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

+ Z t

0

gi(t−s) ∂

∂x

¯

µi(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) = ∂u_{2}

∂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) = ˜ui(x), 1≤i≤N, (1.3)
whereh0≥0,h1≥0 are real numbers and µi, gi,µ_{i},fi, Fi, ˜ui for alli∈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.

2010Mathematics 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

Messaoudi [5] considered an initial boundary value problem related to equation
u_{t}−∆u−

Z t 0

g(t−s)∆u(x, s)ds=|u|^{p−2}u,

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

A(t)|ut|^{p−2}u_{t}−∆u−
Z t

0

g(t−s)∆u(x, s)ds= 0,

form≥2,p≥2,A(t) a bounded and positive definite matrix, andga continuously differentiable decaying function, and proved that, under suitable conditions on g andp, 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

u_{t}− ∂

∂x(µ_{1}(x, t)u_{x}) +
Z t

0

g(t−s) ∂

∂x(µ_{2}(x, s)u_{x}(x, s))ds=f(u) +F(x, t),
where (x, t)∈(0,1)×(0, T), with Robin boundary conditions

u_{x}(0, t)−h_{0}u(0, t) =g_{0}(t), u_{x}(1, t) +h_{1}u(1, t) =g_{1}(t),
and the initial condition

u(x,0) =u_{0}(x),

where h_{0}≥0,h_{1}≥0 are real numbers withh_{0}+h_{1} >0, and µ_{1},g, µ_{2}, f, F, g_{0},
g_{1},u_{0}are given functions, under suitable conditions on µ_{1}, g, µ_{2},f, F,g_{0}, g_{1}, u_{0},
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 µ_{i}(x, t) ≡ µ_{i}(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 solutionublows up in finite time. In the case
ofkFi(t)k small enough, for alli∈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.

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

L^{p}=L^{p}(Ω), W^{k,p}=W^{k,p}(Ω), H^{k} =W^{k,2}, ∀k∈Z+, 1≤p≤ ∞.

We denote the usual norm inL^{2}byk · kand we denotek · kX for the norm in the
Banach spaceX. We will use the notationh·,·ifor either the scalar product inL^{2}
or the dual pairing of a continuous linear functional and an element of a function
space. We call X^{0} the dual space of X. We denote L^{p}(0, T;X), 1 ≤ p ≤ ∞,
the Banach space of measurable functions u: (0, T) → X measurable such that
kuk_{L}p(0,T;X)<+∞, with

kuk_{L}p(0,T;X)=

RT

0 ku(t)k^{p}_{X}dt1/p

<+∞, if 1≤p <∞, ess supku(t)kX, ifp=∞.

OnH^{1}, we use the norm
kvk_{H}1=

q

kvk^{2}+kvxk^{2}, ∀v∈H^{1}.
We define

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

Vi=H_{0}^{1}={v∈H^{1}:v(0) =v(1) = 0}, i= 3, N ,

it is clear that V1, . . . , VN are closed subspaces of H^{1}. Moreover, we have the
following standard lemmas concerning the imbeddings ofH^{1} intoC^{0}(Ω) and ofVi

into C^{0}(Ω), and the equivalence between two norms,v7→ kv_{x}k,v 7→ kvk_{H}1,onV_{i}
for alli∈1, N.

Lemma 2.1. The imbeddingH^{1},→C^{0}(Ω) is compact, and
kvk_{C}0(Ω)≤√

2kvk_{H}1, ∀v∈H^{1}.

Lemma 2.2. For all i∈1, N, the imbedding Vi ,→C^{0}(Ω) is compact. Moreover,
we have

kvk_{C}0(Ω)≤ kvkV_{i}, ∀v∈Vi,

√1

2kvkH^{1}≤ kvxk ≤ kvkH^{1} ∀v∈Vi.

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

bilinear forms{ai(t;·,·)}_{t∈[0,T]},{a^{0}_{i}(t;·,·)}_{t∈[0,T}],{ai(t;·,·)}_{t∈[0,T]} defined by
a1(t;u, v) =hµ1(t)ux, vxi+h0µ1(0, t)u(0)v(0),

a^{0}_{1}(t;u, v) =hµ^{0}_{1}(t)ux, vxi+h0µ^{0}_{1}(0, t)u(0)v(0),

a_{1}(t;u, v) =hµ_{1}(t)u_{x}, v_{x}i+h_{0}µ_{1}(0, t)u(0)v(0), ∀u, v∈V_{1}, t∈[0, T];

a2(t;u, v) =hµ2(t)ux, vxi+h1µ2(1, t)u(1)v(1),
a^{0}_{2}(t;u, v) =hµ^{0}_{2}(t)u_{x}, v_{x}i+h_{1}µ^{0}_{2}(1, t)u(1)v(1),

a2(t;u, v) =hµ_{2}(t)ux, vxi+h1µ_{2}(1, t)u(1)v(1), ∀u, v∈V2, t∈[0, T];

ai(t;u, v) =hµi(t)ux, vxi,
a^{0}_{i}(t;u, v) =hµ^{0}_{i}(t)ux, vxi,

a_{i}(t;u, v) =hµ_{i}(t)u_{x}, v_{x}i, ∀u, v∈V_{i}, 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, µ_{i} ∈ C^{0}(Ω×[0, T]) with µi(x, t) ≥ µ_{i∗} >0 and µ_{i}(x, t)≥
µ_{i∗}>0 for all(x, t)∈Ω×[0, T],i∈1, N; andh_{0}≥0,h_{1}≥0. Then, the families
of symmetric bilinear forms{a_{i}(t;·,·)}_{t∈[0,T}_{]},{a_{i}(t;·,·)}_{t∈[0,T]} defined by (2.1)are
continuous on V_{i}×V_{i} and coercive inV_{i} for alli∈1, N.

Moreover, there existaT >0,a0>0 such that

|ai(t;u, v)| ≤aTkuxkkvxk, ai(t;u, v)| ≤aTkuxkkvxk, for allu, v∈Vi,t∈[0, T], i∈1, N; and

a_{i}(t;v, v)≥a_{0}kv_{x}k^{2}, a_{i}(t;v, v)≥a_{0}kv_{x}k^{2}, ∀v∈V_{i}, t∈[0, T], i∈1, N .
We also have two important lemmas.

Lemma 2.4. Let f ∈C^{0}(R^{N};R), if we set
Φf(r) =

(sup_{|x|}

2≤r|f(x)|, ifr >0,

|f(0)|, ifr= 0,
thenΦ_{f} ∈C^{0}(R+;R+) is nondecreasing and

|f(x)| ≤Φf(|x|_{2}), ∀x∈R^{N},
where|x|2=p

x^{2}_{1}+· · ·+x^{2}_{N} for allx∈R^{N}.
Proof. Withr >0, we denote

B_{r}={x∈R^{N} :|x|2< r}, B¯_{r}={x∈R^{N} :|x|2≤r}.

Letg∈C^{0}(R^{N};R+), we set
ϕ_{g}(r) =

(sup_{|x|}

2≤rg(x), ifr >0,

g(0), ifr= 0.

We claim thatϕg∈C^{0}(R+;R+). It is clear that ϕg(r)≥0 for allr∈R+ andϕg

is nondecreasing inR+.

(i) We prove that ϕg is continuous from right at 0. For all ε > 0, by g ∈
C^{0}(R^{N};R+), there existsδ >0 such that

|g(x)−g(0)|< ε, ∀x∈B¯_{δ}. (2.2)

From (2.2), we have

g(x)< g(0) +ε=ϕg(0) +ε, ∀x∈B¯δ. (2.3)
By the definition ofϕ_{g} and (2.3), it follows that

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

Thereforeϕgis continuous from right at 0.

(ii) For allr0>0. We will prove thatϕg is continuous at r0.

(ii-1) We prove thatϕgis continuous from left atr0. At first, we define a function
ϕ_{g}, withϕ_{g}(r) = sup_{|x|}_{2}_{<r}g(x) for allr >0. Easily to see thatϕ_{g}(r)≤ϕg(r) for
allr >0. We prove thatϕ_{g}(r)≥ϕg(r) for allr >0.

Fixedr >0, by the definition ofϕg, we can assume that ϕg(r) = sup

|x|2<r

g(x) = max

|x|2<rg(x) =g(x0),

where x0 ∈ B¯r. We define the sequence {xn} by xn = (1−_{n}^{1})x0. We will have
{xn} ⊂Brand xn→x_{0}. By the definition ofϕ_{g} and continuity ofg, we obtain

ϕ_{g}(r)≥ lim

n→+∞g(x_{n}) =g(x_{0}) =ϕ_{g}(r).

It is clear that ϕ_{g} is nondecreasing in R+. For all ε >0, by the definition of ϕ_{g},
there existsx_{0}∈B_{r}_{0} such that

ϕ_{g}(r0)−ε < g(x0)≤ϕ_{g}(r0). (2.4)
Putδ=r_{0}− |x_{0}|_{2}>0, for all r∈(r_{0}−δ, r_{0}], we have

ϕ_{g}(r0)−ε < g(x0)≤ϕg(|x0|_{2}) =ϕ_{g}(|x0|_{2})≤ϕ_{g}(r)≤ϕ_{g}(r0). (2.5)
From (2.5), it follows that

ϕg(r0)−ε < ϕg(r)≤ϕg(r0), ∀r∈(r0−δ, r0]. (2.6) Thereforeϕgis continuous from left atr0.

(ii-2) We prove that ϕ_{g} is continuous from right at r_{0}. Byg ∈C^{0}(R^{N};R+), we
have g is uniform continuous on ¯B_{2r}_{0}. For allε >0, there existsδ ∈(0,^{r}_{2}^{0}) such
that

|g(x)−g(y)|< ε, ∀x, y∈B¯_{2r}_{0}, |x−y|_{2}< δ. (2.7)
For allr∈[r_{0}, r_{0}+δ), by the definition ofϕ_{g}, there existsx_{r}∈B¯_{r},y_{r}=^{r}_{r}^{0}x_{r}∈B¯_{r}_{0}
such thatϕ_{g}(r) =g(x_{r}) and

|g(xr)−g(yr)|< ε. (2.8)

From (2.8), we have

ϕ_{g}(r_{0})≤ϕ_{g}(r) =g(x_{r})< g(y_{r}) +ε≤ϕ_{g}(|y_{r}|_{2}) +ε≤ϕ_{g}(r_{0}) +ε, (2.9)
for all r∈ [r0, r0+δ). Therefore ϕg is continuous from right at r0. Finally, with
f ∈C^{0}(R^{N};R), we have

Φf(r) =ϕ_{|f|}(r), ∀r∈R^{+}.

The fact|f| ∈C^{0}(R^{N};R+) leads to Φf ∈C^{0}(R+;R+). For allx∈R^{N}, we have

|f(x)| ≤ϕ_{|f|}(|x|_{2}) = Φ_{f}(|x|_{2}).

Obviously, Φ_{f} is nondecreasing. The proof is complete.

Lemma 2.4 is a slight improvement of a result used in [7, Appendix 1, pp. 2734],
withN = 1 andf ∈C^{0}(R;R).

Lemma 2.5. Letx: [0, T]→R+be a continuous function satisfying the inequality x(t)≤M +

Z t 0

k(s)ω(x(s))ds, ∀t∈[0, T],

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

Ψ(u) = Z u

0

dy

ω(y), u≥0.

(i) If R+∞

0 dy

ω(y)= +∞, then
x(t)≤Ψ^{−1}

Ψ(M) + Z t

0

k(s)ds

, ∀t∈[0, T].

(ii) If R+∞

0 dy

ω(y)<+∞, then there existsT_{∗}∈(0, T]such that
Z T∗

0

k(s)ds≤ Z +∞

0

dy
ω(y),
x(t)≤Ψ^{−1}

Ψ(M) + Z 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 ∈L^{2}(0, T;H)}, (3.1)
satisfying the variational problem

hu^{0}_{i}(t), vii+ai(t;ui(t), vi)−
Z t

0

gi(t−s)ai(s;ui(s), vi)ds

=hfi(~u(t)), vii+hFi(t), vii, ∀vi∈Vi, i∈1, N ,

(3.2) and the initial condition

ui(0) = ˜ui, ∀i∈1, N , (3.3) where

V =V1× · · · ×VN, H = (L^{2})^{N}. (3.4)
We make the following assumptions:

(A1) h0, h1≥0;

(A2) ˜ui∈Vi for alli∈1, N;

(A3) µ_{i} ∈C^{1}(Ω×[0, T]) such that µ_{i}(x, t)≥µ_{i∗} >0 for all (x, t)∈Ω×[0, T],
i∈1, N;

(A4) µ_{i} ∈C^{0}(Ω×[0, T]) such that µ_{i}(x, t)≥µ_{i∗} >0 for all (x, t)∈Ω×[0, T],
i∈1, N;

(A5) f_{i}∈C^{0}(R^{N}) for alli∈1, N;
(A6) gi∈H^{1}(0, T) for all i∈1, N;

(A7) Fi∈L^{2}(QT) fori= 1, N.

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

(i) If

Z +∞

0

dy 1 +y+PN

i=1Φ^{2}_{f}

i(√

y),= +∞

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

(ii) If

Z +∞

0

dy 1 +y+PN

i=1Φ^{2}_{f}

i(√

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 allM >0, there existsLM >0 such that

|fi(x)−fi(y)| ≤LM|x−y|2, ∀x, y∈R^{N}, 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

X

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

hu˙^{(m)}_{i} (t), w_{i}^{(j)}i+ai(t;u^{(m)}_{i} (t), w^{(j)}_{i} )−
Z t

0

gi(t−s)ai(s;u^{(m)}_{i} (s), w^{(j)}_{i} )ds

=hfi(~u^{(m)}(t)), w_{i}^{(j)}i+hFi(t), w_{i}^{(j)}i, j= 1, m, i∈1, N ,

(3.6)

and the initial conditions

u^{(m)}_{i} (0) = ˜u^{(0m)}_{i} , ∀i∈1, N , (3.7)
with

˜
u^{(0m)}_{i} =

m

X

j=1

α^{(mj)}_{i} w^{(j)}_{i} →u˜i strongly inVi fori∈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, T_{m}], for some T_{m} ∈
(0, T]. The proofs are straightforward, so we omit the details.

Step 2: A priori estimates. Taking (w^{(j)}_{1} , . . . , w^{(j)}_{N} ) = ( ˙u^{(m)}_{1} (t), . . . ,u˙^{(m)}_{N} (t)) in
(3.6), and summing overifrom 1 toN, we obtain

N

X

i=1

ku˙^{(m)}_{i} (t)k^{2}+

N

X

i=1

a_{i}(t;u^{(m)}_{i} (t),u˙^{(m)}_{i} (t))

−

N

X

i=1

Z t 0

gi(t−s)ai(s;u^{(m)}_{i} (s),u˙^{(m)}_{i} (t))ds

=

N

X

i=1

hfi(~u^{(m)}(t)),u˙^{(m)}_{i} (t)i+

N

X

i=1

hFi(t),u˙^{(m)}_{i} (t)i.

(3.9)

First, through a direct calculation, we have

d

dtai(t, u^{(m)}_{i} (t), u^{(m)}_{i} (t))

= 2a_{i}(t;u^{(m)}_{i} (t), u^{(m)}_{i} (t)) +a^{0}_{i}(t;u^{(m)}_{i} (t), u^{(m)}_{i} (t)),

(3.10) d

dt Z t

0

gi(t−s)ai(s;u^{(m)}_{i} (s), u^{(m)}_{i} (t))ds

=g_{i}(0)a_{i}(t;u^{(m)}_{i} (t), u^{(m)}_{i} (t)) +
Z t

0

g^{0}_{i}(t−s)a_{i}(s;u^{(m)}_{i} (s), u^{(m)}_{i} (t))ds
+

Z t 0

gi(t−s)ai(s;u^{(m)}_{i} (s),u˙^{(m)}_{i} (t))ds,

(3.11)

∀i∈1, N.

Hence, (3.9) can be rewritten as

2

N

X

i=1

ku˙^{(m)}_{i} (t)k^{2}+ d
dt

N

X

i=1

ai(t;u^{(m)}_{i} (t), u^{(m)}_{i} (t))

=

N

X

i=1

[a^{0}_{i}(t;u^{(m)}_{i} (t), u^{(m)}_{i} (t)) + 2d
dt

Z t 0

gi(t−s)ai(s;u^{(m)}_{i} (s), u^{(m)}_{i} (t))ds]

−2

N

X

i=1

g_{i}(0)a_{i}(t;u^{(m)}_{i} (t), u^{(m)}_{i} (t))

−2

N

X

i=1

Z t 0

g^{0}_{i}(t−s)ai(s;u^{(m)}_{i} (s), u^{(m)}_{i} (t))ds

+ 2

N

X

i=1

hf_{i}(~u^{(m)}(t)),u˙^{(m)}_{i} (t)i+ 2

N

X

i=1

hF_{i}(t),u˙^{(m)}_{i} (t)i.

(3.12)

Next, integrating (3.12), we obtain
S_{m}(t) =S_{m}(0) +

N

X

i=1

Z t 0

a^{0}_{i}(s;u^{(m)}_{i} (s), u^{(m)}_{i} (s))ds

+ 2

N

X

i=1

Z t 0

gi(t−s)ai(s;u^{(m)}_{i} (s), u^{(m)}_{i} (s))ds

−2

N

X

i=1

gi(0) Z t

0

ai(s;u^{(m)}_{i} (s), u^{(m)}_{i} (s))ds

−2

N

X

i=1

Z t 0

ds Z s

0

g^{0}_{i}(s−τ)a_{i}(τ;u^{(m)}_{i} (τ), u^{(m)}_{i} (s))dτ

+ 2

N

X

i=1

Z t 0

hfi(~u^{(m)}(s)),u˙^{(m)}_{i} (s)ids+ 2

N

X

i=1

Z t 0

hFi(s),u˙^{(m)}_{i} (s)ids

=Sm(0) +

6

X

k=1

Jk,

(3.13)

where

Sm(t) =

N

X

i=1

2

Z t 0

ku˙^{(m)}_{i} (s)k^{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

X

i=1

ai(t;u^{(m)}_{i} (t), u^{(m)}_{i} (t))≥a0
N

X

i=1

ku^{(m)}_{ix} (t)k^{2}. (3.15)
Now we estimate the terms Jk on the right-hand side of (3.13) as follows. First
term,J1:

J_{1}=h_{0}
Z t

0

µ^{0}_{1}(0, s)|u^{(m)}_{1} (0, s)|^{2}ds+h_{1}
Z t

0

µ^{0}_{2}(1, s)|u^{(m)}_{2} (1, s)|^{2}ds
+

N

X

i=1

Z t 0

hµ^{0}_{i}(s)u^{(m)}_{ix} (s), u^{(m)}_{ix} (s)ids

=J_{1}^{(1)}+J_{1}^{(2)}+J_{1}^{(3)},

(3.16)

in which

J_{1}^{(1)}=h_{0}
Z t

0

µ^{0}_{1}(0, s)|u^{(m)}_{1} (0, s)|^{2}ds

≤h0kµ^{0}_{1}k_{C}0(Ω×[0,T])

Z t 0

ku^{(m)}_{1x} (s)k^{2}ds

≤ h0

a0

kµ^{0}_{1}k_{C}0(Ω×[0,T])

Z t 0

Sm(s)ds.

(3.17)

Using the same techniques, with appropriate modifications, leads to
J_{1}^{(2)}≤ h1

a0

kµ^{0}_{2}k_{C}0(Ω×[0,T])

Z t 0

Sm(s)ds. (3.18)

Using the Cauchy-Schwarz inequality gives
J_{1}^{(3)}=

N

X

i=1

Z t 0

hµ^{0}_{i}(s)u^{(m)}_{ix} (s), u^{(m)}_{ix} (s)ids

≤ max

1≤i≤Nkµ^{0}_{i}k_{C}0(Ω×[0,T])

Z t 0

N

X

i=1

ku^{(m)}_{ix} (s)k^{2}ds

≤ 1 a0

max

1≤i≤Nkµ^{0}_{i}k_{C}0(Ω×[0,T])

Z t 0

S_{m}(s)ds.

(3.19)

From (3.16)–(3.19), we have

J1≤C1

Z t 0

Sm(s)ds, (3.20)

where C1= 1

a0

h0kµ^{0}_{1}k_{C}0(Ω×[0,T])+h1kµ^{0}_{2}k_{C}0(Ω×[0,T])+ max

1≤i≤Nkµ^{0}_{i}k_{C}0(Ω×[0,T])

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

J_{2}= 2

N

X

i=1

Z t 0

g_{i}(t−s)a_{i}(s;u^{(m)}_{i} (s), u^{(m)}_{i} (t))ds

≤2

N

X

i=1

Z t 0

|gi(t−s)||ai(s;u^{(m)}_{i} (s), u^{(m)}_{i} (t))|ds

≤2

N

X

i=1

ku^{(m)}_{ix} (t)kaTkgik_{L}∞(0,T)

Z t 0

ku^{(m)}_{ix} (s)kds

≤

N

X

i=1

h1

6a_{0}ku^{(m)}_{ix} (t)k^{2}+a^{2}_{T}kgik^{2}_{L}∞(0,T)

6a0

Z t 0

ku^{(m)}_{ix} (s)kds

2

i

≤ 1

6Sm(t) + 1

6a^{2}_{0}T a^{2}_{T} max

1≤i≤Nkgik^{2}_{L}∞(0,T)

Z t 0

Sm(s)ds.

(3.22)

Third term,J_{3}. It is clear that
J_{3}=−2

N

X

i=1

g_{i}(0)
Z t

0

a_{i}(s;u^{(m)}_{i} (s), u^{(m)}_{i} (s))ds

≤2aT N

X

i=1

|gi(0)|

Z t 0

ku^{(m)}_{ix} (s)k^{2}ds≤2aT

a0

1≤i≤Nmax |gi(0)|

Z t 0

Sm(s)ds;

(3.23)

Fourth term,J_{4}.
J_{4}=−2

N

X

i=1

Z t 0

ds Z s

0

g^{0}_{i}(s−τ)a_{i}(τ;u^{(m)}_{i} (τ), u^{(m)}_{i} (s))dτ

≤2aT N

X

i=1

Z t 0

ku^{(m)}_{ix} (s)kds
Z s

0

|g^{0}_{i}(s−τ)|ku^{(m)}_{ix} (τ)kdτ

= 2a_{T}√
T

N

X

i=1

kg_{i}^{0}k_{L}2(0,T)

Z t 0

ku^{(m)}_{ix} (s)k^{2}ds

≤2aT

a0

√ T max

1≤i≤Nkg_{i}^{0}k_{L}2(0,T)

Z t 0

Sm(s)ds.

Fifth term,J5. It is known that

|~u^{(m)}(x, t)|2=
v
u
u
t

N

X

i=1

|u^{(m)}_{i} (x, t)|^{2}≤
v
u
u
t

N

X

i=1

ku^{(m)}_{ix} (t)k^{2}≤

pSm(t)

√a_{0} .
By Lemma 2.4, we have

|fi(~u^{(m)}(x, t))| ≤Φf_{i}

|~u^{(m)}(x, t)|_{2}

≤Φf_{i}( 1

√a_{0}

pSm(t)), ∀i∈1, N , so

kf_{i}(~u^{(m)}(t))k ≤Φ_{f}_{i} 1

√a_{0}

pS_{m}(t)

, ∀i∈1, N; therefore,

J5= 2

N

X

i=1

Z t 0

hfi(~u^{(m)}(s)),u˙^{(m)}_{i} (s)ids

≤

N

X

i=1

Z t 0

[3kfi(~u^{(m)}(s))k^{2}+1

3ku˙^{(m)}_{i} (s)k^{2}]ds

≤3

N

X

i=1

Z t 0

Φ^{2}_{f}_{i}( 1

√a0

pSm(s))ds+1 3

N

X

i=1

Z t 0

ku˙^{(m)}_{i} (s)k^{2}ds

≤ 1

6S_{m}(t) + 3

N

X

i=1

Z t 0

Φ^{2}_{f}

i( 1

√a0

pS_{m}(s))ds.

(3.24)

Sixth term,J6. We have
J_{6}= 2

N

X

i=1

Z t 0

hFi(s),u˙^{(m)}_{i} (s)ids≤1

6S_{m}(t) + 3

N

X

i=1

kFik^{2}_{L}2(QT). (3.25)
Now we estimate the termSm(0). From the convergence in (3.8), we can deduce
the existence of a constantC0>0 such that

Sm(0) =

N

X

i=1

ai 0; ˜u^{(0m)}_{i} ,u˜^{(0m)}_{i}

≤C0, ∀m∈N. (3.26)
From (3.13), (3.20), (3.22)-(3.26), there existM_{T} >0,N_{T} >0 constants indepen-
dent ofmsuch that

S_{m}(t)≤M_{T} +N_{T}
Z t

0

ω(S_{m}(s))ds, ∀t∈[0, T], (3.27)
with

ω(S) = 1 +S+

N

X

i=1

Φ^{2}_{f}_{i} 1

√a0

√ S

. (3.28)

By the same convergence of R+∞

0 dy

ω(y) and R+∞

0

dy 1+y+PN

i=1Φ^{2}

fi(√

y), apply Lemma
2.5 withx(t)≡S_{m}(t), M =M_{T}, k(s)≡N_{T}, ω(S) = 1 +S+PN

i=1Φ^{2}_{f}

i(^{√}^{1}_{a}

0

√S),
we obtain the estimate ofS_{m}(t) in two cases as follows.

Case 1. If

Z +∞

0

dy 1 +y+PN

i=1Φ^{2}_{f}

i(√

y)= +∞

then

Sm(t)≤Ψ^{−1}(Ψ(MT) +NTt)

≤Ψ^{−1}(Ψ(MT) +NTT)≡CT, ∀t∈[0, T], m∈N. (3.29)
Case 2. If

Z +∞

0

dy 1 +y+PN

i=1Φ^{2}_{f}

i(√

y)<+∞

then

Sm(t)≤Ψ^{−1}(Ψ(MT) +NTt)

≤Ψ^{−1}(Ψ(MT) +NTT)≡CT, ∀t∈[0, T∗], m∈N, (3.30)
whereT∗∈(0, T] chosen such thatT∗NT ≤R+∞

0 dy ω(y).

This allows one to take the constantT_{m}=T orT_{m}=T_{∗} for allm∈N. In what
follows, we will writeT_{∗} for bothT andT_{∗}.

Step 3: Limiting process. It follows from (3.14), (3.15) and (3.29) (or (3.30)), that
ku^{(m)}_{i} kL^{∞}(0,T∗;V_{i})≤

rC_{T}
a0

, ku˙^{(m)}_{i} kL^{2}(Q_{T})≤p

C_{T}, ∀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 indexmif necessary) such that

~

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

∂~u^{(m)}

∂t → ∂~u

∂t weakly inL^{2}(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 indexmif necessary) such that

~

u^{(m)}→~u strongly inL^{2}(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 offi for alli∈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,

kf_{i}(~u^{(m)})k_{L}2(Q_{T∗})≤√

T sup

|z|≤q

CT a0

|f_{i}(z)|, ∀i∈1, N .
From [3, Lemma 1.3] we obtain

fi(~u^{(m)})→fi(~u) weakly inL^{2}(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.

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

hu^{0}_{i}(t), v_{i}i+a_{i}(t;u_{i}(t), v_{i})−
Z t

0

g_{i}(t−s)a_{i}(s;u_{i}(s), v_{i})ds

=hfi(~u^{(1)}(t))−fi(~u^{(2)}(t)), vii, ∀v∈Vi, i∈1, N ,

(3.38)

u_{i}(0) = 0, ∀i∈1, N . (3.39)

Takingvi= 2ui(t) in (3.38) and integrating with respect to t, and summing overi from 1 toN, we obtain

N

X

i=1

kui(t)k^{2}+ 2
Z t

0 N

X

i=1

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

= 2

N

X

i=1

Z t 0

ds Z s

0

a_{i}(τ;u_{i}(τ), u_{i}(s))dτ

+ 2

N

X

i=1

Z t 0

hfi(~u^{(1)}(s))−fi(~u^{(2)}(s)), ui(s)ids.

(3.40)

Set %(t) = PN

i=1 kui(t)k^{2}+Rt

0kuix(s)k^{2}ds

. As in Step 2, we can estimate all terms on the right hand side of (3.40) to obtain

%(t)≤DT

Z 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 ∈ C^{0}(R^{N}) 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 equationut−_{∂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 toµ_{i}(x, t)≡µ_{i}(x) andFi(x, t)≡0 for alli∈1, N,

∂u_{i}

∂t − ∂

∂x

µ_{i}(x, t)∂u_{i}

∂x +

Z t 0

g_{i}(t−s) ∂

∂x

µ_{i}(x)∂u_{i}

∂x(x, s) ds

=f_{i}(u_{1}, . . . , u_{N}), (x, t)∈Q_{T}, ∀i∈1, N ,

(4.1)

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,
u_{i}(0, t) =u_{i}(1, t) = 0, 3≤i≤N,

(4.2)

and initial conditions

ui(x,0) = ˜ui(x), ∀i∈1, N . (4.3) We make the following assumptions:

(A3’) µi∈C^{1}(Ω×R+) such thatµi(x, t)≥µ_{i∗}>0, ^{∂µ}_{∂t}^{i}(x, t)≤0 for all (x, t)∈
Ω×R+,i∈1, N;

(A4’) µ_{i}∈C^{0}(Ω) such thatµ_{i}(x)≥µ_{i∗}>0 for allx∈Ω,i= 1, N;

(A5’) f_{i} ∈C^{0}(R^{N}) for alli∈1, N. Furthermore, there existsF ∈C^{1}(R^{N}) such
that

(i) _{∂u}^{∂F}

i =f_{i} for alli∈1, N,

(i) There exists constantd1>2 such thatd1F(~u)≤PN

i=1uifi(~u), for all

~u= (u1, . . . , uN)∈R^{N},

(iii) There exist constantsd_{1}>0,p_{i}>2 for alli∈1, N, such thatF(~u)≥
d1PN

i=1|ui|^{p}^{i}, for all~u= (u1, . . . , uN)∈R^{N};

(A6’) g_{i}∈C^{1}(R+;R+)∩L^{1}(R+) such that 0< g_{i}(t)≤g_{i}(0) andg_{i}^{0}(t)≤0 for all
t≥0,i∈1, N.

Example 4.1. For ~u = (u1, . . . , uN) ∈ R^{N}, we define a function that satisfies
(A5’).

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

N

X

i=1

αi|ui|^{p}^{i}+β|u1|^{q}^{1}. . .|uN|^{q}^{N}ln^{k}(e+|~u|^{2}_{2}),
where β >0, k >1 and α_{i} >0, p_{i}>2, q_{i} >2 for all i∈ 1, N are constants. By
direct calculations, we have

fi(~u) = ∂F

∂ui

(~u)

=piαi|ui|^{p}^{i}^{−2}ui+βqi|u1|^{q}^{1}. . .|uN|^{q}^{N}u^{−1}_{i} ln^{k}(e+|~u|^{2}_{2})
+ 2kβ|u1|^{q}^{1}. . .|uN|^{q}^{N} u_{i}

e+|~u|^{2}_{2}ln^{k−1}(e+|~u|^{2}_{2}), ∀i∈1, N .
It is obvious that (A5’) holds, since

N

X

i=1

uifi(~u) =

N

X

i=1

piαi|ui|^{p}^{i}+β(

N

X

i=1

qi)|u1|^{q}^{1}. . .|uN|^{q}^{N}ln^{k}(e+|~u|^{2}_{2})
+ 2kβ|u1|^{q}^{1}. . .|uN|^{q}^{N} |~u|^{2}_{2}

e+|~u|^{2}_{2}ln^{k−1}(e+|~u|^{2}_{2})

≥

N

X

i=1

piαi|ui|^{p}^{i}+β(

N

X

i=1

qi)|u1|^{q}^{1}. . .|uN|^{q}^{N}ln^{k}(e+|~u|^{2}_{2})

≥d_{1}F(~u),

withd1= min{p1, . . . , pN,PN

i=1qi} and F(~u)≥

N

X

i=1

α_{i}|u_{i}|^{p}^{i} ≥d_{1}

N

X

i=1

|u_{i}|^{p}^{i}, d_{1}= min

1≤i≤Nα_{i}.
Now, onVi×Vi, we consider the following symmetric bilinear forms:

A1(u, v) = Z 1

0

ux(x)vx(x)dx+h0u(0)v(0), a1(u, v) =

Z 1 0

µ_{1}(x)ux(x)vx(x)dx+h0µ_{1}(0)u(0)v(0), ∀u, v∈V1;
A2(u, v) =

Z 1 0

ux(x)vx(x) +h1u(1)v(1), a2(u, v) =

Z 1 0

µ_{2}(x)ux(x)vx(x)dx+h1µ_{2}(1)u(1)v(1), ∀u, v∈V2;
Ai(u, v) =

Z 1 0

ux(v)vx(x)dx, ai(u, v) =

Z 1 0

µ_{i}(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 alli ∈ 1, N. On the other hand, the norm v 7→ kvxk and the normsv7→ kvkAi=p

A_{i}(v, v) andv7→ kvkai =p

a_{i}(v, v) are equivalent.

Lemma 4.2. There exist positive constantsµ_{∗},µ^{∗},µ_{∗},µ^{∗} such that:

(i) A_{i}(v, v)≥ kvxk^{2}, for all v∈V_{i},i∈1, N,

(ii) |Ai(u, v)| ≤(1 + max{h0, h1})kuxkkvxk, for allu, v∈Vi,i= 1, N,
(iii) a_{i}(v, v)≥µ_{∗}kvk^{2}_{A}

i, for allv∈V_{i},i∈1, N,
(iv) |ai(u, v)| ≤µ^{∗}kukA_{i}kvkA_{i}, for allv∈Vi,i= 1, N,

(v) ai(t;v, v)≥µ_{∗}kvk^{2}_{A}

i, for allv∈Vi,i∈1, N,
(vi) |a_{i}(t;u, v)| ≤µ^{∗}kuk_{A}_{i}kvk_{A}_{i}, for allv∈V_{i},i∈1, N,
(vii) a^{0}_{i}(t, v, v)≤0, for allu, v∈Vi,t≥0,i∈1, N.

Lemma 4.3. For i ∈ 1, N, on Vi, the norms v 7→ kvkA_{i} = p

Ai(v, v) and v 7→

kvka_{i} =p

ai(v, v) are equivalent and
pµ_{∗}kvkA_{i} ≤ kvka_{i}≤p

µ^{∗}kvkA_{i}, ∀v∈Vi.
Now we define the modified energy functional related to (4.1)-(4.3),
E(t) =1

2

N

X

i=1

[(g_{i}? u_{i})(t) +a_{i}(t;u_{i}(t), u_{i}(t))−˜g_{i}(t)kuik^{2}_{a}_{i}]−
Z 1

0

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

(gi? ui)(t) = Z t

0

gi(t−s)kui(s)−ui(t)k^{2}_{a}_{i}ds, ˜gi(t) =
Z t

0

gi(s)ds, (4.5)

for alli∈1, N. By multiplying (4.1) byu^{0}_{i}(t), and integrating over Ω, and summing
overifrom 1 toN, we obtain

E^{0}(t) =

N

X

i=1

[−ku^{0}_{i}(t)k^{2}+1

2a^{0}_{i}(t;u_{i}(t), u_{i}(t))−1

2g_{i}(t)ku_{i}k^{2}_{a}

i+1

2(g^{0}_{i}? u_{i})(t)]≤0, (4.6)
for any regular solution. The same result can be established for weak solutions and
for almost everyt, by a denseness argument.

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

1≤i≤Nmax kgik_{L}1(R+)< µ_{∗}
µ^{∗}

1− 1

(d1−1)^{2}

, then for all(˜u1, . . . ,u˜N)∈V such that E(0)<0, we have:

(i) If p1=· · ·=pN, then the weak solution uof (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 PN

i=1ku˜ik^{2} ≥
4^{1+1/p}N, with p = min_{1≤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 ∈L^{2}_{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) thatH^{0}(t)≥0 for allt≥0. This implies that

H(t)≥ H(0) =−E(0)>0, ∀t≥0. (4.9) Set

L_{1}(t) = 1
2

N

X

i=1

ku_{i}(t)k^{2}. (4.10)

By taking the time derivative of (4.10) and using (4.1), we obtain
L^{0}_{1}(t) =

N

X

i=1

(hfi(~u(t)), u_{i}(t)i −a_{i}(t;u_{i}(t), u_{i}(t)) +
Z t

0

g_{i}(t−s)a_{i}(u_{i}(s), u_{i}(t))ds).

Hence
L^{0}_{1}(t)≥

N

X

i=1

(hfi(~u(t)), ui(t)i −ai(t;ui(t), ui(t))) +

N

X

i=1

˜

gi(t)kui(t)k^{2}_{a}_{i}

−

N

X

i=1

Z t 0

g_{i}(t−s)|a_{i}(u_{i}(s)−u_{i}(t), u_{i}(t))|ds.

(4.11)