|u|) + θ(|∇u|,|u|)) (1 + |∇u|)2,(21)
where lim
|∇u|→∞θ(|∇u|,|u|) = 0 and ε(M1)
is a small number. Then, the value
max
{(x, t) : x∈∂Ω, t∈[0, T ]}|∇u (x, t)|estimates by
max
DT|u (x, t)|=M1and functions of structural
coefficients, max
Ω|∇u (x, 0)|and boundary.
Proof. We denote vk(x, t) = uk(x, t) + |u (x, t)|2,
then we equality
∂
∂t vk=aij ∇i∇jvk−
−X
i,j=1,..l X
m=1,...N
2aij ∇ium∇jum−Bj∇jvk−Ck,
where we denote Bj=∂aij
∂uk∇iuk+∂aij
∂xiand Ck=
Pm2bmum+bk.
We change the function v on the function vk=ψwk
where the function ψgiven by
ψ(z) = const ν (M1) ln (z+ 1) .
So, applying the standard arguments we obtain
∂
∂t wk−aij ∇i∇jwk−
−ψ′′
ψ′Pi,j=1,..l Pm=1,...N aij ∇iwm∇jwm≤
≤c1
ψ′+ψ′
∇wk
2
where the constant cis strictly positive. Therefore, there
are some positive constants ⌢
csuch that
∂
∂t wk−aij ∇i∇jwk≤⌢
c,
both functions w and v equal zero on the boundary
{(x, t) : x∈∂Ω, t ∈[0, T ]}and
∂wk
∂n
{(x, t) : x∈∂Ω, t∈[0, T ]}=
=1
ψ′(wk)
∂vk
∂n
{(x, t) : x∈∂Ω, t∈[0, T ]}=
=c
ν
∂uk
∂n
{(x, t) : x∈∂Ω, t∈[0, T ]}
reaches its maximum on {(x, t) : x∈∂Ω, t ∈[0, T ]}
in the same point ˆx, ˆ
tthat ∂uk
∂n .
3. A priori estimations
PROOF
DOI: 10.37394/232020.2023.3.2
Mykola Yaremenko
E-ISSN: 2732-9941
10
Volume 3, 2023
Next, we take the function ˜
ψsuch that
−aij ∇i∇j˜
ψ < −⌢
c
for all x∈Ω and
max
x∈Ωnwk(x, 0) + ˜
ψ(x)o=wk(ˆx, 0) + ˜
ψ(ˆx),
max
{(x, t) : x∈∂Ω, t∈[0, T ]}n˜
ψ(x)o=˜
ψ(ˆx).
Then, we have
∂
∂t wk+˜
ψ< aij ∇i∇jwk+˜
ψ
so that
max
x∈Ωnwk(x, 0) + ˜
ψ(x)o=wk(ˆx, 0) + ˜
ψ(ˆx) = ˜
ψ(ˆx),
thus
∂wk(x, t) + ˜
ψ(x)
∂n
x=ˆx≥0
therefore, there is an estimation
∂˜
ψ(x)
∂n
x=ˆx≥ − ∂wk(x, t)
∂n
(x, t)=(ˆx, ˆ
t)
,
finally, we obtain the value
max
k=1,...,N max
{(x, t) : x∈∂Ω, t∈[0, T ]}
∂uk
∂n
is bounded.
Theorem 1. Let functions aij and
bsatisfy con-
ditions (2)-(5) and
∂aij
∂uk∇iuk∇ju +∂aij
∂xi∇ju +
b
≤µ(|u|) (1 + |∇u|)2,
(22)
∂aij
∂uk∇iuk+∂aij
∂xi
≤µ(|u|) (1 + |∇u|),(23)
b
≤(ε(|u|) + θ(|∇u|,|u|)) (1 + |∇u|)2,(24)
where lim
|∇u|→∞θ(|∇u|,|u|)=0and ε(M1)is a small
number. Let the boundary be smooth enough.
Then, the value max
DT|∇u (x, t)|=M1can be esti-
mated by functions of structural coefficients and
ε(M1),θ.
Proof. In inequality (8), we take φ =
u exp λ|uh|2ξ2(x)hand proceed as (16) we obtain
an estimation R[, T ]RΩξ2|∇u|2dxdt ≤const; next, we
take φ =∇mξ∇mu2, we have
1
2R[0, T ]RΩξ∂tPk=1,...,N Pm=1,...,l ∇muk2dxdt =
=−R[0, T ]RΩξaij ∇m∇iuk∇m∇jukdxdt−
−1
2R[0, T ]RΩaij
∇jξ∇iPk=1,...,N Pm=1,...,l ∇muk2dxdt−
−R[0, T ]RΩ
daij
dxmξ∇iuk∇m∇jukdxdt−
−R[0, T ]RΩ
daij
dxm∇jξ∇iuk∇mukdxdt+
+R[0, T ]RΩbk∆ukξ− ∇muk∇mξdxdt,
(25)
where we denote ∇muk∇mξ=
Pm=1,...,l (∇mξ)∇muk. We take
ξ= 2
X
k=1,...,N X
m=1,...,l ∇muk2
s
η2(x)
where ηis the cutoff for the ball B(ρ)⊂Ω, s ≥0.
Applying conditions, for small ρ, we have
1
2+sRB(ρ)Pk=1,...,N Pm=1,...,l ∇muk2s+1
η2dx
T
0+
+νˆc1R[0, T ]RB(ρ)|∇∇u|2
Pk=1,...,N Pm=1,...,l ∇muk2s
η2dxdt+
+νˆc1R[0, T ]RB(ρ)Pk=1,...,N Pm=1,...,l ∇muk2s+2
η2dxdt ≤
≤˘cR[0, T ]RB(ρ)
1 + Pk=1,...,N Pm=1,...,l ∇muk2s+1|∇η|2dxdt,
therefore, we obtain
max
t∈[0, T ]RB(ρ)Pk=1,...,N Pm=1,...,l ∇muk2s+1
dx ≤
≤c.
Finally, if φ (x, t) =
∇mξ(x, t), ξ|{(x, t) : x∈∂Ω, t∈[0, T ]}= 0 then
we obtain
R[0, T ]RΩξ∇m∂tukdxdt =
=−R[0, T ]RΩaij ∇iξ∇m∇jukdxdt+
+R[0, T ]RΩΘk
mi∇iξdxdt,
where Θk
mi =∂aij
∂xm∇juk+∂aij
∂ud∇mud∇juk+bkδim. We
denote w=∇muk, m = 1, ..., l;k= 1, ..., N a solu-
tion to the system
∂tw=∇jaij ∇iw+ Θk
mi,
applying linear theory, we are proving the theorem.
Remark. If the cylinder intersects the boundary
then we assume that max
∂DT|∇u (x, t)|has already been
estimated. For a domain that intersects we take in (25)
ξ=
2Pk=1,...,N Pm=1,...,l ∇muk2s
η2(x)
for Pk=1,...,N Pm=1,...,l ∇muk2> M2
1;
0,Pk=1,...,N Pm=1,...,l ∇muk2≤M2
1,
so that we are going to obtain the following estimation
max
t∈[0, T ]ZB(ρ)∩Ω
X
k=1,...,N X
m=1,...,l ∇muk2
s+1
dx ≤
≤c.
PROOF
DOI: 10.37394/232020.2023.3.2
Mykola Yaremenko
E-ISSN: 2732-9941
11
Volume 3, 2023
Existence of the solution
We consider a boundary problem for the system (1),
namely, the function u that satisfies
∂
∂tu =X
i,j=1,...,l ∇i(aij (x, t, u)∇ju) +
b(x, t, u, ∇u)
over the closure of DTand on the boundary coincides
with the given function ϕ∈H˜α, ˜α
2(clos (DT)).
Theorem 2. Let functions aij and
bsatisfy
conditions (2)-(5) with γ0, γ2
1, γ2∈P K (β)
and (22)-(24) with lim
|∇u|→∞θ(|∇u|,|u|)=0
and ε(M1)is a small number. Let
ϕ∈C2,1({(x, t) : x∈∂Ω, t ∈[0, T ]}),
max
x∈Ω|∇ϕ(x, 0)|<∞,ϕ∈H˜α, ˜α
2(clos (DT)); and
let
∂
∂t ϕ=X
i,j=1,...,l ∇i(aij (x, t, ϕ)∇jϕ) +
b(x, t, ϕ, ∇ϕ).
Let function aij satisfy the Lipschitz condition at
u on any compact.
Then, there exists a unique solution u ∈
Hα, α
2(clos (DT)) to the problem
u|{(x, t) : x∈∂Ω, t∈[0, T ]}∩{(x, t) : x∈Ω, t=0}=
=ϕ|{(x, t) : x∈∂Ω, t∈[0, T ]}∩{(x, t) : x∈Ω, t=0}
for the system (1).
This theorem can be proven by the Leray–Schauder
method with the application estimations obtained in pre-
vious chapters. A linearized system is given as
∂
∂t w = (τaij (x, t, v) + (1 −τ)δij )∇i∇jw+
−τB (x, t, v, ∇v) + (1 −τ)∂
∂t ϕ−∆ϕ, τ ∈[0,1] ,
where we denote
B(x, t, v, ∇v) =
=−
b(x, t, v, ∇v)−∂aij (x, t, v)
∂vk∇jv∇ivk−
−∂aij (x, t, v)
∂xi∇jv,
and we consider function w to be unknown and v to be
given.
The linearized system defines the nonlinear operator
Φ (τ) : v 7→ w given by w = Φ (v, τ ), where the func-
tion w is a solution to the linearized system for each given
parameter τ∈[0,1]. The fixed point of the operator Φ
at the point τ= 1 is a solution to the boundary problem
for system (1). The existence of such a fixed point is
guaranteed by the Leray–Schauder theorem, uniqueness
follows from the Lipschitz condition straightforwardly by
the contradiction method.
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