Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem
  with a Mixed Nonhomogeneous Condition   We study the initial-boundary value problem for a nonlinear wave equation
given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u)
, 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t),
u(x,0)=\^u_{0}(x), u_{t}(x,0)={\^u}_{1}(x), where \eta \geq 0, q\geq 2 are
given constants {\^u}_{0}, {\^u}_{1}, g, k, f are given functions. In part I
under a certain local Lipschitzian condition on f, a global existence and
uniqueness theorem is proved. The proof is based on the paper [10] associated
to a contraction mapping theorem and standard arguments of density. In Part} 2,
under more restrictive conditions it is proved that the solution u(t) and its
derivative u_{x}(t) decay exponentially to 0 as t tends to infinity.

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The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin