Abstract
In this paper, we consider the initial-boundary value problem of nonlinear viscoelastic plate equations with dissipative terms. We prove that, for certain initial data in the stable set, the decay rate estimate of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using Nakao’s method. Conversely, for certain initial data in the unstable set, we use the perturbed energy method to show that the solution blows up in finite time when the initial energy is not larger than some positive number. This improves earlier results in the literature.
Highlights
In this paper, we consider the following initial-boundary value problem of the nonlinear viscoelastic plate equations with dissipative terms: ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨utt – γ u – = f (u, v), vtt – δ v – ut + u – utt +t g (t s) u(s) ds + |ut|p– ut (x, t) ∈ × (, T), vt + v – vtt +v(s) ds + |vt|q– vt ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩= f (u, v), (x, t) ∈ × (, T), u(x, ) = u (x), ut(x, ) = u (x), v(x, ) = v (x), vt(x, ) = v (x), u(x, t) = ∂νu(x, t) =, v(x, t) = ∂ν x∈ x∈ v(x, t), =
A solution with positive initial energy blows up in finite time when the initial data is inside the unstable set
For h = |ut|m– ut and h = |vt|r– vt, Han and Wang [ ] showed several results concerned with local existence, global existence, and finitetime blow-up with negative initial energy. The latter blow-up result has been improved by Messaoudi, Said-Houari, and Guesmia [, ] by studying a larger class of initial data for which the initial energy can take positive values and obtained that the rate of decay of the total energy depends on those of the relaxation functions
Summary
We consider the following initial-boundary value problem of the nonlinear viscoelastic plate equations with dissipative terms: ). Kafini and Messaoudi [ ] considered a nonlinear wave equation and obtained a finite-time blow-up result with arbitrary positive initial energy. Xu and Yang [ ] established a blow-up result for certain solutions of
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