Abstract
In this paper, we establish a general decay rate properties of solutions for a coupled system of viscoelastic wave equations in IRn under some assumptions on g1; g2 and linear forcing terms. We exploit a density function to introduce weighted spaces for solutions and using an appropriate perturbed energy method. The questions of global existence in the nonlinear cases is also proved in Sobolev spaces using the well known Galerkin method.
Highlights
Which means that, our energy is uniformly bounded and decreasing along the trajectories
We start with some results related to viscoelastic plate equations with strong damping in [23]: t utt + ∆2u − ∆pu − g(t − s)∆u(s, x)ds − ∆ut + f (u) = 0, supplemented with the following conditions: x ∈ Ω × R+, u(t, x) = ∆u = 0, on ∂Ω × R+, u(0, x) = u0, ut(0, t) = u1, on Ω
The problem related to (1.1) in a bounded domain Ω ⊂ Rn, (n ≥ 1) with a smooth boundary ∂Ω and g is a positive nonincreasing function was considered as equation in [15], where they established an explicit and very general decay rate result for relaxation functions satisfying: g′(t) ≤ −H(g(t)), t ≥ 0, H(0) = 0, for a positive function H ∈ C1(R+) and H is linear or strictly increasing and strictly convex C2 function on
Summary
We introduce some notation and results needed for our work. We omit the space variable x of u(x, t), u′(x, t) and for simplicity reason denotes u(x, t) = u and u′(x, t) = u′, when no confusion arises. (A1 ) We assume that the function gi : R+ −→ R+(for i = 1, 2) is of class C1 satisfying:. We define the function spaces of our problem and its norm as follows: D(Rn) = f ∈ L2n/(n−2)(Rn) : ∇f ∈ (L2(Rn))n ,. The spaces L2ρ(Rn) to be the closure of C0∞(Rn) functions with respect to the inner product:. L2ρ(Rn) are defined with respect to the inner product (2.5), we may consider equation (2.10) as operator equation:. (u, v)E = ∇u∇vdx, Rn and the energy space is the completion of D(∆0) with respect to (u, v)E. Its domain is a Hilbert space with respect to the graph scalar product (u, v)D(∆1) = (u, v)L2ρ + (∆1u, ∆1v)L2ρ , for all u, v ∈ D(∆1). We are ready to state and prove our existence results
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