| Summary: | This paper is focused on energy decay rates for the viscoelastic wave equation that includes nonlinear time-varying delay, nonlinear damping at the boundary, and acoustic boundary conditions. We derive general decay rate results without requiring the condition <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and without imposing any restrictive growth assumption on the damping term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> using the multiplier method and some properties of the convex functions. Here we investigate the relaxation function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ψ</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>≤</mo><mo>−</mo><mi>μ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>G</mi><mrow><mo>(</mo><mi>ψ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <i>G</i> is a convex and increasing function near the origin, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> is a positive nonincreasing function. Moreover, the energy decay rates depend on the functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula> and <i>G</i>, as well as the function <i>F</i> defined by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>0</mn></msub></semantics></math></inline-formula>, which characterizes the growth behavior of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>f</mi><mn>1</mn></msub></semantics></math></inline-formula> at the origin.
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