In this article, we study oscillatory properties of solutions to neu- tral nonlinear impulsive hyperbolic partial differential equations with several delays

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Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 27, pp. 1–10.

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

OSCILLATION OF SOLUTIONS TO NEUTRAL NONLINEAR IMPULSIVE HYPERBOLIC EQUATIONS WITH

SEVERAL DELAYS

JICHEN YANG, ANPING LIU, GUANGJIE LIU

Abstract. In this article, we study oscillatory properties of solutions to neutral nonlinear impulsive hyperbolic partial differential equations with several delays. We establish sufficient conditions for oscillation of all solutions.

1. Introduction

The theory of partial differential equations can be applied to many fields, such as to biology, population growth, engineering, generic repression, control theory and climate model. In the last few years, the fundamental theory of partial differential equations with deviating argument has undergone intensive development. The qualitative theory of this class of equations, however, is still in an initial stage of development. Many srudies have been done under the assumption that the state variables and system parameters change continuously. However, one may easily visualize situations in nature where abrupt change such as shock and disasters may occur. These phenomena are short-time perturbations whose duration is negligi- ble in comparison with the duration of the whole evolution process. Consequently, it is natural to assume, in modeling these problems, that these perturbations act instantaneously, that is, in the form of impulses.

In 1991, the first paper [4] on this class of equations was published. However, on oscillation theory of impulsive partial differential equations only a few of papers have been published. Recently, Bainov, Minchev, Liu and Luo [1, 2, 6, 7, 8, 9, 10, 11]

investigated the oscillation of solutions of impulsive partial differential equations with or without deviating argument. But there is a scarcity in the study of oscillation theory of nonlinear impulsive hyperbolic equations of neutral type with several delays.

2000Mathematics Subject Classification. 35B05, 35L70, 35R10, 35R12.

Key words and phrases. Oscillation; hyperbolic equation; impulsive; neutral type; delay.

c

2013 Texas State University - San Marcos.

Submitted October 1, 2012. Published January 27, 2013.

1

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In this article, we discuss oscillatory properties of solutions for the nonlinear impulsive hyperbolic equation of neutral type with several delays.

∂²

∂t² h

u(t, x) +

m

X

i=1

giu(t−τi, x)i

=a(t)h(u)∆u−q(t, x)f(u(t, x)) +

l

X

r=1

ar(t)hr(u(t−σr, x))∆u(t−σr, x)

−

n

X

j=1

qj(t, x)fj(u(t−ρj, x)), t6=tk, (t, x)∈R⁺×Ω =G,

(1.1)

u(t⁺_k, x) =hk(tk, x, u(tk, x)), k= 1,2, . . . , (1.2) ut(t⁺_k, x) =pk(tk, x, ut(tk, x)), k= 1,2, . . . , (1.3) with the boundary conditions

u= 0, (t, x)∈R⁺×∂Ω, (1.4)

∂u

∂n+ϕ(t, x)u= 0, (t, x)∈R⁺×∂Ω, (1.5)

and the initial condition

u(t, x) = Φ(t, x), ∂u(t, x)

∂t = Ψ(t, x), (t, x)∈[−δ,0]×Ω.

Here Ω⊂R^N is a bounded domain with boundary ∂Ω smooth enough and nis a unit exterior normal vector of∂Ω, δ= max{τi, σ_r, ρ_j}, Φ(t, x)∈C²([−δ,0]×Ω,R), Ψ(t, x)∈C¹([−δ,0]×Ω,R).

This article is organized as follows. in Section 2, we study the oscillatory properties of solutions for problems (1.1), (1.4). In Section 3, we discuss oscillatory properties of solutions for problems (1.1), (1.5).

We will use the following conditions:

(H1) a(t), ai(t) ∈ P C(R⁺,R⁺), τi, σr, ρj are positive constants, q(t, x), qj(t, x) are functions inC(R⁺×Ω,¯ (0,∞)),g_iis a non-negative constant,Pm

i=1g_i<

1,i= 1,2, . . . , m,j= 1,2, . . . , n,r= 1,2, . . . , l; whereP C denote the class of functions which are piecewise continuous intwith discontinuities of first kind only att=tk,k= 1,2, . . ., and left continuous att=tk,k= 1,2, . . .. (H2) h(u), hr(u) ∈ C¹(R,R), f(u), fr(u) ∈ C(R,R); f(u)/u ≥ C a positive constant, fj(u)/u ≥ Cj a positive constant, u 6= 0; h(0) = 0, hr(0) = 0, uh⁰(u) ≥ 0, uh⁰_r(u) ≥ 0, ϕ(t, x) ∈ C(R⁺×∂Ω,R), h(u)ϕ(t, x) ≥ 0, hr(u)ϕ(t−σr, x) ≥ 0, r = 1,2, . . . , l, 0 < t1 < t2 < · · · < tk < . . ., limk→∞tk=∞.

(H3) u(t, x) and their derivativesut(t, x) are piecewise continuous int with discontinuities of first kind only att=t_k,k= 1,2, . . ., and left continuous at t=t_k,u(t_k, x) =u(t⁻_k, x),u_t(t_k, x) =u_t(t⁻_k, x),k= 1,2, . . ..

(H4) hk(tk, x, u(tk, x)), pk(tk, x, ut(tk, x)) ∈ P C(R⁺ ×Ω¯ ×R,R), k = 1,2, . . ., and there exist positive constants a_k, a_k, b_k, b_k and b_k ≤a_k such that for k= 1,2, . . . ,

a_k≤ hk(tk, x, η)

η ≤a_k, b_k≤ pk(tk, x, φ) φ ≤b_k.

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Let us construct the sequence {t¯k} ={tk} ∪ {tki} ∪ {tkr} ∪ {tkj}, where tki =tk+τi, tkr =tk+σr, tkj =tk+ρj and ¯tk <¯tk+1, i= 1,2, . . . , m, r= 1,2, . . . , l,j= 1,2, . . . , n, k= 1,2, . . ..

Definition 1.1. By a solution of problems (1.1), (1.4) ((1.5)) with initial condition, we mean that any functionu(t, x) for which the following conditions are valid:

(1) If−δ≤t≤0, thenu(t, x) = Φ(t, x), ^∂u(t,x)_∂t = Ψ(t, x).

(2) If 0≤t≤¯t₁=t₁, then u(t, x) coincides with the solution of the problems (1.1)–(1.3) and (1.4) ((1.5)) with initial condition.

(3) If ¯tk < t≤¯tk+1, ¯tk ∈ {tk} \({tki} ∪ {tkr} ∪ {tkj}), thenu(t, x) coincides with the solution of the problems (1.1)–(1.3) and (1.4) ((1.5)).

(4) If ¯tk< t≤t¯k+1, ¯tk ∈ {tki} ∪ {tkr} ∪ {tkj}, thenu(t, x) satisfies (1.4) ((1.5)) and coincides with the solution of the problem

∂²

∂t² h

u(t⁺, x) +

m

X

i=1

g_iu((t−τ_i)⁺, x)i

=a(t)h(u(t⁺, x))∆u(t⁺, x)−q(t, x)f(u(t⁺, x)) +

l

X

r=1

a_r(t)h_r(u((t−σ_r)⁺, x))∆u((t−σ_r)⁺, x)

−

n

X

j=1

qj(t, x)fj(u((t−ρj)⁺, x)), t6=tk, (t, x)∈R⁺×Ω =G, u(¯t⁺_k, x) =u(¯tk, x), ut(¯t⁺_k, x) =ut(¯tk, x),

for¯t_k ∈({tki} ∪ {tkr} ∪ {tkj})\ {tk}, or

u(¯t⁺_k, x) =hk_i(¯tk, x, u(¯tk, x)), ut(¯t⁺_k, x) =pk_i(¯tk, x, ut(¯tk, x)), for ¯tk ∈({tki} ∪ {tkr} ∪ {tkj})∩ {tk},

where the numberk_i is determined by the equality ¯t_k=t_k_i. We introduce the notation:

Γ_k={(t, x) :t∈(t_k, t_k+1), x∈Ω}, Γ =∪^∞_k=0Γ_k, Γ¯k={(t, x) :t∈(tk, tk+1), x∈Ω},¯ Γ =¯ ∪^∞_k=0Γ¯k, v(t) =

Z

Ω

u(t, x)dx, q(t) = min

x∈Ω¯

q(t, x), qj(t) = min

x∈Ω¯

qj(t, x).

Definition 1.2. The solutionu∈C²(Γ)∩C¹(¯Γ) of problems (1.1), (1.4) ((1.5)) is called non-oscillatory in the domainGif it is either eventually positive or eventually negative. Otherwise, it is called oscillatory.

2. Oscillation properties for (1.1),(1.4) For the main result of this article, we need following lemmas.

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Lemma 2.1. Let u∈ C²(Γ)∩C¹(¯Γ) be a positive solution of (1.1), (1.4) in G, then functionw(t) satisfies the impulsive differential inequality

w⁰⁰(t) +C 1−

m

X

i=1

gi

q(t)w(t) +

n

X

j=1

Cj

1−

m

X

i=1

gi

qj(t)w(t−ρj)≤0, t6=tk, (2.1) a_k≤ w(t⁺_k)

w(t_k) ≤ak, k= 1,2, . . . , (2.2) b_k≤ w⁰(t⁺_k)

w⁰(t_k) ≤b_k, k= 1,2, . . . , (2.3) wherew(t) =v(t) +Pm

i=1g_iv(t−τ_i).

Proof. Letu(t, x) be a positive solution of the problem (1.1), (1.4) inG. Without loss of generality, we may assume that there exists a T > 0, t₀ > T such that u(t, x) > 0, u(t−τ_i, x) > 0, i = 1,2, . . . , m, u(t−σ_r, x) > 0, r = 1,2, . . . , l, u(t−ρ_j, x)>0,j= 1,2, . . . n, for any (t, x)∈[t₀,∞)×Ω.

Fort≥t₀,t6=t_k, k= 1,2, . . ., integrating (1.1) with respect toxover Ω yields d²

dt² hZ

Ω

u(t, x)dx+

m

X

i=1

g_i Z

Ω

u(t−τ_i, x)dxi

=a(t) Z

Ω

h(u)∆u dx− Z

Ω

q(t, x)f(u(t, x))dx +

l

X

r=1

ar(t) Z

Ω

hr(u(t−σr, x))∆u(t−σr, x)dx

−

n

X

j=1

Z

Ω

qj(t, x)fj(u(t−ρj, x))dx.

By Green’s formula and the boundary condition, we have Z

Ω

h(u)∆u dx= Z

∂Ω

h(u)∂u

∂nds− Z

Ω

h⁰(u)|gradu|²dx

=− Z

Ω

h⁰(u)|gradu|²dx≤0, Z

Ω

h_r(u(t−σ_r, x))∆u(t−σ_r, x)dx≤0.

From condition (H2), we can easily obtain Z

Ω

q(t, x)f(u(t, x))dx≥Cq(t) Z

Ω

u(t, x)dx, Z

Ω

qj(t, x)fj(u(t−ρj, x))dx≥Cjqj(t) Z

Ω

u(t−ρj, x)dx.

From the above it follows that d²

dt² h

v(t) +

m

X

i=1

g_iv(t−τ_i)i

+Cq(t)v(t) +

n

X

j=1

C_jq_j(t)v(t−ρ_j)≤0.

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Setw(t) =v(t) +Pm

i=1giv(t−τi), we havew(t)≥v(t), then we can obtain w⁰⁰(t) +Cq(t)v(t) +

n

X

j=1

Cjqj(t)v(t−ρj)≤0, t6=tk.

From this inequality, we have w⁰⁰(t)<0, and w⁰(t) >0, w(t)> 0 for t ≥t0. In fact, ifw⁰(t)<0, there existst1≥t0 such thatw⁰(t1)<0. Hence we have

w(t)−w(t1) = Z t

t₁

w⁰(s)ds≤ Z t

t₁

w⁰(t1)ds=w⁰(t1)(t−t1),

t→+∞lim w(t) =−∞.

This is a contradiction, sow⁰(t)>0. Because v(t) =w(t)−

m

X

i=1

giv(t−τi)

=w(t)−

m

X

i=1

gi

h

w(t−τi)−

m

X

i=1

giv(t−2τi)i

=w(t)−

m

X

i=1

giw(t−τi) +

m

X

i=1

gi

hX^m

i=1

giv(t−2τi)i

≥ 1−

m

X

i=1

g_i w(t),

v(t−ρ_j)≥ 1−

m

X

i=1

g_i

w(t−ρ_j).

Hence, we obtain w⁰⁰(t) +C

1−

m

X

i=1

gi

q(t)w(t) +

n

X

j=1

Cj

1−

m

X

i=1

gi

qj(t)w(t−ρj)≤0, t6=tk. Fort≥t₀, t=t_k, k= 1,2, . . ., from (1.2), (1.3) and condition (H4), we obtain

a_k≤ u(t⁺_k, x)

u(t_k, x) ≤a_k, (2.4)

b_k≤ ut(t⁺_k, x)

ut(tk, x) ≤b_k. (2.5)

According to thev(t) =R

Ωu(t, x)dx, we obtain a_k≤ v(t⁺_k)

v(tk) ≤ak, b_k≤ v⁰(t⁺_k) v⁰(tk) ≤bk. Becausew(t) =v(t) +Pm

i=1giv(t−τi),we finally have a_k ≤ w(t⁺_k)

w(t_k) ≤a_k, b_k≤ w⁰(t⁺_k) w⁰(t_k) ≤b_k.

Hence we obtain thatw(t) is a solution of impulsive differential inequality (2.1)–

(2.3). This completes the proof.

Lemma 2.2 ([5, Theorem 1.4.1]). Assume that

(A1) the sequence {tk} satisfies0< t₀< t₁< t₂< . . .,lim_k→∞t_k=∞;

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(A2) m(t)∈P C¹[R⁺,R]is left continuous at tk fork= 1,2, . . .; (A3) fork= 1,2, . . . andt≥t0,

m⁰(t)≤p(t)m(t) +q(t), t6=tk, m(t⁺_k)≤dkm(tk) +ek,

where p(t), q(t)∈C(R⁺,R), d_k ≥0 and e_k are constants. P C denote the class of piecewise continuous function from R⁺ toR, with discontinuities of the first kind only at t=tk, k= 1,2, . . ..

Then

m(t)≤m(t₀) Y

t0<tk<t

d_kexpZ t t₀

p(s)ds +

Z t

t₀

Y

s<tk<t

d_kexpZ t s

p(r)dr q(s)ds

+ X

t₀<t_k<t

Y

t_k<t_j<t

djexpZ t t_k

p(s)ds ek.

Lemma 2.3 ([11]). Let w(t) be an eventually positive (negative) solution of the differential inequality (2.1)–(2.3). Assume that there existsT ≥t₀such thatw(t)>

0 (w(t)<0)fort≥T. If

t→+∞lim Z t

t₀

Y

t₀<t_k<s

b_k ak

ds= +∞ (2.6)

hold, thenw⁰(t)≥0 (w⁰(t)≤0)fort∈[T, t_l]∪ ∪^+∞_k=l(t_k, t_k+1]

, wherel= min{k: t_k ≥T}.

The following theorem is the first main result of this article.

Theorem 2.4. If condition (2.6)and the following condition holds,

t→+∞lim Z t

t₀

Y

t0<t_k<s

a_k

b_kF(s)ds= +∞, (2.7)

where

F(s) =C 1−

m

X

i=1

g_i q(s) +

n

X

j=1

C_j 1−

m

X

i=1

g_i

q_j(s) exp(−δz(t₀)).

Then each solution of (1.1)–(1.3),(1.4)oscillates inG.

Proof. Let u(t, x) be a non-oscillatory solution of (1.1), (1.4). Without loss of generality, we can assume that there exists T >0, t₀ ≥T, such thatu(t, x)>0, u(t−τ_i, x)>0,i = 1,2, . . . , m, u(t−σ_r, x)>0,r = 1,2, . . . , l, u(t−ρ_j, x)>0, j= 1,2, . . . , nfor any (t, x)∈[t₀,∞)×Ω. From Lemma 2.1, we know thatw(t) is a solution of (2.1)–(2.3).

Fort≥t₀,t6=t_k, k= 1,2, . . ., define z(t) = w⁰(t)

w(t), t≥t0. (2.8)

From Lemma 2.3, we havez(t)≥0,t ≥t0, w⁰(t)−z(t)w(t) = 0. We may assume thatw(t0) = 1, thus in view of (2.1)–(2.3) we have that fort≥t0,

w(t) = expZ t t₀

z(s)ds

, (2.9)

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w⁰(t) =z(t) expZ t t0

z(s)ds

, (2.10)

w⁰⁰(t) =z²(t) expZ t t0

z(s)ds

+z⁰(t) expZ t t0

z(s)ds

. (2.11)

We substitute (2.9)–(2.11) into (2.1) and obtain z²(t) expZ t

t₀

z(s)ds

+z⁰(t) expZ t t₀

z(s)ds +C

1−

m

X

i=1

g_i

q(t) expZ t t₀

z(s)ds

+

n

X

j=1

Cj

1−

m

X

i=1

gi

qi(t) expZ t−ρj

t0

z(s)ds

≤0.

Hence we have

z²(t) +z⁰(t) +C 1−

m

X

i=1

gi

q(t)

+

n

X

j=1

C_j 1−

m

X

i=1

g_i

q_j(t) exp

− Z t

t−ρj

z(s)ds

≤0, t6=t_k, then we have

z⁰(t) +C 1−

m

X

i=1

gi

q(t)

+

n

X

j=1

Cj

1−

m

X

i=1

gi

qj(t) exp

− Z t

t−ρj

z(s)ds

≤0, t6=tk.

From above inequality and conditionbk ≤a_k, we know thatz(t) is non-increasing, thenz(t)≤z(t0), fort≥t0, we obtain

z⁰(t) +C 1−

m

X

i=1

g_i q(t) +

n

X

j=1

C_j 1−

m

X

i=1

g_i

q_j(t) exp(−δz(t₀))≤0, t6=t_k. From (2.2), (2.3) and (2.8), we obtain

z(t⁺_k) =w⁰(t⁺_k)

w(t⁺_k) ≤ bkw⁰(tk) a_kw(t_k) = bk

a_kz(t_k), so we can easily obtain

z⁰(t)≤ −C 1−

m

X

i=1

g_i q(t)−

n

X

j=1

C_j 1−

m

X

i=1

g_i

q_j(t) exp(−δz(t₀)) t6=t_k,

z(t⁺_k)≤ bk

a_kz(tk).

Let

−F(t) =−C 1−

m

X

i=1

g_i q(t)−

n

X

j=1

C_j 1−

m

X

i=1

g_i

q_j(t) exp(−δz(t₀)).

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Then according to Lemma 2.2, we have z(t)≤z(t0) Y

t₀<t_k<t

bk

a_k + Z t

t₀

Y

s<t_k<t

bk

a_k (−F(s))ds

= Y

t0<tk<t

b_k a_k

hz(t₀)− Z t

t₀

Y

t0<tk<s

a_k bk

F(s)dsi

<0.

Sincez(t)≥0, this is a contradiction. The proof is complete.

3. Oscillation properties of the problem (1.1),(1.5) For the second main theorem, we need following lemma.

Lemma 3.1. Let u∈ C²(Γ)∩C¹(¯Γ) be a positive solution of (1.1), (1.5) in G, then functionw(t) satisfies the impulsive differential inequality

w⁰⁰(t) +C 1−

m

X

i=1

gi

q(t)w(t) +

n

X

j=1

Cj

1−

m

X

i=1

gi

qj(t)w(t−ρj)≤0, t6=tk, (3.1) a_k≤ w(t⁺_k)

w(tk) ≤ak, k= 1,2, . . . , (3.2) b_k≤ w⁰(t⁺_k)

w⁰(t_k) ≤bk, k= 1,2, . . . , (3.3) wherew(t) =v(t) +Pm

i=1giv(t−τi).

Proof. Letu(t, x) be a positive solution of the problem (1.1), (1.5) inG. Without loss of generality, we may assume that there exists a T > 0, t0 > T such that u(t, x) > 0, u(t−τi, x) > 0, i = 1,2, . . . , m, u(t−σr, x) > 0, r = 1,2, . . . , l, u(t−ρj, x) > 0, j = 1,2, . . . n, for any (t, x) ∈ [t0,∞)×Ω. For t ≥ t0, t 6= tk, k= 1,2, . . ., integrating (1.1) with respect toxover Ω yields

d² dt²

hZ

Ω

u(t, x)dx+

m

X

i=1

gi

Z

Ω

u(t−τi, x)dxi

=a(t) Z

Ω

h(u)∆u dx− Z

Ω

q(t, x)f(u(t, x))dx +

l

X

r=1

a_r(t) Z

Ω

h_r(u(t−σ_r, x))∆u(t−σ_r, x)dx

−

n

X

j=1

Z

Ω

qj(t, x)fj(u(t−ρj, x))dx.

By Green’s formula and the boundary condition, we have Z

Ω

h(u)∆u dx= Z

∂Ω

h(u)∂u

∂nds− Z

Ω

h⁰(u)|gradu|²dx

=− Z

∂Ω

h(u)ϕ(t, x)u ds− Z

Ω

h⁰(u)|gradu|²dx

≤ − Z

Ω

h⁰(u)|gradu|²dx≤0,

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Z

Ω

hr(u(t−σr, x))∆u(t−σr, x)dx≤0.

The rest of the proof is similar to the one in Lemma 2.1. We omit it.

The following theorem is the second main result of this article.

Theorem 3.2. If conditions(2.6)and (2.7)hold, then each solution of (1.1)–(1.3), (1.5)oscillates inG.

The proof of the above theorem is similar to that of Theorem 2.4. We omit it.

4. Examples Example 4.1. Consider the equation

∂²

∂t² h

u(t, x) +1

2u(t−π 2, x)i

=u²∆u−ue^u²+e^tu²(t−π

2, x)∆u(t−π 2, x)

−(x²+ 1)e^tu(t−3π

2 , x)e^u²^(t−^3π²^,x), t >1, t6= 2^k, (t, x)∈R⁺×Ω =G,

u((2^k)⁺, x) = (4 + sin 2^kcosx)u(2^k, x), k= 1,2, . . . , u_t((2^k)⁺, x) = (2 + sin 2^kcosx)u_t(2^k, x), k= 1,2, . . . , with the boundary condition

u= 0, (t, x)∈R⁺×∂Ω,

where a(t) = 1, a1(t) = e^t, τ1 = ^π₂, σ1 = ^π₂, ρ1 = ^3π₂, h(u) = u², h1(u) = u², f(u) =ue^u²,f1(u) =ue^u²,q(t, x) = 1,q1(t, x) = (x²+ 1)e^t, g1= ¹₂,tk = 2^k. It is easy to verify that the condition (H1)–(H4) and the conditions of Theorem 2.4 are satisfied. Hence the all solutions of above problem oscillate.

Example 4.2. Consider the equation

∂²

∂t²

hu(t, x) +1

2u(t−π 2, x)i

=u²∆u−ue^u²+e^tu²(t−π

2, x)∆u(t−π

2, x)−(x²+ 1)e^tu(t−3π

2 , x)e^u²^(t−^3π²^,x), t >1, t6= 3k, (t, x)∈R⁺×Ω =G,

u((3k)⁺, x) = (4 + sin 3kcosx)u(3k, x), k= 1,2, . . . , ut((3k)⁺, x) = (2 + sin 3kcosx)ut(3k, x), k= 1,2, . . . , with the boundary condition

∂u

∂n+t²x²u= 0, (t, x)∈R⁺×∂Ω,

where a(t) = 1, a1(t) = e^t, τ1 = ^π₂, σ1 = ^π₂, ρ1 = ^3π₂, h(u) = u², h1(u) = u², f(u) = ue^u², f1(u) = ue^u², q(t, x) = 1, q1(t, x) = (x²+ 1)e^t, g1 = ¹₂, tk = 3k, ϕ(t, x) =t²x². It is easy to verify that the condition H) and condition of Theorem 3.2 are satisfied. Hence the all solutions of the above problem oscillate.

Acknowledgments. This research was partially supported by grants from the Na- tional Basic Research Program of China, nos. 2010CB950904 and 2011CB710604.

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Jichen Yang

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei, 430074, China

E-mail address: yjch [email protected]

Anping Liu (corresponding author)

E-mail address:wh [email protected]

Guangjie Liu

E-mail address:[email protected]