Blow up of Solutions to a Class of Damped Viscoelastic Inverse Source Problem
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Blow up of Solutions to a Class of Damped Viscoelastic Inverse Source Problem Mohammad Shahrouzi1
© Foundation for Scientific Research and Technological Innovation 2017
Abstract This article is devoted to the study blow up of solutions to a quasilinear inverse source problem with memory and damping terms. We obtain sufficient conditions on initial functions for which the solutions blow up in a finite time. Estimates of the lifespan of solutions are also given. Keywords Inverse problem · Blow up · Viscoelastic
Introduction We consider the following inverse problem of determining a pair of functions {u(x, t), f (t)} that satisfy u tt − ∇[(a0 + a|∇u|m )∇u] + +|u| p u
t 0
eλ(t−τ ) g(t − τ )Δu(τ )dτ + bu t = h(x, t, u, ∇u)
+ f (t)ω(x), x ∈ Ω, t > 0
(1.1)
u(x, t) = 0, x ∈ Γ, t > 0
(1.2)
u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ Ω Ω u(x, t)ω(x)d x = 1, t > 0
(1.3) (1.4)
where Ω is a bounded (or unbounded) domain of R n (n ≥ 1) with smooth boundary Γ . Here p > m ≥ 0 and a, b, λ, a0 are positive numbers. Moreover, h(x, t, u, ∇u), g(t) and ω(x) are functions that satisfy specific conditions that will be enunciated later. Without memory term and in direct type, that is, for the wave equations, there are many results about existence, uniqueness, blowing-up, global existence and other properties of the solution, see [9,13–15] and references therein.
B 1
Mohammad Shahrouzi [email protected] Department of Mathematics, Jahrom University, P.O. Box: 74137-66171, Jahrom, Iran
123
Differ Equ Dyn Syst
Recently, Bilgin and Kalantarov [1], studied Eq. (1.1) when g(t) = f (t) = 0. They proved blow-up of solutions with strongly damping term when p > m ≥ 0. In direct problems, it is worth mentioning some papers in connecting with blow up of solutions for viscoelastic equations. Messaoudi [16] showed that, concerning nonexistence, Georgiev and Todorova’s results [9] can be extended to the following equation using the concavity method with a modification in energy functional due to the different nature of the problem: t u tt − Δu + g(t − τ )Δu(τ )dτ + au t |u t |m−2 = bu|u| p−2 . (1.5) 0
Wu in [25] studied (1.5) when a = 0 and proved that there are solutions, under some conditions on the initial data, which blow up in finite time with nonpositive initial energy as well as positive initial energy. Later, in [24], Wang considered (1.5) with a = 1, m = 2 and Dirichlet boundary conditions. He established a blow-up result with arbitrary positive initial energy for 2 < p < 2n−2 n−2 when the initial data and kernel of the memory g have been chosen appropriately (see [4,5,18,23,26]). In contrast with the extensive literature on global behavior of solutions in direct problems, less is known about inverse type. We next recall some existing results of the global existence and blow up of solutions for inverse problems. In [7], Eden and Kalantarov applied the modified concavity method to the problem u t − Δu − |u| p u + b(x, t, u, ∇u) = F(t)ω(x), x ∈ Ω, t > 0 u(x, t) = 0, x ∈ ∂Ω, t > 0
u(x, 0) = u 0 (x), x ∈ Ω Ω
u(x,
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