| Summary: | The main goal of the present paper is to study the existence, uniqueness and behavior of a solution for a coupled system of nonlinear viscoelastic wave equations with the presence of weak and strong damping terms. Owing to the Faedo-Galerkin method combined with the contraction mapping theorem, we established a local existence in <inline-formula> <math display="inline"> <semantics> <mrow> <mo>[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo>]</mo> </mrow> </semantics> </math> </inline-formula>. The local solution was made global in time by using appropriate a priori energy estimates. The key to obtaining a novel decay rate is the convexity of the function <inline-formula> <math display="inline"> <semantics> <mi>χ</mi> </semantics> </math> </inline-formula>, under the special condition of the initial energy <inline-formula> <math display="inline"> <semantics> <mrow> <mi>E</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. The condition of the weights of weak and strong damping has a fundamental role in the proof. The existence of both three different damping mechanisms and strong nonlinear sources make the paper very interesting from a mathematics point of view, especially when it comes to unbounded spaces such as <inline-formula> <math display="inline"> <semantics> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </semantics> </math> </inline-formula>.
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