Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 106, pp. 1–26.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXPONENTIAL DECAY AND BLOW-UP FOR NONLINEAR

HEAT EQUATIONS WITH VISCOELASTIC TERMS AND

ROBIN-DIRICHLET CONDITIONS

LE THI PHUONG NGOC, NGUYEN THANH LONG

Abstract. In this article, we consider a system of nonlinear heat equations

with viscoelastic terms and Robin-Dirichlet conditions. First, we prove exis-
tence and uniqueness of a weak solution. Next, we prove a blow up result of

weak solutions with negative initial energy. Also, we give a sufficient condition

that guarantees the existence and exponential decay of global weak solutions.
The main tools are the Faedo-Galerkin method, a Lyapunov functional, and a

suitable energy functional.

1. Introduction

In this article, we consider the system of nonlinear heat equations containing
viscoelastic terms

∂ui
∂t
− ∂

∂x

(
µi(x, t)

∂ui
∂x

)
+

∫ t

0

gi(t− s)
∂

∂x

(
µ̄i(x, s)

∂ui
∂x

(x, s)
)
ds

= fi(u1, . . . , uN ) + Fi(x, t),

(1.1)

where 0 < x < 1, t > 0, 1 ≤ i ≤ N , with N ∈ N and N ≥ 2, associated with
boundary conditions

∂u1

∂x
(0, t)− h0u1(0, t) = u1(1, t) = 0,

u2(0, t) =
∂u2

∂x
(1, t) + h1u2(1, t) = 0,

ui(0, t) = ui(1, t) = 0, 3 ≤ i ≤ N,

(1.2)

and initial conditions

ui(x, 0) = ũi(x), 1 ≤ i ≤ N, (1.3)

where h0 ≥ 0, h1 ≥ 0 are real numbers and µi, gi, µi, fi, Fi, ũi for all i ∈ 1, N are
given functions satisfying conditions specified later.

System (1.1) arises naturally within frameworks of mathematical models in engi-
neering and physical sciences, which have been studied by many authors and several
results concerning existence, nonexistence, regularity, exponential decay, blow-up
in finite time and asymptotic behavior have been established, see [4, 5, 6] and
references therein.

2010 Mathematics Subject Classification. 34B60, 35K55, 35Q72, 80A30.

Key words and phrases. Nonlinear heat equations; blow up; exponential decay.
c©2020 Texas State University.

Submitted November 20, 2019. Published October 26, 2020.

1



2 L. T. P. NGOC, N. T. LONG EJDE-2020/106

Messaoudi [5] considered an initial boundary value problem related to equation

ut −∆u−
∫ t

0

g(t− s)∆u(x, s)ds = |u|p−2u,

and proved a blow-up result for certain solutions with positive initial energy, under
suitable conditions on g and p. In [6], the authors considered a quasilinear parabolic
system of the form

A(t)|ut|p−2ut −∆u−
∫ t

0

g(t− s)∆u(x, s)ds = 0,

for m ≥ 2, p ≥ 2, A(t) a bounded and positive definite matrix, and g a continuously
differentiable decaying function, and proved that, under suitable conditions on g
and p, a general decay of the energy function for the global solution and a blow-up
result for the solution with both positive and negative initial energy.

Long, Y, and Ngoc [4] considered a nonlinear heat equation with a viscoelastic
term

ut −
∂

∂x
(µ1(x, t)ux) +

∫ t

0

g(t− s) ∂
∂x

(µ2(x, s)ux(x, s))ds = f(u) + F (x, t),

where (x, t) ∈ (0, 1)× (0, T ), with Robin boundary conditions

ux(0, t)− h0u(0, t) = g0(t), ux(1, t) + h1u(1, t) = g1(t),

and the initial condition

u(x, 0) = u0(x),

where h0 ≥ 0, h1 ≥ 0 are real numbers with h0 + h1 > 0, and µ1, g, µ2, f , F , g0,
g1, u0 are given functions, under suitable conditions on µ1, g, µ2, f , F , g0, g1, u0,
a exponential decay of the energy function for the global solution and a blow-up
result for the solution have been established.

Motivated by the above mentioned works, we study the blow-up and exponen-
tial decay estimates for problem (1.1)-(1.3). This article is organized as follows. In
Section 2, we present some preliminaries and notations. In Section 3, by applying
the Faedo-Galerkin method and the weak compact method, we establish the ex-
istence of a unique weak solution u of (1.1)-(1.3) on (0, T ), for every T > 0. In
Sections 4 and 5, problem (1.1)-(1.3) is considered with µi(x, t) ≡ µi(x), for all
i ∈ 1, N . In the case of Fi ≡ 0, for all i ∈ 1, N , when some auxiliary conditions
are satisfied, we prove that the weak solution u blows up in finite time. In the case
of ‖Fi(t)‖ small enough, for all i ∈ 1, N , we verify that if the initial energy is also
small enough, then the energy of the solution decays exponentially as t → +∞.
For the proof of the blow up result, we divide it into two steps. First, we show
that the weak solution obtained here is not a global solution in R+. Second, we
prove that this solution blows up at finite time T∞, where [0, T∞) is a maximal
interval on which the solution of (1.1)-(1.3) exists. For the proof of exponential
decay result, a Lyapunov functional is constructed via defining a suitable energy
functional. The results obtained here is a relative generalization of [4, 7, 8], by
improving and developing these previous works, essentially.



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 3

2. Preliminary results and notation

First, we put Ω = (0, 1), QT = Ω× (0, T ), T > 0, and denote the usual function
spaces used throughout the paper by the notation

Lp = Lp(Ω), W k,p = W k,p(Ω), Hk = W k,2, ∀k ∈ Z+, 1 ≤ p ≤ ∞.

We denote the usual norm in L2 by ‖ · ‖ and we denote ‖ · ‖X for the norm in the
Banach space X. We will use the notation 〈·, ·〉 for either the scalar product in L2

or the dual pairing of a continuous linear functional and an element of a function
space. We call X ′ the dual space of X. We denote Lp(0, T ;X), 1 ≤ p ≤ ∞,
the Banach space of measurable functions u : (0, T ) → X measurable such that
‖u‖Lp(0,T ;X) < +∞, with

‖u‖Lp(0,T ;X) =


(∫ T

0
‖u(t)‖pXdt

)1/p
< +∞, if 1 ≤ p <∞,

ess sup ‖u(t)‖X , if p =∞.

On H1, we use the norm

‖v‖H1 =

√
‖v‖2 + ‖vx‖2, ∀v ∈ H1.

We define

V1 = {v ∈ H1 : v(1) = 0},
V2 = {v ∈ H1 : v(0) = 0},

Vi = H1
0 = {v ∈ H1 : v(0) = v(1) = 0}, i = 3, N,

it is clear that V1, . . . , VN are closed subspaces of H1. Moreover, we have the
following standard lemmas concerning the imbeddings of H1 into C0(Ω) and of Vi
into C0(Ω), and the equivalence between two norms, v 7→ ‖vx‖, v 7→ ‖v‖H1 , on Vi
for all i ∈ 1, N .

Lemma 2.1. The imbedding H1 ↪→ C0(Ω) is compact, and

‖v‖C0(Ω) ≤
√

2‖v‖H1 , ∀v ∈ H1.

Lemma 2.2. For all i ∈ 1, N , the imbedding Vi ↪→ C0(Ω) is compact. Moreover,
we have

‖v‖C0(Ω) ≤ ‖v‖Vi
, ∀v ∈ Vi,

1√
2
‖v‖H1 ≤ ‖vx‖ ≤ ‖v‖H1 ∀v ∈ Vi.

Let µi, µi ∈ C0(Ω× [0, T ]) with µi(x, t) ≥ µi∗ > 0 and µi(x, t) ≥ µi∗ > 0 for
all (x, t) ∈ Ω × [0, T ] and for all i ∈ 1, N . We consider the families of symmetric



4 L. T. P. NGOC, N. T. LONG EJDE-2020/106

bilinear forms {ai(t; ·, ·)}t∈[0,T ], {a′i(t; ·, ·)}t∈[0,T ], {ai(t; ·, ·)}t∈[0,T ] defined by

a1(t;u, v) = 〈µ1(t)ux, vx〉+ h0µ1(0, t)u(0)v(0),

a′1(t;u, v) = 〈µ′1(t)ux, vx〉+ h0µ
′
1(0, t)u(0)v(0),

a1(t;u, v) = 〈µ1(t)ux, vx〉+ h0µ1(0, t)u(0)v(0), ∀u, v ∈ V1, t ∈ [0, T ];

a2(t;u, v) = 〈µ2(t)ux, vx〉+ h1µ2(1, t)u(1)v(1),

a′2(t;u, v) = 〈µ′2(t)ux, vx〉+ h1µ
′
2(1, t)u(1)v(1),

a2(t;u, v) = 〈µ2(t)ux, vx〉+ h1µ2(1, t)u(1)v(1), ∀u, v ∈ V2, t ∈ [0, T ];

ai(t;u, v) = 〈µi(t)ux, vx〉,
a′i(t;u, v) = 〈µ′i(t)ux, vx〉,

ai(t;u, v) = 〈µi(t)ux, vx〉, ∀u, v ∈ Vi, t ∈ [0, T ], i = 3, N.

(2.1)

Then we have the following lemma, whose proof is straightforward so we omit.

Lemma 2.3. Let µi, µi ∈ C0(Ω× [0, T ]) with µi(x, t) ≥ µi∗ > 0 and µi(x, t) ≥
µi∗ > 0 for all (x, t) ∈ Ω× [0, T ], i ∈ 1, N ; and h0 ≥ 0, h1 ≥ 0. Then, the families
of symmetric bilinear forms {ai(t; ·, ·)}t∈[0,T ], {ai(t; ·, ·)}t∈[0,T ] defined by (2.1) are

continuous on Vi × Vi and coercive in Vi for all i ∈ 1, N .

Moreover, there exist aT > 0, a0 > 0 such that

|ai(t;u, v)| ≤ aT ‖ux‖‖vx‖, ai(t;u, v)| ≤ aT ‖ux‖‖vx‖,

for all u, v ∈ Vi, t ∈ [0, T ], i ∈ 1, N ; and

ai(t; v, v) ≥ a0‖vx‖2, ai(t; v, v) ≥ a0‖vx‖2, ∀v ∈ Vi, t ∈ [0, T ], i ∈ 1, N.

We also have two important lemmas.

Lemma 2.4. Let f ∈ C0(RN ;R), if we set

Φf (r) =

{
sup|x|2≤r |f(x)|, if r > 0,

|f(0)|, if r = 0,

then Φf ∈ C0(R+;R+) is nondecreasing and

|f(x)| ≤ Φf (|x|2), ∀x ∈ RN ,

where |x|2 =
√
x2

1 + · · ·+ x2
N for all x ∈ RN .

Proof. With r > 0, we denote

Br = {x ∈ RN : |x|2 < r}, B̄r = {x ∈ RN : |x|2 ≤ r}.
Let g ∈ C0(RN ;R+), we set

ϕg(r) =

{
sup|x|2≤r g(x), if r > 0,

g(0), if r = 0.

We claim that ϕg ∈ C0(R+;R+). It is clear that ϕg(r) ≥ 0 for all r ∈ R+ and ϕg
is nondecreasing in R+.

(i) We prove that ϕg is continuous from right at 0. For all ε > 0, by g ∈
C0(RN ;R+), there exists δ > 0 such that

|g(x)− g(0)| < ε, ∀x ∈ B̄δ. (2.2)



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 5

From (2.2), we have

g(x) < g(0) + ε = ϕg(0) + ε, ∀x ∈ B̄δ. (2.3)

By the definition of ϕg and (2.3), it follows that

ϕg(0) ≤ ϕg(r) ≤ ϕg(δ) ≤ ϕg(0) + ε, ∀r ∈ [0, δ].

Therefore ϕg is continuous from right at 0.
(ii) For all r0 > 0. We will prove that ϕg is continuous at r0.
(ii-1) We prove that ϕg is continuous from left at r0. At first, we define a function

ϕg, with ϕg(r) = sup|x|2<r g(x) for all r > 0. Easily to see that ϕg(r) ≤ ϕg(r) for

all r > 0. We prove that ϕg(r) ≥ ϕg(r) for all r > 0.
Fixed r > 0, by the definition of ϕg, we can assume that

ϕg(r) = sup
|x|2<r

g(x) = max
|x|2<r

g(x) = g(x0),

where x0 ∈ B̄r. We define the sequence {xn} by xn = (1− 1
n )x0. We will have

{xn} ⊂ Br and xn → x0. By the definition of ϕg and continuity of g, we obtain

ϕg(r) ≥ lim
n→+∞

g(xn) = g(x0) = ϕg(r).

It is clear that ϕg is nondecreasing in R+. For all ε > 0, by the definition of ϕg,
there exists x0 ∈ Br0 such that

ϕg(r0)− ε < g(x0) ≤ ϕg(r0). (2.4)

Put δ = r0 − |x0|2 > 0, for all r ∈ (r0 − δ, r0], we have

ϕg(r0)− ε < g(x0) ≤ ϕg(|x0|2) = ϕg(|x0|2) ≤ ϕg(r) ≤ ϕg(r0). (2.5)

From (2.5), it follows that

ϕg(r0)− ε < ϕg(r) ≤ ϕg(r0), ∀r ∈ (r0 − δ, r0]. (2.6)

Therefore ϕg is continuous from left at r0.
(ii-2) We prove that ϕg is continuous from right at r0. By g ∈ C0(RN ;R+), we

have g is uniform continuous on B̄2r0 . For all ε > 0, there exists δ ∈ (0, r02 ) such
that

|g(x)− g(y)| < ε, ∀x, y ∈ B̄2r0 , |x− y|2 < δ. (2.7)

For all r ∈ [r0, r0 + δ), by the definition of ϕg, there exists xr ∈ B̄r, yr = r0
r xr ∈ B̄r0

such that ϕg(r) = g(xr) and

|g(xr)− g(yr)| < ε. (2.8)

From (2.8), we have

ϕg(r0) ≤ ϕg(r) = g(xr) < g(yr) + ε ≤ ϕg(|yr|2) + ε ≤ ϕg(r0) + ε, (2.9)

for all r ∈ [r0, r0 + δ). Therefore ϕg is continuous from right at r0. Finally, with
f ∈ C0(RN ;R), we have

Φf (r) = ϕ|f |(r), ∀r ∈ R+.

The fact |f | ∈ C0(RN ;R+) leads to Φf ∈ C0(R+;R+). For all x ∈ RN , we have

|f(x)| ≤ ϕ|f |(|x|2) = Φf (|x|2).

Obviously, Φf is nondecreasing. The proof is complete. �

Lemma 2.4 is a slight improvement of a result used in [7, Appendix 1, pp. 2734],
with N = 1 and f ∈ C0(R;R).



6 L. T. P. NGOC, N. T. LONG EJDE-2020/106

Lemma 2.5. Let x : [0, T ]→ R+ be a continuous function satisfying the inequality

x(t) ≤M +

∫ t

0

k(s)ω(x(s))ds, ∀t ∈ [0, T ],

where M ≥ 0, k : [0, T ] → R+ is continuous and ω : R+ → (0,+∞) is continuous
and nondecreasing. Set

Ψ(u) =

∫ u

0

dy

ω(y)
, u ≥ 0.

(i) If
∫ +∞

0
dy
ω(y) = +∞, then

x(t) ≤ Ψ−1
(

Ψ(M) +

∫ t

0

k(s)ds
)
, ∀t ∈ [0, T ].

(ii) If
∫ +∞

0
dy
ω(y) < +∞, then there exists T∗ ∈ (0, T ] such that∫ T∗

0

k(s)ds ≤
∫ +∞

0

dy

ω(y)
,

x(t) ≤ Ψ−1
(

Ψ(M) +

∫ t

0

k(s)ds
)
, ∀t ∈ [0, T∗].

For a proof of the above lemma, see [1].

3. Existence and uniqueness of a weak solution to (1.1)-(1.3)

Definition 3.1. A weak solution to (1.1)-(1.3) is a function ~u = (u1, . . . , uN )
belonging to the functional space

W (T ) = {~u ∈ L∞(0, T ;V ) :
∂~u

∂t
∈ L2(0, T ;H)}, (3.1)

satisfying the variational problem

〈u′i(t), vi〉+ ai(t;ui(t), vi)−
∫ t

0

gi(t− s)ai(s;ui(s), vi)ds

= 〈fi(~u(t)), vi〉+ 〈Fi(t), vi〉, ∀vi ∈ Vi, i ∈ 1, N,

(3.2)

and the initial condition

ui(0) = ũi, ∀i ∈ 1, N, (3.3)

where

V = V1 × · · · × VN , H = (L2)N . (3.4)

We make the following assumptions:

(A1) h0, h1 ≥ 0;
(A2) ũi ∈ Vi for all i ∈ 1, N ;
(A3) µi ∈ C1(Ω× [0, T ]) such that µi(x, t) ≥ µi∗ > 0 for all (x, t) ∈ Ω × [0, T ],

i ∈ 1, N ;
(A4) µi ∈ C0(Ω× [0, T ]) such that µi(x, t) ≥ µi∗ > 0 for all (x, t) ∈ Ω × [0, T ],

i ∈ 1, N ;
(A5) fi ∈ C0(RN ) for all i ∈ 1, N ;
(A6) gi ∈ H1(0, T ) for all i ∈ 1, N ;
(A7) Fi ∈ L2(QT ) for i = 1, N .

Theorem 3.2. Let T > 0 and (A1)–(A7) hold.



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 7

(i) If ∫ +∞

0

dy

1 + y +
∑N
i=1 Φ2

fi
(
√
y)
,= +∞

then (1.1)-(1.3) has a global weak solution ~u ∈W (T ) satisfying (3.2)-(3.3).
(ii) If ∫ +∞

0

dy

1 + y +
∑N
i=1 Φ2

fi
(
√
y)

< +∞

then (1.1)-(1.3) has a local weak solution ~u ∈W (T∗) satisfying (3.2))-(3.3)
with a certain T∗ small enough.

In addition if

(A5*) For all M > 0, there exists LM > 0 such that

|fi(x)− fi(y)| ≤ LM |x− y|2, ∀x, y ∈ RN , i ∈ 1, N,

then the solution is unique.

Proof. It consists of four steps.

Step 1: Faedo-Galerkin approximation (introduced by Lions [3]). Let {w(j)
i }j∈N

be a denumerable base of Vi for i = 1, N . We find an approximate solution of
(1.1)-(1.3) in the form

u
(m)
i (t) =

m∑
j=1

c
(mj)
i (t)w

(j)
i , ∀i ∈ 1, N, (3.5)

where the coefficient functions c
(mj)
i , 1 ≤ j ≤ m, i ∈ 1, N , satisfy the system of

ordinary differential equations

〈u̇(m)
i (t), w

(j)
i 〉+ ai(t;u

(m)
i (t), w

(j)
i )−

∫ t

0

gi(t− s)ai(s;u(m)
i (s), w

(j)
i )ds

= 〈fi(~u(m)(t)), w
(j)
i 〉+ 〈Fi(t), w(j)

i 〉, j = 1,m, i ∈ 1, N,

(3.6)

and the initial conditions

u
(m)
i (0) = ũ

(0m)
i , ∀i ∈ 1, N, (3.7)

with

ũ
(0m)
i =

m∑
j=1

α
(mj)
i w

(j)
i → ũi strongly in Vi for i ∈ 1, N. (3.8)

By the above assumptions, we can prove the existence of a solution ~u(m) =

(u
(m)
1 , . . . , u

(m)
N ) for the system (3.6)-(3.8) on the interval [0, Tm], for some Tm ∈

(0, T ]. The proofs are straightforward, so we omit the details.



8 L. T. P. NGOC, N. T. LONG EJDE-2020/106

Step 2: A priori estimates. Taking (w
(j)
1 , . . . , w

(j)
N ) = (u̇

(m)
1 (t), . . . , u̇

(m)
N (t)) in

(3.6), and summing over i from 1 to N , we obtain

N∑
i=1

‖u̇(m)
i (t)‖

2
+

N∑
i=1

ai(t;u
(m)
i (t), u̇

(m)
i (t))

−
N∑
i=1

∫ t

0

gi(t− s)ai(s;u(m)
i (s), u̇

(m)
i (t))ds

=

N∑
i=1

〈fi(~u(m)(t)), u̇
(m)
i (t)〉+

N∑
i=1

〈Fi(t), u̇(m)
i (t)〉.

(3.9)

First, through a direct calculation, we have

d

dt
ai(t, u

(m)
i (t), u

(m)
i (t))

= 2ai(t;u
(m)
i (t), u

(m)
i (t)) + a′i(t;u

(m)
i (t), u

(m)
i (t)),

(3.10)

d

dt

∫ t

0

gi(t− s)ai(s;u(m)
i (s), u

(m)
i (t))ds

= gi(0)ai(t;u
(m)
i (t), u

(m)
i (t)) +

∫ t

0

g′i(t− s)ai(s;u
(m)
i (s), u

(m)
i (t))ds

+

∫ t

0

gi(t− s)ai(s;u(m)
i (s), u̇

(m)
i (t))ds,

(3.11)

∀i ∈ 1, N .
Hence, (3.9) can be rewritten as

2

N∑
i=1

‖u̇(m)
i (t)‖

2
+
d

dt

N∑
i=1

ai(t;u
(m)
i (t), u

(m)
i (t))

=

N∑
i=1

[a′i(t;u
(m)
i (t), u

(m)
i (t)) + 2

d

dt

∫ t

0

gi(t− s)ai(s;u(m)
i (s), u

(m)
i (t))ds]

− 2

N∑
i=1

gi(0)ai(t;u
(m)
i (t), u

(m)
i (t))

− 2

N∑
i=1

∫ t

0

g′i(t− s)ai(s;u
(m)
i (s), u

(m)
i (t))ds

+ 2

N∑
i=1

〈fi(~u(m)(t)), u̇
(m)
i (t)〉+ 2

N∑
i=1

〈Fi(t), u̇(m)
i (t)〉.

(3.12)



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 9

Next, integrating (3.12), we obtain

Sm(t) = Sm(0) +

N∑
i=1

∫ t

0

a′i(s;u
(m)
i (s), u

(m)
i (s))ds

+ 2

N∑
i=1

∫ t

0

gi(t− s)ai(s;u(m)
i (s), u

(m)
i (s))ds

− 2

N∑
i=1

gi(0)

∫ t

0

ai(s;u
(m)
i (s), u

(m)
i (s))ds

− 2

N∑
i=1

∫ t

0

ds

∫ s

0

g′i(s− τ)ai(τ ;u
(m)
i (τ), u

(m)
i (s))dτ

+ 2

N∑
i=1

∫ t

0

〈fi(~u(m)(s)), u̇
(m)
i (s)〉ds+ 2

N∑
i=1

∫ t

0

〈Fi(s), u̇(m)
i (s)〉ds

= Sm(0) +

6∑
k=1

Jk,

(3.13)

where

Sm(t) =

N∑
i=1

(
2

∫ t

0

‖u̇(m)
i (s)‖

2
ds+ ai(t;u

(m)
i (t), u

(m)
i (t))

)
. (3.14)

By (A1)–(A7), and using Lemmas 2.3 and 2.4, we estimate the terms on both
sides of (3.13) as follows. At first, we note that

Sm(t) ≥
N∑
i=1

ai(t;u
(m)
i (t), u

(m)
i (t)) ≥ a0

N∑
i=1

‖u(m)
ix (t)‖

2
. (3.15)

Now we estimate the terms Jk on the right-hand side of (3.13) as follows. First
term, J1:

J1 = h0

∫ t

0

µ′1(0, s)|u(m)
1 (0, s)|

2
ds+ h1

∫ t

0

µ′2(1, s)|u(m)
2 (1, s)|

2
ds

+

N∑
i=1

∫ t

0

〈µ′i(s)u
(m)
ix (s), u

(m)
ix (s)〉ds

= J
(1)
1 + J

(2)
1 + J

(3)
1 ,

(3.16)

in which

J
(1)
1 = h0

∫ t

0

µ′1(0, s)|u(m)
1 (0, s)|

2
ds

≤ h0‖µ′1‖C0(Ω×[0,T ])

∫ t

0

‖u(m)
1x (s)‖

2
ds

≤ h0

a0
‖µ′1‖C0(Ω×[0,T ])

∫ t

0

Sm(s)ds.

(3.17)

Using the same techniques, with appropriate modifications, leads to

J
(2)
1 ≤ h1

a0
‖µ′2‖C0(Ω×[0,T ])

∫ t

0

Sm(s)ds. (3.18)



10 L. T. P. NGOC, N. T. LONG EJDE-2020/106

Using the Cauchy-Schwarz inequality gives

J
(3)
1 =

N∑
i=1

∫ t

0

〈µ′i(s)u
(m)
ix (s), u

(m)
ix (s)〉ds

≤ max
1≤i≤N

‖µ′i‖C0(Ω×[0,T ])

∫ t

0

N∑
i=1

‖u(m)
ix (s)‖

2
ds

≤ 1

a0
max

1≤i≤N
‖µ′i‖C0(Ω×[0,T ])

∫ t

0

Sm(s)ds.

(3.19)

From (3.16)–(3.19), we have

J1 ≤ C1

∫ t

0

Sm(s)ds, (3.20)

where

C1 =
1

a0

(
h0‖µ′1‖C0(Ω×[0,T ]) + h1‖µ′2‖C0(Ω×[0,T ]) + max

1≤i≤N
‖µ′i‖C0(Ω×[0,T ])

)
. (3.21)

Second term, J2. By the Cauchy-Schwarz inequality, we obtain

J2 = 2

N∑
i=1

∫ t

0

gi(t− s)ai(s;u(m)
i (s), u

(m)
i (t))ds

≤ 2

N∑
i=1

∫ t

0

|gi(t− s)||ai(s;u(m)
i (s), u

(m)
i (t))|ds

≤ 2

N∑
i=1

‖u(m)
ix (t)‖aT ‖gi‖L∞(0,T )

∫ t

0

‖u(m)
ix (s)‖ds

≤
N∑
i=1

[1

6
a0‖u(m)

ix (t)‖
2

+
a2
T ‖gi‖

2
L∞(0,T )

6a0

(∫ t

0

‖u(m)
ix (s)‖ds

)2]
≤ 1

6
Sm(t) +

1

6a2
0

Ta2
T max

1≤i≤N
‖gi‖2L∞(0,T )

∫ t

0

Sm(s)ds.

(3.22)

Third term, J3. It is clear that

J3 = −2

N∑
i=1

gi(0)

∫ t

0

ai(s;u
(m)
i (s), u

(m)
i (s))ds

≤ 2aT

N∑
i=1

|gi(0)|
∫ t

0

‖u(m)
ix (s)‖

2
ds ≤2aT

a0
max

1≤i≤N
|gi(0)|

∫ t

0

Sm(s)ds;

(3.23)

Fourth term, J4.

J4 = −2

N∑
i=1

∫ t

0

ds

∫ s

0

g′i(s− τ)ai(τ ;u
(m)
i (τ), u

(m)
i (s))dτ

≤ 2aT

N∑
i=1

∫ t

0

‖u(m)
ix (s)‖ds

∫ s

0

|g′i(s− τ)|‖u(m)
ix (τ)‖dτ

= 2aT
√
T

N∑
i=1

‖g′i‖L2(0,T )

∫ t

0

‖u(m)
ix (s)‖

2
ds



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 11

≤ 2aT
a0

√
T max

1≤i≤N
‖g′i‖L2(0,T )

∫ t

0

Sm(s)ds.

Fifth term, J5. It is known that

|~u(m)(x, t)|2 =

√√√√ N∑
i=1

|u(m)
i (x, t)|

2
≤

√√√√ N∑
i=1

‖u(m)
ix (t)‖2 ≤

√
Sm(t)
√
a0

.

By Lemma 2.4, we have

|fi(~u(m)(x, t))| ≤ Φfi

(
|~u(m)(x, t)|2

)
≤ Φfi(

1
√
a0

√
Sm(t)), ∀i ∈ 1, N,

so

‖fi(~u(m)(t))‖ ≤ Φfi

( 1
√
a0

√
Sm(t)

)
, ∀i ∈ 1, N ;

therefore,

J5 = 2

N∑
i=1

∫ t

0

〈fi(~u(m)(s)), u̇
(m)
i (s)〉ds

≤
N∑
i=1

∫ t

0

[3‖fi(~u(m)(s))‖
2

+
1

3
‖u̇(m)

i (s)‖
2
]ds

≤ 3

N∑
i=1

∫ t

0

Φ2
fi(

1
√
a0

√
Sm(s))ds+

1

3

N∑
i=1

∫ t

0

‖u̇(m)
i (s)‖

2
ds

≤ 1

6
Sm(t) + 3

N∑
i=1

∫ t

0

Φ2
fi(

1
√
a0

√
Sm(s))ds.

(3.24)

Sixth term, J6. We have

J6 = 2

N∑
i=1

∫ t

0

〈Fi(s), u̇(m)
i (s)〉ds ≤ 1

6
Sm(t) + 3

N∑
i=1

‖Fi‖2L2(QT ). (3.25)

Now we estimate the term Sm(0). From the convergence in (3.8), we can deduce
the existence of a constant C0 > 0 such that

Sm(0) =

N∑
i=1

ai
(
0; ũ

(0m)
i , ũ

(0m)
i

)
≤ C0, ∀m ∈ N. (3.26)

From (3.13), (3.20), (3.22)-(3.26), there exist MT > 0, NT > 0 constants indepen-
dent of m such that

Sm(t) ≤MT +NT

∫ t

0

ω(Sm(s))ds, ∀t ∈ [0, T ], (3.27)

with

ω(S) = 1 + S +

N∑
i=1

Φ2
fi

( 1
√
a0

√
S
)
. (3.28)

By the same convergence of
∫ +∞

0
dy
ω(y) and

∫ +∞
0

dy

1+y+
∑N

i=1 Φ2
fi

(
√
y)

, apply Lemma

2.5 with x(t) ≡ Sm(t), M = MT , k(s) ≡ NT , ω(S) = 1 + S +
∑N
i=1 Φ2

fi
( 1√

a0

√
S),

we obtain the estimate of Sm(t) in two cases as follows.



12 L. T. P. NGOC, N. T. LONG EJDE-2020/106

Case 1. If ∫ +∞

0

dy

1 + y +
∑N
i=1 Φ2

fi
(
√
y)

= +∞

then
Sm(t) ≤ Ψ−1(Ψ(MT ) +NT t)

≤ Ψ−1(Ψ(MT ) +NTT ) ≡ CT , ∀t ∈ [0, T ], m ∈ N.
(3.29)

Case 2. If ∫ +∞

0

dy

1 + y +
∑N
i=1 Φ2

fi
(
√
y)

< +∞

then
Sm(t) ≤ Ψ−1(Ψ(MT ) +NT t)

≤ Ψ−1(Ψ(MT ) +NTT ) ≡ CT , ∀t ∈ [0, T∗], m ∈ N,
(3.30)

where T∗ ∈ (0, T ] chosen such that T∗NT ≤
∫ +∞

0
dy
ω(y) .

This allows one to take the constant Tm = T or Tm = T∗ for all m ∈ N. In what
follows, we will write T∗ for both T and T∗.

Step 3: Limiting process. It follows from (3.14), (3.15) and (3.29) (or (3.30)), that

‖u(m)
i ‖L∞(0,T∗;Vi) ≤

√
CT
a0

, ‖u̇(m)
i ‖L2(QT ) ≤

√
CT , ∀m ∈ N, ∀i ∈ 1, N. (3.31)

Applying the Banach-Alaoglu theorem and Kakuntani theorem, the above uniform
bounds with respect to m imply that one can extract a subsequence (which we
relabel with the index m if necessary) such that

~u(m) → ~u weak* in L∞(0, T∗;V ), (3.32)

∂~u(m)

∂t
→ ∂~u

∂t
weakly in L2(0, T∗;H). (3.33)

By Aubin-Lions compactness theorem and Riesz-Fisher theorem, it is straight-
forward to go on extracting, from weak convergence results (3.32) and (3.33), a
subsequence (which we relabel with the index m if necessary) such that

~u(m) → ~u strongly in L2(0, T∗;H),

~u(m)(x, t)→ ~u(x, t) a.e. (x, t) ∈ QT∗ .
(3.34)

It remains to show the convergence of the nonlinear terms. Using the continuity
argument of fi for all i ∈ 1, N and (3.34), one deduces that

fi(~u
(m)(x, t))→ fi(~u(x, t)) a.e. (x, t) ∈ QT∗ , ∀i ∈ 1, N. (3.35)

On the other hand,

‖fi(~u(m))‖L2(QT∗ ) ≤
√
T sup
|z|≤

√
CT
a0

|fi(z)|, ∀i ∈ 1, N.

From [3, Lemma 1.3] we obtain

fi(~u
(m))→ fi(~u) weakly in L2(QT∗), ∀i ∈ 1, N. (3.36)

Combining (3.32), (3.33), (3.36) and (3.8), it is enough to pass to the limit in (3.6)
and (3.7) to show that ~u satisfies (3.2) and (3.3). In addition, from (3.32) and
(3.33), we have ~u ∈ W (T∗) and the proof of the existence of a weak solution is
complete.



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 13

Step 4: Uniqueness of the solution. Suppose ~u(1) and ~u(2) are two solutions of
(1.1)-(1.3) on the interval [0, T∗] such that

~u(i) ∈W (T∗), i = 1, 2. (3.37)

Then ~u = ~u(1) − ~u(2) = (u1, . . . , uN ) ∈W (T∗) satisfies

〈u′i(t), vi〉+ ai(t;ui(t), vi)−
∫ t

0

gi(t− s)ai(s;ui(s), vi)ds

= 〈fi(~u(1)(t))− fi(~u(2)(t)), vi〉, ∀v ∈ Vi, i ∈ 1, N,

(3.38)

ui(0) = 0, ∀i ∈ 1, N. (3.39)

Taking vi = 2ui(t) in (3.38) and integrating with respect to t, and summing over i
from 1 to N , we obtain

N∑
i=1

‖ui(t)‖2 + 2

∫ t

0

N∑
i=1

ai(s;ui(s), ui(s))ds

= 2

N∑
i=1

∫ t

0

ds

∫ s

0

ai(τ ;ui(τ), ui(s))dτ

+ 2

N∑
i=1

∫ t

0

〈fi(~u(1)(s))− fi(~u(2)(s)), ui(s)〉ds.

(3.40)

Set %(t) =
∑N
i=1

(
‖ui(t)‖2 +

∫ t
0
‖uix(s)‖2ds

)
. As in Step 2, we can estimate all

terms on the right hand side of (3.40) to obtain

%(t) ≤ DT

∫ t

0

%(s)ds, ∀t ∈ [0, T∗], (3.41)

where DT > 0. Applying Gronwall’s lemma, (3.41) leads to %(t) ≡ 0; i.e., ~u(1) =
~u(2). Theorem 3.2 is proved. �

Lemma 2.4 is a powerful and efficient tool for estimate the nonlinear terms.
By Lemma 2.4, we can relax assumptions for fi ∈ C0(RN ) for all i ∈ 1, N , that
is, fi can be bounded by the polynomial of |~u|2 for all i ∈ 1, N or not. It is an
improvement of the assumptions in [8], here the authors had to suppose that f is
bounded by the polynomial of |u| for the initial boundary problem for a nonlinear
heat equation ut− ∂

∂x (µ(x, t)ux)+f(u) = f1(x, t), 0 < x < 1, 0 < t < T , associated
with Robin boundary conditions.

4. Blow-up of solutions

In this section we study the blow up in finite time of the solution of (1.1)-(1.3)
corresponding to µi(x, t) ≡ µi(x) and Fi(x, t) ≡ 0 for all i ∈ 1, N ,

∂ui
∂t
− ∂

∂x

(
µi(x, t)

∂ui
∂x

)
+

∫ t

0

gi(t− s)
∂

∂x

(
µi(x)

∂ui
∂x

(x, s)
)
ds

= fi(u1, . . . , uN ), (x, t) ∈ QT , ∀i ∈ 1, N,

(4.1)



14 L. T. P. NGOC, N. T. LONG EJDE-2020/106

with boundary conditions

∂u1

∂x
(0, t)− h0u1(0, t) = u1(1, t) = 0,

u2(0, t) =
∂u2

∂x
(1, t) + h1u2(1, t) = 0,

ui(0, t) = ui(1, t) = 0, 3 ≤ i ≤ N,

(4.2)

and initial conditions

ui(x, 0) = ũi(x), ∀i ∈ 1, N. (4.3)

We make the following assumptions:

(A3’) µi ∈ C1(Ω× R+) such that µi(x, t) ≥ µi∗ > 0, ∂µi

∂t (x, t) ≤ 0 for all (x, t) ∈
Ω× R+, i ∈ 1, N ;

(A4’) µi ∈ C0(Ω) such that µi(x) ≥ µi∗ > 0 for all x ∈ Ω, i = 1, N ;
(A5’) fi ∈ C0(RN ) for all i ∈ 1, N . Furthermore, there exists F ∈ C1(RN ) such

that
(i) ∂F

∂ui
= fi for all i ∈ 1, N ,

(i) There exists constant d1 > 2 such that d1F(~u) ≤
∑N
i=1 uifi(~u), for all

~u = (u1, . . . , uN ) ∈ RN ,
(iii) There exist constants d1 > 0, pi > 2 for all i ∈ 1, N , such that F(~u) ≥

d1

∑N
i=1 |ui|

pi , for all ~u = (u1, . . . , uN ) ∈ RN ;
(A6’) gi ∈ C1(R+;R+) ∩ L1(R+) such that 0 < gi(t) ≤ gi(0) and g′i(t) ≤ 0 for all

t ≥ 0, i ∈ 1, N .

Example 4.1. For ~u = (u1, . . . , uN ) ∈ RN , we define a function that satisfies
(A5’).

F(~u) = F(u1, . . . , uN ) =

N∑
i=1

αi|ui|pi + β|u1|q1 . . . |uN |qN ln
k
(e+|~u|22),

where β > 0, k > 1 and αi > 0, pi > 2, qi > 2 for all i ∈ 1, N are constants. By
direct calculations, we have

fi(~u) =
∂F
∂ui

(~u)

= piαi|ui|pi−2ui + βqi|u1|q1 . . . |uN |qNu−1
i ln

k
(e+|~u|22)

+ 2kβ|u1|q1 . . . |uN |qN
ui

e+|~u|22
lnk−1(e+|~u|22), ∀i ∈ 1, N.

It is obvious that (A5’) holds, since

N∑
i=1

uifi(~u) =

N∑
i=1

piαi|ui|pi + β(

N∑
i=1

qi)|u1|q1 . . . |uN |qN ln
k
(e+|~u|22)

+ 2kβ|u1|q1 . . . |uN |qN
|~u|22
e+|~u|22

lnk−1(e+|~u|22)

≥
N∑
i=1

piαi|ui|pi + β(

N∑
i=1

qi)|u1|q1 . . . |uN |qN ln
k
(e+|~u|22)

≥ d1F(~u),



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 15

with d1 = min{p1, . . . , pN ,
∑N
i=1 qi} and

F(~u) ≥
N∑
i=1

αi|ui|pi ≥ d1

N∑
i=1

|ui|pi , d1 = min
1≤i≤N

αi.

Now, on Vi × Vi, we consider the following symmetric bilinear forms:

A1(u, v) =

∫ 1

0

ux(x)vx(x)dx+ h0u(0)v(0),

a1(u, v) =

∫ 1

0

µ1(x)ux(x)vx(x)dx+ h0µ1(0)u(0)v(0), ∀u, v ∈ V1;

A2(u, v) =

∫ 1

0

ux(x)vx(x) + h1u(1)v(1),

a2(u, v) =

∫ 1

0

µ2(x)ux(x)vx(x)dx+ h1µ2(1)u(1)v(1), ∀u, v ∈ V2;

Ai(u, v) =

∫ 1

0

ux(v)vx(x)dx,

ai(u, v) =

∫ 1

0

µi(x)ux(x)vx(x)dx, ∀u, v ∈ Vi, i = 3, N.

It is easy to show that the forms Ai(·, ·), ai(·, ·) are continuous on Vi × Vi and
coercive on Vi for all i ∈ 1, N . On the other hand, the norm v 7→ ‖vx‖ and the

norms v 7→ ‖v‖Ai =
√
Ai(v, v) and v 7→ ‖v‖ai =

√
ai(v, v) are equivalent.

Lemma 4.2. There exist positive constants µ∗, µ
∗, µ∗, µ

∗ such that:

(i) Ai(v, v) ≥ ‖vx‖2, for all v ∈ Vi, i ∈ 1, N ,
(ii) |Ai(u, v)| ≤ (1 + max{h0, h1})‖ux‖‖vx‖, for all u, v ∈ Vi, i = 1, N ,

(iii) ai(v, v) ≥ µ∗‖v‖2Ai
, for all v ∈ Vi, i ∈ 1, N ,

(iv) |ai(u, v)| ≤ µ∗‖u‖Ai
‖v‖Ai

, for all v ∈ Vi, i = 1, N ,
(v) ai(t; v, v) ≥ µ∗‖v‖2Ai

, for all v ∈ Vi, i ∈ 1, N ,

(vi) |ai(t;u, v)| ≤ µ∗‖u‖Ai‖v‖Ai , for all v ∈ Vi, i ∈ 1, N ,
(vii) a′i(t, v, v) ≤ 0, for all u, v ∈ Vi, t ≥ 0, i ∈ 1, N .

Lemma 4.3. For i ∈ 1, N , on Vi, the norms v 7→ ‖v‖Ai
=
√
Ai(v, v) and v 7→

‖v‖ai =
√
ai(v, v) are equivalent and√

µ∗‖v‖Ai ≤ ‖v‖ai ≤
√
µ∗‖v‖Ai

, ∀v ∈ Vi.

Now we define the modified energy functional related to (4.1)-(4.3),

E(t) =
1

2

N∑
i=1

[(gi ? ui)(t) +ai(t;ui(t), ui(t))− g̃i(t)‖ui‖2ai ]−
∫ 1

0

F(~u(x, t))dx, (4.4)

where

(gi ? ui)(t) =

∫ t

0

gi(t− s)‖ui(s)− ui(t)‖2aids, g̃i(t) =

∫ t

0

gi(s)ds, (4.5)



16 L. T. P. NGOC, N. T. LONG EJDE-2020/106

for all i ∈ 1, N . By multiplying (4.1) by u′i(t), and integrating over Ω, and summing
over i from 1 to N , we obtain

E′(t) =

N∑
i=1

[−‖u′i(t)‖
2
+

1

2
a′i(t;ui(t), ui(t))−

1

2
gi(t)‖ui‖2ai +

1

2
(g′i ? ui)(t)] ≤ 0, (4.6)

for any regular solution. The same result can be established for weak solutions and
for almost every t, by a denseness argument.

Theorem 4.4. Let assumptions (A1), (A3’)–(A6’), (A5*) hold. If

max
1≤i≤N

‖gi‖L1(R+) <
µ∗
µ∗

(
1− 1

(d1 − 1)
2

)
,

then for all (ũ1, . . . , ũN ) ∈ V such that E(0) < 0, we have:

(i) If p1 = · · · = pN , then the weak solution u of (4.1)-(4.3) blows up in finite
time.

(ii) If there exist i, j = 1, N , i 6= j such that pi 6= pj and
∑N
i=1 ‖ũi‖

2 ≥
41+1/pN , with p = min1≤i≤N pi, then the weak solution u of (4.1)-(4.3)
blows up in finite time.

Proof. It consists of two steps.
Step1. First, we prove that

Problem (4.1)-(4.3) has no global weak solution. (4.7)

Indeed, by contradiction we assume that

~u ∈W (R+) = {~u ∈ L∞loc(R+;V ) ∩ C(R+;H) :
∂~u

∂t
∈ L2

loc(R+;H)},

is a global weak solution of (4.1)-(4.3). We define

H(t) = −E(t), t ≥ 0. (4.8)

Then it follows from (4.6) that H′(t) ≥ 0 for all t ≥ 0. This implies that

H(t) ≥ H(0) = −E(0) > 0, ∀t ≥ 0. (4.9)

Set

L1(t) =
1

2

N∑
i=1

‖ui(t)‖2. (4.10)

By taking the time derivative of (4.10) and using (4.1), we obtain

L′1(t) =

N∑
i=1

(〈fi(~u(t)), ui(t)〉 − ai(t;ui(t), ui(t)) +

∫ t

0

gi(t− s)ai(ui(s), ui(t))ds).

Hence

L′1(t) ≥
N∑
i=1

(〈fi(~u(t)), ui(t)〉 − ai(t;ui(t), ui(t))) +

N∑
i=1

g̃i(t)‖ui(t)‖2ai

−
N∑
i=1

∫ t

0

gi(t− s)|ai(ui(s)− ui(t), ui(t))|ds.

(4.11)



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 17

By using the Schwarz inequality and Young inequality, for all δ1 > 0, we obtain

N∑
i=1

∫ t

0

gi(t− s)|ai(ui(s)− ui(t), ui(t))|ds

≤
N∑
i=1

[
1

2δ1
g̃i(t)‖ui(t)‖2ai +

δ1
2

(gi ? ui)(t)].

(4.12)

From (4.4) and (4.9), we obtain∫ 1

0

F(~u(x, t))dx ≥ 1

2

N∑
i=1

[(gi ? ui)(t) + ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai ]. (4.13)

Since ϕ(x) = µ∗
µ∗ (1− 1

(x−1)2
) is continuous and nondecreasing on (2, d1), it follows

that

0 = ϕ(2) < max
1≤i≤N

‖gi‖L1(R+) <
µ∗
µ∗

(
1− 1

(d1 − 1)
2

)
= ϕ(d1).

Then there exists a unique constant p̂ ∈ (2, d1) such that

max
1≤i≤N

‖gi‖L1(R+) = ϕ(p̂). (4.14)

Set δ1 = p̂ and δ2 = δ1
d1

. From (4.11)-(4.14) we deduce that

L′1(t) ≥ (1− δ2)

N∑
i=1

〈fi(~u(t)), ui(t)〉+ δ2

N∑
i=1

〈fi(~u(t)), ui(t)〉

−
N∑
i=1

ai(t;ui(t), ui(t)) +
(
1− 1

2δ1

) N∑
i=1

g̃i(t)‖ui(t)‖2ai −
δ1
2

N∑
i=1

(gi ? ui)(t)

≥ (1− δ2)

N∑
i=1

〈fi(~u(t)), ui(t)〉+
δ2d1

2

N∑
i=1

(gi ? ui)(t)

+
δ2d1

2

N∑
i=1

ai(t;ui(t), ui(t))

− δ2d1

2

N∑
i=1

g̃i(t)‖ui(t)‖2ai −
N∑
i=1

ai(t;ui(t), ui(t))

+
(
1− 1

2δ1

) N∑
i=1

g̃i(t)‖ui(t)‖2ai −
δ1
2

N∑
i=1

(gi ? ui)(t)

=
(
1− p̂

d1

) N∑
i=1

〈fi(~u(t)), ui(t)〉+
( p̂

2
− 1
) N∑
i=1

ai(t;ui(t), ui(t))

− (p̂− 1)2

2p̂

N∑
i=1

g̃i(t)‖ui(t)‖2ai

≥
(
1− p̂

d1

) N∑
i=1

〈fi(~u(t)), ui(t)〉+
( p̂

2
− 1
)µ∗
µ∗

N∑
i=1

‖ui(t)‖2ai



18 L. T. P. NGOC, N. T. LONG EJDE-2020/106

− (p̂− 1)2

2p̂
max

1≤i≤N
‖gi‖L1(R+)

N∑
i=1

‖ui(t)‖2ai

=
(
1− p̂

d1

) N∑
i=1

〈fi(~u(t)), ui(t)〉

+
(p̂− 1)2

2p̂
(ϕ(p̂)− max

1≤i≤N
‖gi‖L1(R+))

N∑
i=1

‖ui(t)‖2ai

=
(
1− p̂

d1

) N∑
i=1

〈fi(~u(t)), ui(t)〉

≥ (1− δ1
d1

)d1

N∑
i=1

‖ui(t)‖piLpi

≥ (1− δ1
d1

)d1

N∑
i=1

‖ui(t)‖pi ≡ θ
N∑
i=1

‖ui(t)‖pi .

We consider the following cases:

(i) If p1 = · · · = pN = p, then from the inequality (
∑N
i=1 yi)

α≤ Nα−1∑N
i=1 y

α
i ,

for all α ≥ 1, y1, . . . , yN ≥ 0, we obtain

L′1(t) ≥ θ
N∑
i=1

‖ui(t)‖p ≥ N1− p
2 θ
( N∑
i=1

‖ui(t)‖2
)p/2

≥ Nθ

(
√

2N)p
Lp/21 (t) ≡ θ1Lp/21 (t).

(4.15)

A direct integration of (4.15) yields

L
p
2−1
1 (t) ≥ 2

(p− 2)θ1(T∗ − t)
, ∀t ∈ [0, T∗),

with T∗ = 2
(p−2)θ1

L1− p
2

1 (0). Therefore, limt→T−∗ L1(t) = +∞. This is a contradic-

tion with ~u ∈ C([0, T∗];H). Thus, (4.7) holds.
(ii) If there exist i, j ∈ 1, N , i 6= j such that pi 6= pj . We put p = min1≤i≤N pi,

using the inequality xp ≤ xpi + 1, for all x ≥ 0, i ∈ 1, N , we obtain

L′1(t) ≥ θ
N∑
i=1

‖ui(t)‖pi ≥θ(
N∑
i=1

‖ui(t)‖p −N)

≥ θ[N1− p
2 (

N∑
i=1

‖ui(t)‖2)
p
2 −N ] =

Nθ

(
√

2N)p
(Lp/21 (t)− (

√
2N)p).

(4.16)

From L′1(t) ≥ 0, for all t ≥ 0, we have L1(t) ≥ L1(0) = 1
2

∑N
i=1 ‖ũi‖

2
, for all t ≥ 0.

It follows from
∑N
i=1 ‖ũi‖

2 ≥ 41+1/pN , that 1
2L

p/2
1 (t) ≥ 1

2L
p/2
1 (0) ≥ (

√
2N)p, for

all t ≥ 0. Therefore, from (4.16) we deduce that

L′1(t) ≥ Nθ

(
√

2N)p
(
1

2
Lp/21 (t) +

1

2
Lp/21 (t)− (

√
2N)p)

≥ Nθ

2(
√

2N)p
L

p
2
1 (t) ≡ θ2L

p
2
1 (t), ∀t ≥ 0.

(4.17)



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 19

A direct integration of (4.17) gives

L
p
2−1
1 (t) ≥ 2

(p− 2)θ2(T∗ − t)
, ∀t ∈ [0, T∗),

with T∗ = 2
(p−2)θ2

L1− p
2

1 (0).

Therefore limt→T−∗ L1(t) = +∞. This is a contradiction with ~u ∈ C([0, T∗];H).

Thus, (4.7) holds.

Step 2. Next, we put

T∞ = sup{T > 0 : (4.1)-(4.3) has a unique solution ~u ∈W (T )}.
By (4.7), we have T∞ < +∞. We now prove that

lim
t→T−∞

‖~ux(t)‖ = +∞. (4.18)

Indeed, assume that (4.18) is not true, then there exists a constant M > 0 and a
sequence {tm} with {tm} ⊂ (0, T∞), tm → T∞ such that

‖~ux(tm)‖2 ≤M, ∀m ∈ N.

Following the argument as above, for each m ∈ N, there exists a unique weak
solution

~u∗ ∈ {~u ∈ L∞(tm, tm + η;V ) ∩ C([tm, tm + η];H) :
∂~u

∂t
∈ L2(tm, tm + η;H)}

of (4.1)-(4.3) with the initial data

~u∗(tm) = ~u(tm),

with η > 0 independent of m ∈ N. By tm → T∞, we can get tm + η > T∞ for
m ∈ N sufficiently large. It is clear that the function

~U(t) =

{
~u(t), 0 ≤ t ≤ tm,
~u∗(t), tm ≤ t ≤ tm + η,

is a weak solution of (4.1)-(4.3) on [0, tm+η], tm+η > T∞, we obtain a contradiction
to the maximality of T∞. Thus, (4.18) holds. Theorem 4.4 is proved . �

5. Exponential decay of solutions

In this section, we study the global solution of (1.1)-(1.3), corresponding to
µi(x, t) ≡ µi(x) as in Section 4. We shall make suitable and necessary assumptions,
for which the solution obtained here decays exponentially, these assumptions are
as follows.

(A5”) fi ∈ C0(RN ) for all i ∈ 1, N . Furthermore, there exists F ∈ C1(RN ;R)
such that

(i) ∂F
∂ui

= fi for all i ∈ 1, N ,

(ii) There exists a nondecreasing function G : R+ → R+ such that

lim
z→0+

G(z) = 0, F(~u) ≤ G(|~u|2)|~u|22, ∀~u ∈ RN ,

(iii) There exists a constant d2 > 2 such that d2F(~u) ≥
∑N
i=1 uifi(~u), for

all ~u ∈ RN ;
(A6”) gi ∈ C1(R+,R+) ∩ L1(R+) such that

(i) 0 < gi(t) ≤ gi(0), g′i(t) ≤ 0 for all t ≥ 0, i ∈ 1, N ,



20 L. T. P. NGOC, N. T. LONG EJDE-2020/106

(ii) L ≡ µ∗ − µ∗ max
1≤i≤N

‖gi‖L1(R+) > 0,

(iii) There exist constants ξi > 0 for all i ∈ 1, N such that

g′i(t) ≤ −ξigi(t), ∀t ≥ 0, i ∈ 1, N ;

(A7”) Fi ∈ L2(R+;L2) and there exist constants Ci > 0, γi > 0 for all i ∈ 1, N
such that

‖Fi(t)‖ ≤ Ci exp(−γit), ∀t ≥ 0, i ∈ 1, N.

Example 5.1. We note that the function F given in Example 4.1 also satisfies
(A5”). Indeed, we have

F(~u) ≤
N∑
i=1

αi|~u|pi2 + β|~u|q1+···+qN
2 lnk(e+|~u|22) ≤ G(|~u|2)|~u|22, ∀~u ∈ RN ,

where G(z) =
∑N
i=1 αiz

pi−2 + βzq1+···+qN−2lnk(e+ z2) → 0 as z → 0+. From
Example 4.1, we have

N∑
i=1

uifi(~u) =

N∑
i=1

piαi|ui|pi + β
( N∑
i=1

qi

)
|u1|q1 . . . |uN |qN ln

k
(e+|~u|22)

+ 2kβ|u1|q1 . . . |uN |qN
|~u|22
e+|~u|22

lnk−1(e+|~u|22)

≤
N∑
i=1

piαi|ui|pi + β
( N∑
i=1

qi + 2k
)
|u1|q1 . . . |uN |qN ln

k
(e+|~u|22)

≤ d2F(~u),

where d2 = max{p1, . . . , pN , 2k +
∑N
i=1 qi} > 2. Hence, (A5”) holds.

Now, for δ > 0 to be chosen later, we define

L(t) = E(t) +
δ

2

N∑
i=1

‖ui(t)‖2 = E(t) + δL1(t), (5.1)

where

E(t) =
1

2

N∑
i=1

((gi ? ui)(t) + ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai)−
∫ 1

0

F(~u(x, t))dx.

With p ∈ (2, d2), we can rewrite the energy functional E(t) as follows

E(t) = (
1

2
− 1

p
)

N∑
i=1

[(gi ? ui)(t) + ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai ] +
1

p
I(t),

where

I(t) = I(~u(t))

=

N∑
i=1

[
(gi ? ui)(t) + ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai

]
− p

∫ 1

0

F(~u(x, t))dx.



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 21

Lemma 5.2. Assume that (A1), (A3’), (A4’), (A5”)–(A7”) hold. Then

E′(t) ≤ −(1− ε1

2
)

N∑
i=1

‖u′i(t)‖
2 − 1

2

N∑
i=1

ξi(gi ? ui)(t) +
1

2ε1

N∑
i=1

‖Fi(t)‖2, (5.2)

for all ε1 > 0.

.

Proof. Multiplying ith equation in (1.1) by u′i, and integrating over Ω, and summing
over i from 1 to N ; we obtain

E′(t) = −
N∑
i=1

‖u′i(t)‖
2

+
1

2

N∑
i=1

a′i(t;ui(t), ui(t))−
1

2

N∑
i=1

gi(t)‖ui(t)‖2ai

+
1

2

N∑
i=1

(g′i ? ui)(t) +

N∑
i=1

〈Fi(t), u′i(t)〉,

(5.3)

for any regular solution u. We can extend (5.3) to weak solutions by using denseness
arguments.

On the other hand, we have

N∑
i=1

〈Fi(t), u′i(t)〉 ≤
ε1

2

N∑
i=1

‖u′i(t)‖
2

+
1

2ε1

N∑
i=1

‖Fi(t)‖2,

1

2

N∑
i=1

(g′i ? ui)(t) ≤ −
1

2

N∑
i=1

ξi(gi ? ui)(t),

1

2

N∑
i=1

a′i(t;ui(t), ui(t)) ≤ 0.

(5.4)

From (5.3) and (5.4), we obtain (5.2). Lemma 5.2 is proved. �

Lemma 5.3. Assume that (A1), (A3’), (A4’), (A5”)–(A7”) hold. Suppose I(0) > 0
and

R∗ =

√√√√ 2p

(p− 2)L
(E(0) +

1

2

N∑
i=1

‖Fi‖2L2(R+;L2)) is small enough (5.5)

such that

η∗ = L− pG(R∗) >
d2 − p
d2

µ∗. (5.6)

Then I(t) > 0 for all t ≥ 0.

Proof. By the continuity of I(t) and I(0) > 0, there exists T1 > 0 such that

I(t) = I(~u(t)) > 0, ∀t ∈ [0, T1]. (5.7)

This gives

E(t) ≥
(1

2
− 1

p

) N∑
i=1

(
ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai

)
≥
(1

2
− 1

p

)(
µ∗ − µ∗ max

1≤i≤N
‖gi‖L1(R+)

) N∑
i=1

‖ui(t)‖2Ai



22 L. T. P. NGOC, N. T. LONG EJDE-2020/106

=
(p− 2)L

2p

N∑
i=1

‖ui(t)‖2Ai
, ∀t ∈ [0, T1].

Hence
N∑
i=1

‖ui(t)‖2Ai
≤ 2p

(p− 2)L
E(t), ∀t ∈ [0, T1]. (5.8)

From (5.2) with ε1 = 1 and (5.8), we obtain

N∑
i=1

‖ui(t)‖2Ai
≤ 2p

(p− 2)L
E(t)

≤ 2p

(p− 2)L

(
E(0) +

1

2

N∑
i=1

‖Fi‖2L2(R+;L2)

)
≡ R2

∗,

(5.9)

for all t ∈ [0, T1]; so

|~u(x, t)|2 =

√√√√ N∑
i=1

|ui(x, t)|2 ≤

√√√√ N∑
i=1

‖uix(t)‖2 ≤

√√√√ N∑
i=1

‖ui(t)‖2Ai
≤ R∗.

By (A5”), we have∫ 1

0

F(~u(x, t))dx ≤
∫ 1

0

G(|~u(x, t)|2)|~u(x, t)|22dx ≤ G(R∗)

N∑
i=1

‖ui(t)‖2Ai
. (5.10)

Consequently

I(~u(t)) =

N∑
i=1

[(gi ? ui)(t) + ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai ]− p
∫ 1

0

F(~u(x, t))dx

≥
N∑
i=1

(gi ? ui)(t) + (L− pG(R∗))

N∑
i=1

‖ui(t)‖2Ai

≡
N∑
i=1

((gi ? ui)(t) + η∗‖ui(t)‖2Ai
) > 0, ∀t ∈ [0, T1].

Now, we set T∞ = sup{T > 0 : I(t) > 0, ∀t ∈ [0, T ]}. If T∞ < +∞, then by the
continuity of I(t), we have I(T∞) ≥ 0.

In the case I(T∞) > 0, by the same arguments as above, we can deduce that
there exists T ′∞ > T∞ such that I(t) > 0, for all t ∈ [0, T ′∞]. We obtain a
contradiction to the definition of T∞.

In the case I(T∞) = 0, it follows that

0 = I(T∞) ≥
N∑
i=1

(
(gi ? ui)(T∞) + η∗‖ui(T∞)‖2Ai

)
≥ 0.

Therefore

ui(T∞) = (gi ? ui)(T∞) = 0, ∀i ∈ 1, N.

From the fact that g(T∞ − s) > 0, for all s ∈ [0, T∞], we have

(gi ? ui)(T∞) =

∫ T∞

0

gi(T∞−s)‖ui(s)‖2aids = 0,



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 23

it follows that ‖ui(s)‖ ≤ ‖ui(s)‖ai = 0, a.e. s ∈ [0, T∞]. By ui ∈ C([0, T∞];L2), we
deduce that ui(s) = 0, for all s ∈ [0, T∞], i.e. ui(0) = 0. This leads to I(0) = 0. We
get in contradiction with I(0) > 0. Consequently, T∞ = +∞, i.e. I(t) > 0, for all
t ≥ 0. This completes the proof. �

Lemma 5.4. Let I(0) > 0 and (5.5), (5.6) hold. Set

E1(t) =

N∑
i=1

(
(gi ? ui)(t) + ‖ui(t)‖2Ai

)
+ I(t). (5.11)

Then there exist positive constants β1, β2 such that

β1E1(t) ≤ L(t) ≤ β2E1(t), ∀t ≥ 0. (5.12)

Proof. It is not difficult to see that

L(t) =
(1

2
− 1

p

) N∑
i=1

(
(gi ? ui)(t) + ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai

)
+
I(t)

p
+ δL1(t)

≥ p− 2

2p

N∑
i=1

(
(gi ? ui)(t) + L‖ui(t)‖2Ai

)
+
I(t)

p
≥ β1E1(t),

where β1 = min{p−2
2p ,

(p−2)L
2p , 1

p}. Similarly,

L(t) ≤ p− 2

2p

N∑
i=1

((gi ? ui)(t) + µ∗‖ui(t)‖2Ai
) +

I(t)

p
+
δ

2

N∑
i=1

‖ui(t)‖2Ai

=
p− 2

2p

N∑
i=1

(gi ? ui)(t) +
( (p− 2)µ∗

2p
+
δ

2

) N∑
i=1

‖ui(t)‖2Ai
+
I(t)

p
≤ β2E1(t),

where β2 = max{p−2
2p ,

(p−2)µ∗

2p + δ
2 ,

1
p}. The proof is complete. �

Lemma 5.5. Suppose I(0) > 0 (5.5), (5.6) hold. Then

L′1(t) ≤
( 1

2ε2
+
d2

p

) N∑
i=1

(gi ? ui)(t)−
ε3d2

p
I(t) +

1

2ε2

N∑
i=1

‖Fi(t)‖2

−
(d2

p
− 1− ε2

2

) N∑
i=1

g̃i(t)‖ui(t)‖2ai

−
[ (1− ε3)d2η∗

p
− (

d2

p
− 1)µ∗ − ε2

2

] N∑
i=1

‖ui(t)‖2Ai
,

(5.13)

for all ε2 > 0, ε3 ∈ (0, 1).



24 L. T. P. NGOC, N. T. LONG EJDE-2020/106

Proof. Multiplying the ith equation in (1.1) by u′i, and integrating over Ω, and
summing over i from 1 to N , we obtain

L′1(t) = −
N∑
i=1

ai(t;ui(t), ui(t)) +

N∑
i=1

g̃i(t)‖ui(t)‖2ai +

N∑
i=1

〈fi(~u(t)), ui(t)〉

+

N∑
i=1

〈Fi(t), ui(t)〉+

N∑
i=1

∫ t

0

gi(t− s)ai(ui(s)− ui(t), ui(t))ds.

(5.14)

For all ε2 > 0, we have∫ t

0

gi(t− s)ai(ui(s)− ui(t), ui(t))ds ≤
ε2

2
g̃i(t)‖ui(t)‖2ai +

1

2ε2
(gi ? ui)(t), (5.15)

N∑
i=1

〈Fi(t), ui(t)〉 ≤
1

2ε2

N∑
i=1

‖Fi(t)‖2 +
ε2

2

N∑
i=1

‖ui(t)‖2Ai
. (5.16)

For each ε3 ∈ (0, 1), we have

N∑
i=1

〈fi(~u(t)), ui(t)〉

≤ d2

∫ 1

0

F(~u(x, t))dx

=
d2

p

[ N∑
i=1

((gi ? ui)(t) + ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai)− I(t)
]

=
d2

p

N∑
i=1

[
(gi ? ui)(t) + ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai

]
− ε3d2

p
I(t)− (1− ε3)d2

p
I(t)

≤ d2

p

N∑
i=1

[
(gi ? ui)(t) + ai(t;ui(t), ui(t))− g̃i(t)‖ui(t)‖2ai

]
− ε3d2

p
I(t)− (1− ε3)d2η∗

p

N∑
i=1

‖ui(t)‖2Ai
.

(5.17)

Since d2
p − 1 > 0, we have

(d2

p
− 1
) N∑
i=1

ai(t;ui(t), ui(t)) ≤
(d2

p
− 1
)
µ∗

N∑
i=1

‖ui(t)‖2Ai
. (5.18)

From (5.14)–(5.18), we obtain (5.13). The proof is complete. �

Theorem 5.6. Assume (A1), (A3’), (A4’), (A5”)–(A7”) hold, and (ũ1, . . . , ũN ) ∈
V . If I(0) > 0 and the initial energy E(0) satisfies (5.5) and (5.6), Then there
exist positive constants C, γ such that

E1(t) ≤ C exp(−γt), ∀t ≥ 0. (5.19)



EJDE-2020/106 NONLINEAR HEAT EQUATIONS 25

Proof. From (5.1), (5.2) and (5.13) it follows that

L′(t) ≤ −
(
1− ε1

2

) N∑
i=1

‖u′i(t)‖
2 − 1

2

N∑
i=1

[
ξi−δ

( 1

ε2
+

2d2

p

)]
(gi ? ui)(t)

− δε3d2

p
I(t)− δ

[ (1− ε3)d2η∗
p

−
(d2

p
− 1
)
µ∗ − ε2

2

] N∑
i=1

‖ui(t)‖2Ai

− δ
(d2

p
− 1− ε2

2

) N∑
i=1

g̃i(t)‖ui(t)‖2ai + ρ(t),

(5.20)

for all δ, ε1, ε2 > 0 and ε3 ∈ (0, 1), where

ρ(t) =
1

2

( 1

ε1
+

δ

ε2

) N∑
i=1

‖Fi(t)‖2. (5.21)

As η∗ >
d2−p
d2

µ∗, we can choose ε2 > 0 and ε3 ∈ (0, 1) such that

σ1 =
d2(1−
ε

3)η∗p−
(d2

p
− 1
)
µ∗ − ε2

2
> 0, σ2 =

d2

p
− 1− ε2

2
> 0. (5.22)

We continue by choosing δ and ε1 > 0 such that

1− ε1

2
> 0, σ3 =

1

2

[
min

1≤i≤N
ξi−δ

( 1

ε2
+

2d2

p

)]
> 0. (5.23)

From (5.21), we have

ρ(t) =
1

2

( 1

ε1
+

δ

ε2

) N∑
i=1

‖Fi(t)‖2 ≤ C0 exp(−2γ0t), ∀t ≥ 0, (5.24)

where C0 = 1
2

(
1
ε1

+ δ
ε2

)
min1≤i≤N C

2
i , γ0 = max1≤i≤N γi. Then, we deduce from

(5.20) -(5.24), that there exists γ∗ > 0 such that

L′(t) ≤ −γ∗
[ N∑
i=1

((gi ? ui)(t) + ‖ui(t)‖2Ai
)+I(t)

]
+ C0 exp(−2γ0t)

≤ −γ∗
β2
L(t) + C0 exp(−2γ0t),

(5.25)

where γ∗ = min{ ε3d2δp , δσ1, σ3}, 0 < γ < min{γ∗β2
, 2γ0}. By integrating (5.25), we

deduce

E1(t) ≤ 1

β1
L(t) ≤ 1

β1

(
L(0) +

C0

2γ0 − γ

)
exp(−γt) ≡ C exp(−γt), ∀t ≥ 0.

This implies (5.19), and completes the proof. �

Acknowledgements. The authors wish to express their sincere thanks to the
referees and the editor for their valuable comments and remarks. This research was
funded by the Vietnam National University Ho Chi Minh City (VNU-HCM) under
Grant no. B2020-18-01.



26 L. T. P. NGOC, N. T. LONG EJDE-2020/106

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Le Thi Phuong Ngoc

University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam.
Department of Mathematics and Computer Science, University of Science, Ho Chi Minh

City, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam.

Vietnam National University, Ho Chi Minh City, Vietnam
Email address: ngoc1966@gmail.com

Nguyen Thanh Long
Department of Mathematics and Computer Science, University of Science, Ho Chi Minh

City, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam.

Vietnam National University, Ho Chi Minh City, Vietnam
Email address: longnt2@gmail.com


	1. Introduction
	2. Preliminary results and notation
	3. Existence and uniqueness of a weak solution to (??)-(??)
	4. Blow-up of solutions
	5. Exponential decay of solutions
	Acknowledgements

	References