∂x µi(x, t)∂ui ∂x + Z t 0 gi(t−s

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) = ∂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) = ˜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−2u,

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−2u_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(µ₁(x, t)u_x) + Z t

0

g(t−s) ∂

∂x(µ₂(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)−h₀u(0, t) =g₀(t), u_x(1, t) +h₁u(1, t) =g₁(t), and the initial condition

u(x,0) =u₀(x),

where h₀≥0,h₁≥0 are real numbers withh₀+h₁ >0, and µ₁,g, µ₂, f, F, g₀, g₁,u₀are 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 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 inL²byk · kand we denotek · kX for the norm in the Banach spaceX. We will use the notationh·,·ifor either the scalar product inL² or the dual pairing of a continuous linear functional and an element of a function space. We call X⁰ 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_Lp(0,T;X)<+∞, with

kuk_Lp(0,T;X)=





 RT

0 ku(t)k^p_Xdt1/p

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

OnH¹, we use the norm kvk_H1=

q

kvk²+kvxk², ∀v∈H¹. We define

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

Vi=H₀¹={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 ofH¹ intoC⁰(Ω) and ofVi

into C⁰(Ω), and the equivalence between two norms,v7→ kv_xk,v 7→ kvk_H1,onV_i for alli∈1, N.

Lemma 2.1. The imbeddingH¹,→C⁰(Ω) is compact, and kvk_C0(Ω)≤√

2kvk_H1, ∀v∈H¹.

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

kvk_C0(Ω)≤ kvkV_i, ∀v∈Vi,

√1

2kvkH¹≤ kvxk ≤ kvkH¹ ∀v∈Vi.

Let µi, µ_i ∈ C⁰(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],{a⁰_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),

a⁰₁(t;u, v) =hµ⁰₁(t)ux, vxi+h0µ⁰₁(0, t)u(0)v(0),

a₁(t;u, v) =hµ₁(t)u_x, v_xi+h₀µ₁(0, t)u(0)v(0), ∀u, v∈V₁, t∈[0, T];

a2(t;u, v) =hµ2(t)ux, vxi+h1µ2(1, t)u(1)v(1), a⁰₂(t;u, v) =hµ⁰₂(t)u_x, v_xi+h₁µ⁰₂(1, t)u(1)v(1),

a2(t;u, v) =hµ₂(t)ux, vxi+h1µ₂(1, t)u(1)v(1), ∀u, v∈V2, t∈[0, T];

ai(t;u, v) =hµi(t)ux, vxi, a⁰_i(t;u, v) =hµ⁰_i(t)ux, vxi,

a_i(t;u, v) =hµ_i(t)u_x, v_xi, ∀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 ∈ C⁰(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; andh₀≥0,h₁≥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)≥a₀kv_xk², a_i(t;v, v)≥a₀kv_xk², ∀v∈V_i, t∈[0, T], i∈1, N . We also have two important lemmas.

Lemma 2.4. Let f ∈C⁰(R^N;R), if we set Φf(r) =

(sup_|x|

2≤r|f(x)|, ifr >0,

|f(0)|, ifr= 0, thenΦ_f ∈C⁰(R+;R+) is nondecreasing and

|f(x)| ≤Φf(|x|₂), ∀x∈R^N, where|x|2=p

x²₁+· · ·+x²_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∈C⁰(R^N;R+), we set ϕ_g(r) =

(sup_|x|

2≤rg(x), ifr >0,

g(0), ifr= 0.

We claim thatϕg∈C⁰(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 ∈ C⁰(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<rg(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−_n¹)x0. We will have {xn} ⊂Brand xn→x₀. By the definition ofϕ_g and continuity ofg, we obtain

ϕ_g(r)≥ lim

n→+∞g(x_n) =g(x₀) =ϕ_g(r).

It is clear that ϕ_g is nondecreasing in R+. For all ε >0, by the definition of ϕ_g, there existsx₀∈B_r0 such that

ϕ_g(r0)−ε < g(x0)≤ϕ_g(r0). (2.4) Putδ=r₀− |x₀|₂>0, for all r∈(r₀−δ, r₀], 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 r₀. Byg ∈C⁰(R^N;R+), we have g is uniform continuous on ¯B_2r0. For allε >0, there existsδ ∈(0,^r₂⁰) such that

|g(x)−g(y)|< ε, ∀x, y∈B¯_2r0, |x−y|₂< δ. (2.7) For allr∈[r₀, r₀+δ), by the definition ofϕ_g, there existsx_r∈B¯_r,y_r=^r_r⁰x_r∈B¯_r0 such thatϕ_g(r) =g(x_r) and

|g(xr)−g(yr)|< ε. (2.8)

From (2.8), we have

ϕ_g(r₀)≤ϕ_g(r) =g(x_r)< g(y_r) +ε≤ϕ_g(|y_r|₂) +ε≤ϕ_g(r₀) +ε, (2.9) for all r∈ [r0, r0+δ). Therefore ϕg is continuous from right at r0. Finally, with f ∈C⁰(R^N;R), we have

Φf(r) =ϕ_|f|(r), ∀r∈R⁺.

The fact|f| ∈C⁰(R^N;R+) leads to Φf ∈C⁰(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 ∈C⁰(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 ∈L²(0, T;H)}, (3.1) satisfying the variational problem

hu⁰_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 = (L²)^N. (3.4) We make the following assumptions:

(A1) h0, h1≥0;

(A2) ˜ui∈Vi for alli∈1, N;

(A3) µ_i ∈C¹(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 ∈C⁰(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∈C⁰(R^N) for alli∈1, N; (A6) gi∈H¹(0, T) for all i∈1, N;

(A7) Fi∈L²(QT) fori= 1, N.

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

(i) If

Z +∞

0

dy 1 +y+PN

i=1Φ²_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=1Φ²_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)k²+

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)) +a⁰_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

g⁰_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)k²+ d dt

N

X

i=1

ai(t;u^(m)_i (t), u^(m)_i (t))

=

N

X

i=1

[a⁰_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

g⁰_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

a⁰_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

g⁰_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)k²ds+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)k². (3.15) Now we estimate the terms Jk on the right-hand side of (3.13) as follows. First term,J1:

J₁=h₀ Z t

0

µ⁰₁(0, s)|u^(m)₁ (0, s)|²ds+h₁ Z t

0

µ⁰₂(1, s)|u^(m)₂ (1, s)|²ds +

N

X

i=1

Z t 0

hµ⁰_i(s)u^(m)_ix (s), u^(m)_ix (s)ids

=J₁⁽¹⁾+J₁⁽²⁾+J₁⁽³⁾,

(3.16)

in which

J₁⁽¹⁾=h₀ Z t

0

µ⁰₁(0, s)|u^(m)₁ (0, s)|²ds

≤h0kµ⁰₁k_C0(Ω×[0,T])

Z t 0

ku^(m)_1x (s)k²ds

≤ h0

a0

kµ⁰₁k_C0(Ω×[0,T])

Z t 0

Sm(s)ds.

(3.17)

Using the same techniques, with appropriate modifications, leads to J₁⁽²⁾≤ h1

a0

kµ⁰₂k_C0(Ω×[0,T])

Z t 0

Sm(s)ds. (3.18)

Using the Cauchy-Schwarz inequality gives J₁⁽³⁾=

N

X

i=1

Z t 0

hµ⁰_i(s)u^(m)_ix (s), u^(m)_ix (s)ids

≤ max

1≤i≤Nkµ⁰_ik_C0(Ω×[0,T])

Z t 0

N

X

i=1

ku^(m)_ix (s)k²ds

≤ 1 a0

max

1≤i≤Nkµ⁰_ik_C0(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

h0kµ⁰₁k_C0(Ω×[0,T])+h1kµ⁰₂k_C0(Ω×[0,T])+ max

1≤i≤Nkµ⁰_ik_C0(Ω×[0,T])

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

J₂= 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

6a₀ku^(m)_ix (t)k²+a²_Tkgik²_L∞(0,T)

6a0

Z t 0

ku^(m)_ix (s)kds

2

i

≤ 1

6Sm(t) + 1

6a²₀T a²_T max

1≤i≤Nkgik²_L∞(0,T)

Z t 0

Sm(s)ds.

(3.22)

Third term,J₃. It is clear that J₃=−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)k²ds≤2aT

a0

1≤i≤Nmax |gi(0)|

Z t 0

Sm(s)ds;

(3.23)

Fourth term,J₄. J₄=−2

N

X

i=1

Z t 0

ds Z s

0

g⁰_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

|g⁰_i(s−τ)|ku^(m)_ix (τ)kdτ

= 2a_T√ T

N

X

i=1

kg_i⁰k_L2(0,T)

Z t 0

ku^(m)_ix (s)k²ds

≤2aT

a0

√ T max

1≤i≤Nkg_i⁰k_L2(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)k²≤

pSm(t)

√a₀ . By Lemma 2.4, we have

|fi(~u^(m)(x, t))| ≤Φf_i

|~u^(m)(x, t)|₂

≤Φf_i( 1

√a₀

pSm(t)), ∀i∈1, N , so

kf_i(~u^(m)(t))k ≤Φ_fi 1

√a₀

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))k²+1

3ku˙^(m)_i (s)k²]ds

≤3

N

X

i=1

Z t 0

Φ²_fi( 1

√a0

pSm(s))ds+1 3

N

X

i=1

Z t 0

ku˙^(m)_i (s)k²ds

≤ 1

6S_m(t) + 3

N

X

i=1

Z t 0

Φ²_f

i( 1

√a0

pS_m(s))ds.

(3.24)

Sixth term,J6. We have J₆= 2

N

X

i=1

Z t 0

hFi(s),u˙^(m)_i (s)ids≤1

6S_m(t) + 3

N

X

i=1

kFik²_L2(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

Φ²_fi 1

√a0

√ S

. (3.28)

By the same convergence of R+∞

0 dy

ω(y) and R+∞

0

dy 1+y+PN

i=1Φ²

fi(√

y), apply Lemma 2.5 withx(t)≡S_m(t), M =M_T, k(s)≡N_T, ω(S) = 1 +S+PN

i=1Φ²_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=1Φ²_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=1Φ²_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 kL²(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 inL²(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 inL²(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_L2(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 inL²(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

hu⁰_i(t), v_ii+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)k²+ 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)k²+Rt

0kuix(s)k²ds

. 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 ∈ C⁰(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(u₁, . . . , 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∈C¹(Ω×R+) such thatµi(x, t)≥µ_i∗>0, ^∂µ_∂tⁱ(x, t)≤0 for all (x, t)∈ Ω×R+,i∈1, N;

(A4’) µ_i∈C⁰(Ω) such thatµ_i(x)≥µ_i∗>0 for allx∈Ω,i= 1, N;

(A5’) f_i ∈C⁰(R^N) for alli∈1, N. Furthermore, there existsF ∈C¹(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 constantsd₁>0,p_i>2 for alli∈1, N, such thatF(~u)≥ d1PN

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

(A6’) g_i∈C¹(R+;R+)∩L¹(R+) such that 0< g_i(t)≤g_i(0) andg_i⁰(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|^pi+β|u1|^q1. . .|uN|^qNln^k(e+|~u|²₂), 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|^pi−2ui+βqi|u1|^q1. . .|uN|^qNu⁻¹_i ln^k(e+|~u|²₂) + 2kβ|u1|^q1. . .|uN|^qN u_i

e+|~u|²₂ln^k−1(e+|~u|²₂), ∀i∈1, N . It is obvious that (A5’) holds, since

N

X

i=1

uifi(~u) =

N

X

i=1

piαi|ui|^pi+β(

N

X

i=1

qi)|u1|^q1. . .|uN|^qNln^k(e+|~u|²₂) + 2kβ|u1|^q1. . .|uN|^qN |~u|²₂

e+|~u|²₂ln^k−1(e+|~u|²₂)

≥

N

X

i=1

piαi|ui|^pi+β(

N

X

i=1

qi)|u1|^q1. . .|uN|^qNln^k(e+|~u|²₂)

≥d₁F(~u),

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

i=1qi} and F(~u)≥

N

X

i=1

α_i|u_i|^pi ≥d₁

N

X

i=1

|u_i|^pi, d₁= 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

µ₁(x)ux(x)vx(x)dx+h0µ₁(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

µ₂(x)ux(x)vx(x)dx+h1µ₂(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)≥ kvxk², 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_∗kvk²_A

i, for allv∈V_i,i∈1, N, (iv) |ai(u, v)| ≤µ^∗kukA_ikvkA_i, for allv∈Vi,i= 1, N,

(v) ai(t;v, v)≥µ_∗kvk²_A

i, for allv∈Vi,i∈1, N, (vi) |a_i(t;u, v)| ≤µ^∗kuk_Aikvk_Ai, for allv∈V_i,i∈1, N, (vii) a⁰_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)kuik²_ai]− Z 1

0

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

(gi? ui)(t) = Z t

0

gi(t−s)kui(s)−ui(t)k²_aids, ˜gi(t) = Z t

0

gi(s)ds, (4.5)

for alli∈1, N. By multiplying (4.1) byu⁰_i(t), and integrating over Ω, and summing overifrom 1 toN, we obtain

E⁰(t) =

N

X

i=1

[−ku⁰_i(t)k²+1

2a⁰_i(t;u_i(t), u_i(t))−1

2g_i(t)ku_ik²_a

i+1

2(g⁰_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_L1(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˜ik² ≥ 4^1+1/pN, with p = min_1≤i≤Npi, 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) thatH⁰(t)≥0 for allt≥0. This implies that

H(t)≥ H(0) =−E(0)>0, ∀t≥0. (4.9) Set

L₁(t) = 1 2

N

X

i=1

ku_i(t)k². (4.10)

By taking the time derivative of (4.10) and using (4.1), we obtain L⁰₁(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 L⁰₁(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)k²_ai

−

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)