Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Global existence and Asymptotic behavior for a system of wave equation
  in presence of distributed delay term   In this paper, we consider the following viscoelastic coupled wave equation
with a delay term: $$ \begin{gathered} u_{tt}(x,t)-Lu(x,t)-\int_0^t
g_1(t-\sigma)L u(x,\sigma)d\sigma + \mu_{1}u_{t}(x,t) + \int_{\tau_1}^{\tau_2}
\mu_2(s)u_{t}(x,t-s)ds + f_1(u,\upsilon)=0, \\ \upsilon_{tt}(x,t) -
L\upsilon(x,t) - \int_0^t g_{2}(t-\sigma)L \upsilon(x,\sigma)d\sigma +
\mu_3\upsilon_t(x,t) + \int_{\tau_1}^{\tau_2} \mu_4(s)\upsilon_{t}(x,t-s)ds +
f_{2}(u,\upsilon)=0, \end{gathered} $$ in a bounded domain. Under appropriate
conditions on $\mu_{1}$, $\mu_{2}$, $\mu_{3}$ and $\mu_{4}$, we prove global
existence result by combining the energy method with the Faedo-Galerkin's
procedure. In addition , we focus on asymptotic behavior by using an
appropriate Lyapunov functional.

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The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat