Global Existence and Intrinsic Decay Rates for the Energy of a Kirchhoff Type in a Nonlinear Viscoelastic Equation

Draifia Alaeddine

I. INTRODUCTION

In this works, we study the global existence and intrinsic decay rates for the energy of a kirchhoff type in a nonlinear viscoelastic equation

with initial data

and boundary conditions

where $\Omega$ is a bounded domain in $\mathbb{R}^n$ , $h(.): \mathbb{R}_+ \longrightarrow \mathbb{R}_+$ are given functions which will be spaced later and $u_0(x)$ , $u_1(x)$ are given initial data belonging to appropriate space. All the parameter $b$ are assumed to be positive constants. The function $\Phi(x)$ is the density, $(\rho(x))^{-1} = \Phi(x)$ , $\Phi(x) > 0$ , for all $x \in \Omega$ , and

where s > 0, $\xi_{0} > 0$ , $\xi_{1} > 0$ and $\gamma \geq 1$ . For more information on using Kirchhoff type, see [4 - 6].

Also, a result of local existence for problem (1.1) - (1.3) for $\xi_1 = 0$ has been proved in [1], for $\xi_1 \neq 0$ , in the same way as [1], we get the same basic results for the local existence of problem (1.1) - (1.3) with a slight change in some calculations that do not affect the basic results.

The motivation of our work is due to some results regarding the following research papers: Boumaza, N and Boulaaras, S. [2] studied the general decay for Kirchhoff type in viscoelasticity with not necessarily decreasing kernel of (1.1)–(1.3). Marcelo M. Irena Lasiecka and Claudete M. Webler. [3] studied the intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density. M. M. Cavalcanti, V. N. Domingos Cavalcanti, I. Lasiecka and F. A. Falcao Nascimento. [7] studied the intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects.

I. Lasiecka, S. A. Messaoudi and M.

I. Mustafa. [8] studied the note on intrinsic decay rates for abstract wave equations with memory.

I. Lasiecka and X. Wang. [9] studied the intrinsic decay rate estimates for semilinear abstract second order equations with memory. Cavalcanti M. Filho VND. Cavalcanti JSP. Soriano JA. [10] studied the existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping. For more results in this direction, see $[11-15]$ .

However, $[1-3]$ , $[4-6]$ and $[11-15]$ did not study the intrinsic decay rates for the energy of problem (1.1)–(1.3) of relaxation kernels described by the inequality $h'(t) \leq -H(h(t))$ for all $t \geq 0$ , with H convex. Motivated by the above research, we will consider the intrinsic decay rates for the energy of relaxation kernels described by the inequality $h'(t) \leq -H(h(t))$ for all $t \geq 0$ of the model (1.1)–(1.3) in this paper.

The outline of the paper is as follows. In the second section we define the energy $E(t)$ associated to (1.1) – (1.3) and show that it is a non-increasing function of t. In section 3, we prove global existence of solution of (1.1) – (1.3). Finally, in section 4, we prove the intrinsic decay rates for the energy of the posed problem.

II. ASSUMPTIONS AND MAIN RESULTS

In this section, we define the energy $E(t)$ associated to (1.1)–(1.3) and show that it is a non-increasing function of t. We suppose that the kernel $h(t)$ is a function satisfying

Assumptions 1

The relaxation function $h:\mathbb{R}_{+}\longrightarrow \mathbb{R}_{+}$ is a $C^1\cap L^1$ decreasing function and satisfies

Assumptions 2 (i) In addition to Assumption 1, we require

where $H \in C^1(R_+)$ which $H(0) = 0$ is a given strictly increasing and convex function. Moreover,

(ii) With reference to the function $H$ introduced above, let $y(t)$ be the solution of the ODE

(iii) We assume that there exists $\alpha_0\in [0,1)$ such that $y^{1 - \alpha_0}\in L_1(1,\infty)$ . In order to formulate the long-time behavior results, we recall the binary notation

Note that, in view of (2.1), we have that

The energy satisfies the following identity

Lemma 1. We have the identity

Proof. Multiplying (1.1) by $\rho(x)u_{t}$ and integration over $\Omega$ , we have

We have

And by using integration by parts, we have

Notes

(2.7) And by using integration by parts, we have

and using

then

by using (2.1), we get

Notes

and

By replacement (2.9) and (2.10) into (2.8), we get

By combining (2.6), (2.7) and (2.11) into (2.5), we get

Notes then

by using (2.2) into (2.12), we get (2.4).

The proof of Lemma 1 is completes.

III. GLOBAL EXISTENCE

In this section we show that any solution of $(1.1)-(1.3)$ is bounded and global, provided that $E(0)$ is positive and small enough.

Theorem 1. Assume that (2.3) holds. Then the solution to problem (1.1) - (1.3) is bounded and global.

Proof. It suffices to show that $\| u_t(t)\|_{L_\rho^2 (\Omega)}^2 +\| \nabla u(t)\|_{L^2 (\Omega)}^2$ is bounded independently of $t$ .

By using (2.3) and (A1) into (2.12), we get

where $\omega_{1} > 0$ and $\omega_{2} > 0$ , then

where $\omega_{3} > 0$ .

Then the solution to problem (1.1) - (1.3) is bounded and global.

The proof of Theorem 1 is completes.

Now, we are in a position to state our main result.

Lemma 2. Let us assume that Assumption 1 and Assumption 2 are the place. Then, there exists a positive constant $T_{0} > 0$ such that

for all $T > T_{0}$ and all $n \in N$ , where $\tilde{H}$ is given in (4.49) and $C_{9}$ is given in (4.52).

Proof. For this purpose, a by now standard procedure is to multiply (1.1) by the viscoelastic multiplier

and integrating over $\Omega \times (nT,(n + 1)T)$ , we infer that

We shall analyze the above terms separately.

Direct calculation give

For the second term, by using integration by parts, we have

Notes For the third term, by using integration by parts, we have

Combining (4.2) - (4.4) into (4.1), we arrive at

\begin{array}{l} \left(u _ {t} (t), \int_ {0} ^ {t} h (t - s) (u (t) - u (s)) d s\right) _ {L _ {\rho} ^ {2} (\Omega)} \Bigg | _ {n T} ^ {(n + 1) T} \\- \int_ {n T} ^ {(n + 1) T} \left(u _ {t} (t), \int_ {0} ^ {t} h ^ {\prime} (t - s) (u (t) - u (s)) d s\right) _ {L _ {\rho} ^ {2} (\Omega)} d t \\- \int_ {n T} ^ {(n + 1) T} \left(\int_ {0} ^ {t} h (s) d s\right) \| u _ {t} (x, t) \| _ {L _ {\rho} ^ {2} (\Omega)} ^ {2} d t \\+ \int_ {n T} ^ {(n + 1) T} \left(\xi_ {0} + \xi_ {1} \| \nabla u \| _ {L ^ {2} (\Omega)} ^ {2 \gamma}\right) \left(\nabla u (t), \int_ {0} ^ {t} h (t - s) (\nabla u (t) - \nabla u (s)) d s\right) _ {L ^ {2} (\Omega)} d t \\+ \int_ {n T} ^ {(n + 1) T} \left\| \int_ {0} ^ {t} h (t - s) (\nabla u (t) - \nabla u (s)) d s \right\| _ {L ^ {2} (\Omega)} ^ {2} d t \\- \int_ {n T} ^ {(n + 1) T} \left(\int_ {0} ^ {t} h (t - s) \nabla u (t) d s, \int_ {0} ^ {t} h (t - s) (\nabla u (t) - \nabla u (s)) d s\right) _ {L ^ {2} (\Omega)} d t \\+ b \int_ {n T} ^ {(n + 1) T} \left(u _ {t} (t), (h \diamond u) (t)\right) _ {L ^ {2} (\Omega)} d t \\= 0, \end{array} \tag {4.}\tag{4.5}

then (4.5) is equivalent

Notes

Estimate for $|J_1|$ , where

Now, let $m \in N$ be an arbitrary, natural number.

By using Young's inequality (for $\varepsilon = 1$ ), we get

Notes by replacement (4.9) into (4.7) and using $\int_0^{mT}h(mT - s)ds = \int_0^{mT}h(s)ds$ , we get

by using (2.2), we get

then, by combining (4.11) into (4.10), we get

Estimate for $|J_2|$ , where

By using Young's inequality (for $\varepsilon = \frac{\varepsilon_1}{2}$ ), we get

Notes By using (4.8), Cauchy-Schwarz inequality and (2.1), we get

By replacement (4.14) into (4.13), we get

Estimate $|J_3|$ , where

By using Young's inequality (for $\varepsilon = \frac{\varepsilon_2}{2}$ ), we get

by using $\| \nabla u(t)\|_{L^2 (\Omega)}^{2\gamma}\leq \left(\frac{2(\gamma + 1)}{\xi_1} E(0)\right)^{\frac{2\gamma}{2(\gamma + 1)}}$ , we get

by using Cauchy-Schwarz inequality and (2.1), we get

Notes

by replacement (4.17) and (4.18) into (4.16), we get

Estimate $|J_4|$ , where

By using (4.18), we get

Now, estimate $|J_5|$ , where

By using Young's inequality (for $\varepsilon = \frac{\varepsilon_3}{2}$ ), Cauchy-Schwarz inequality and (4.18), we get

Notes

Now, estimate $|J_6|$ , where

By using Young's inequality (for $\varepsilon = \frac{\varepsilon_1}{2}$ ), (4.8) and (4.18), we get

Combining (4.12), (4.15) and (4.19)-(4.22) into (4.6), and recalling that $\| h\|_{L^1 (0,\infty)} < \xi_0$ , we write

Notes

(4.23)

Since $h(0) > 0$ , we can select a points $t_{1} < T$ with $t_{1}$ close to zero such that for all $t \geq t_{1}$

Then (4.23) is equivalent

Now, multiplying (1.1) by $\rho(x)u(x,t)$ and integrating over $\Omega\times(nT,(n+1)T)$ , we infer that

Notes

By using

we get

By using integration by parts, we get

Notes By combining (4.26) - (4.29) into (4.25), we get

Then (4.30) is equivalent

By using Young's inequality (for $\varepsilon = 1$ ), (4.8), $\frac{1}{2} \| u_t(t)\|_{L_\rho^2 (\Omega)}^2\leq E(t)$ and $\frac{1}{2}\| \nabla u(t)\|_{L^2 (\Omega)}^2\leq l^{-1}E(t)$ , we get

Notes

then

where

To estimate the term

By using Young's inequality (for $\varepsilon = \frac{\varepsilon_4}{2}$ ) and (2.1), we get

\begin{array}{r l} & {\mathrm{and} \frac {1}{2} \| \nabla u (t) \| _ {L ^ {2} (\Omega)} ^ {2} \leq l ^ {- 1} E (t), \mathrm{weget}} \\& {\qquad - \frac {b}{2} \left\{\| u ((n + 1) T) \| _ {L _ {\rho} ^ {2} (\Omega)} ^ {2} - \| u (n T) \| _ {L _ {\rho} ^ {2} (\Omega)} ^ {2} \right\}} \\& {\leq \frac {b}{2} \| \rho \| _ {L ^ {2} (\Omega)} ^ {2} \left\{\| \nabla u ((n + 1) T) \| _ {L ^ {2} (\Omega)} ^ {2} + \| \nabla u (n T) \| _ {L ^ {2} (\Omega)} ^ {2} \right\}} \\& {\leq b \| \rho \| _ {L ^ {2} (\Omega)} ^ {2} l ^ {- 1} \left\{E ((n + 1) T) + E (n T) \right\}} \\& {= C _ {2} ^ {\prime} \left\{E ((n + 1) T) + E (n T) \right\},} \end{array} \tag {4.8}\tag{4.34}

By combining (4.32) - (4.34) into (4.31), we can write

Notes

(4.35) On multiplied (4.24) by $\gamma_{1}$ and multiplied (4.35) by $\gamma_{2}$ and combining suitably, we get

Let

by using (4.37) and (4.38) into (4.36), we get

where

Adding and subtracting in (4.39) the term

in order to recover the energy $E(t)$ , we obtain

(4.40) where

From (4·40), choosing $\varepsilon$ sufficiently small, $k_{1} > 0$ and T large enough and using

where

we get

In the last step, we need to relate the viscoelastic energy to the viscoelastic damping. In the case when the relaxation function obeys a linear equation, this relation is straightforward and is expressed by a suitable multiplication. However, in the case of general decays, additional arguments are used. Here, we follow $[17]$ . From the assumption 2 made on the viscoelastic kernel h and from $[17, Lemma 4]$ we obtain

Next, we shall employ the following version of Jensen's inequality applied to measures and convex functions $F$ . Let $F$ be a convex increasing function on $[\alpha, b]$ , let $f: \Omega \longmapsto [\alpha, b]$ , and let $h$ be an integrable function such that $h(x) \geq 0$ and

Then, we have

We shall use (4.44) in order to bring the functions $H$ in front of the integrals. Let us denote

We note that the function $\hat{H}^{-1} + k_{1}$ is concave. Let

thus, we have

On the other hand, from the identity $(2.4)$ for the energy, we can write

where

En replacement (4.45) into (4.43) and using

we get

and using (4.46) we get

thus, we get

Notes

where

By integrating $t$ to $(n + 1)T$ on both sides of the inequality $\frac{d}{dt}\{E(t)\} \leq 0$ yields

integrating (4.50) from $nT$ to $(n + 1)T$ yields

by replacement (4.51) into (4.48), we get

For T large enough, where $C_{6}$ is a positive constant, which implies that

where

which gives that

Notes

by using (4.47) into (4.53), we get

from the above we have

Then the Proof of Lemma 2 is complete.

Lemma 3. Let p be a positive, increasing function such that $p(0) = 0$ . Since p is increasing, we can define an increasing function q, $q(x) \equiv x - (I + p)^{-1}(x)$ . Consider a sequence $F_{n}$ of positive numbers which satisfies

Then $F_{m} \leq S(m)$ where $S(t)$ is a solution of the differential equation

Moreover, if $p(x) > 0$ for $x > 0$ then $\lim_{t \to \infty} S(t) = 0$ .

Proof. Proof of the Lemma use the proof retraction. Assume $F_{m} \leq S(m)$ and prove that $F_{m+1} \leq S(m+1)$ .

Inequality (4.54) is equivalent to

and since $(I+p)^{-1}$ is monotone increasing, $F_{m+1} \leq (I+p)^{-1} F_m$ , and using

On the other hand, since q is an increasing function, the solution $S(t)$ of equation (4.55) is described by a nonlinear contraction.

In particular integrating $\frac{d}{dt}\left\{S(t)\right\} \leq 0$ from $m$ to $\tau$ yields

Integrating equation (4.55) from $m$ to $(m + 1)$ yields

Since q is increasing, by using (4.57) we obtain for all $m \leq \tau \leq m + 1$

Notes

then

by replacement (4.59) into (4.58) and using the inductive assumption $F_{m} \leq S(m)$ , we get

comparing (4.60) with (4.56) yields

Then the Proof of Lemma 3 is complete.

Theorem 2. Let us assume that Assumption 1 and Assumption 2 are the place. Then there exist positive constants $c_1, c_2$ and $T_0$ such that the solution of problem (1.1) - (1.3) satisfies $E(t) \leq s(t)$ , where $s(t)$ verifies the ODE

with $\hat{H}(s) = c_1H(c_2s)$ .

Proof. Thus, we are in a position to apply the result of Lemma 2 with

This yields

Setting $t = mT + \tau$ and recalling the evolution property gives

which completes the proof of Theorem 2.