In this paper, we consider
the initial-boundary value problem for a generalized Kelvin–Voight equation with
p-Laplacian and a damping term:
class="formula" id="fm0010">
Here
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the velocity field,
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305261&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305261&_rdoc=1&_issn=0022247X&md5=1fcbee39e622ec7032e7a8eebb6e969b" title="Click to view the MathML source">P(x,t)class="mathContainer hidden">class="mathCode"> is
the pressure,
ν is
the viscosity kinematic coefficient, and
ϰ is
the viscosity relaxation coefficient (is a length scale parameter characterizing
the elasticity of
the fluid). The coefficient
γ and
the exponents
p,
m are given constants. Under appropriate conditions on
the data, we prove
the existence and uniqueness of
the global and local weak solutions. Under several assumptions on
the exponents
p,
m,
the coefficients
ν,
ϰ, and specified initial data, a finite time blow-up and
the behavior of
the solutions for large times are also established.