Asymptotic Behavior for a Nondissipative and Nonlinear System of the Kirchhoff Viscoelastic Type

Theorem 1.1. Assume that $$ and h(t) is a nonnegative summable kernel. If 0 < p < 2/(n − 2) when n ≥ 3 and p > 0 when n = 1,2, then there exists a unique solution u to problem (1.1) such that

for T small enough.

Proposition 2.1. There exist ρ_i > 0,   i = 1,2 such that

for all t ≥ 0 and small λ_i,   i = 1,2.

Proof. By the inequalities

where C_p is the Poincaré constant, we have On the other hand, Therefore, ρ₁[ℰ(t) + Φ₃(t) + Φ₄(t)] ≤ L(t) ≤ ρ₂[ℰ(t) + Φ₃(t) + Φ₄(t)] for some constant ρ_i > 0,   i = 1,2 and small λ_i,   i = 1,2 such that λ₁ < min {1, (1 − κ)/C_p} and λ₂ < min {1/C_pκ, 1 − λ₁}.

The identity to follow is easy to justify and is helpful to prove our result.

Lemma 2.2. One has for h ∈ C(0, ∞) and v ∈ C((0, ∞); L²(Ω))

Lemma 2.3. Let a(t) be a positive continuous function in J : = [α, β),   k_j(t),   j = 1, …, n are nonnegative continuous functions, g_j(u),   j = 1, …, n are nondecreasing continuous functions in R₊, with g_j(u) > 0 for u > 0, and u(t) is a nonnegative continuous functions in J. If g₁ ∝ g₂ ∝ ⋯∝g_n in (0, ∞), then the inequality

implies that where c₀(t): = sup _0≤s≤ta(s), and β₀ is chosen so that the functions c_j(t),   j = 1, …, n are defined for α ≤ t < β₀.

Lemma 2.4. Assume that 2 ≤ q < +∞ if n = 1,2 or 2 ≤ q < 2n/(n − 2) if n ≥ 3. Then there exists a positive constant C_e = C_e(Ω, q) such that

for $$.

Lemma 3.1. One has for t ≥ t_* and δ_i > 0,   i = 1, …, 5

where BV[h] is the total variation of h.

Proof. This lemma is proved by a direct differentiation of Φ₂(t) along solutions of (1.1) and estimation of the different terms in the obtained expression of the derivative. Indeed, we have

or Therefore, For all measurable sets 𝒜 and 𝒬 such that 𝒜 = R⁺∖𝒬, it is clear that For δ₁ > 0, the first term in the right-hand side of (3.8) satisfies and the third one fulfills

Back to (3.8) we may write

The last term in the right-hand side of (3.7) will be estimated as follows: For the fourth term in (3.7), it holds that Moreover, from Lemma 2.4, for p > 0 if n = 1,2 and 0 < p < 2/(n − 2) if n ≥ 3, we find The definition of ℰ(t) in (2.5) allows us to write

Gathering all the relations (3.11)–(3.15) together with (3.7), we obtain for t ≥ t_*

Theorem 3.2. Assume that the hypotheses (H1)–(H3), (A)–(C) hold and ℛ_h < 1/4. If $$, then, for global solutions and small $$, there exist T₁ > 0 and U > 0 such that L(t) > U,   t ≥ T₁ or

for some positive constants M₁ and ν₁ as long as (D) holds. If $$, then there exist T₂ > 0 and V > 0 such that L(t) > V,   t ≥ T₂ or for some positive constants M₂ and ν₂ as long as (E) holds.

Proof. A differentiation of Φ₁(t) with respect to t along trajectories of (1.1) gives

and Lemma 2.2 implies Next, a differentiation of Φ₃(t) and Φ₄(t) yields

Taking into account Lemma 3.1 and the relations (2.6), (3.20)-(3.21), we see that

Next, as in [17], we introduce the sets and observe that where 𝒩_h is the null set where h^′ is not defined and 𝒬_h is as in (3.2). Furthermore, if we denote 𝒬_n : = R⁺∖𝒜_n, then $$ because 𝒬_n+1 ⊂ 𝒬_n for all n and $$. Moreover, we designate by $$ the sets In (3.22), we take $$ and $$. Choosing λ₁ = (h_* − ε)λ₂, it is clear that for small ε and δ₂, large n and t_*, if $$. We deduce that Furthermore, if $$, then with and a small β > 0. Pick and H_γ(0) such that Note that this is possible if t_* is so large that h_* > 7κ/(8 − κ) even though

Taking the relations (3.22)–(3.30) into account and selecting λ₂ < δ₃/C_pBV[h] so that

and small enough so that we find for δ₃ = ε/2, large δ₄, small Ψ_γ(0), and t ≥ t_* for some positive constant C₁. Take $$ small, and (i.e., A = 2(q + 1)(λ₁ − α) for some 0 < α < λ₁) and (i.e., B = (p + 2)(λ₁ − β) for some 0 < β < λ₁) to derive that for some positive constant C₂.

If lim _t→∞η(t) ≠ 0, then there exist a $$ and C₃ > 0 such that η(t) ≥ C₃ for $$. Thus, in virtue of Proposition 2.1, for C₃ > 0, we have

where If there exists a $$ such that $$, where $$, then from (3.39) and it follows that for t ≥ T with $$. Now we apply Lemma 2.3 to get where $$ and $$. If, in addition, $$, then $$ is uniformly bounded by a positive constant C₄. Thus and by continuity (3.44) holds for all t ≥ 0.

If lim _t→∞η(t) = 0, then for any C > 0 there exists a $$ such that η(t) ≤ C for $$. Therefore,

for some C₅ > 0. The previous argument carries out with $$ replaced by $$.

In case that q < p, we reverse the roles of p and q in the argument above. The case p = q is clear.

Remark 3.3. The case where the derivative of the kernel does not approach zero on 𝒜 (as is the case, for instance, when h^′ ≤ −Ch on 𝒜) is interesting. Indeed, the right-hand side in condition (3.34) will be replaced by C/4 with a possibly large constant C.

Remark 3.4. The argument clearly works for all kinds of kernels previously treated where derivatives cannot be positive or even take the value zero. In these cases there will be no need for the smallness conditions on the kernels. This work shows that derivatives may be positive (i.e., kernels may be increasing) on some “small” subintervals and open the door for (optimal) estimations and improvements of these sets.

Remark 3.5. The assumptions a^′(t) < 2(q + 1)(λ₁ − α)a(t) and b^′(t) < (p + 2)(λ₁ − β)b(t) may be relaxed to a^′(t) < 2(q + 1)λ₁a(t) and b^′(t) < (p + 2)λ₁b(t), respectively. In this case α = α(t) and β = β(t) would depend on t.

Remark 3.6. The assertion in Theorem 3.2 is an “alternative” statement. As a next step it would be nice to discuss the (sufficient conditions of) occurrence of each case in addition to the global existence.