Asymptotic stability and blow up of solutions for a Petrovsky inverse source problem with dissipative boundary condition

Abstract

Communicated by S. ChenThis paper is concerned with global in time behavior of solutions for a quasilinear Petrovsky inverse source problem with boundary dissipation. We establish a stability result when the integral constraint vanishes as time goes to infinity. We also show that the smooth solutions blow up when the data is chosen appropriately.(1.5) has been considered by Amroun and Benaissa in [1], where the authors proved: global existence of solutions by means of the stable set method in H 2 0 . / combined with the Faedo-Galerkin procedure. In [5], Messaoudi studied problem (1.5) when h.u t / D aju t j m 2 u t and proved the existence of a local weak solution and showed that this solution blows up in finite time with negative initial energy if p > m.

General decay and blow up of solutions for a class of inverse problem with elasticity term and variable‐exponent nonlinearities

“…They found conditions on data guaranteeing global nonexistence of solutions when 𝜙(t) ≡ 1, also established a stability result with the opposite sign on the power type nonlinearity and b(x, t, u, ∇ u) ≡ 0. Next Tahamtani and Shahrouzi 4 extend previous results to a Petrovsky inverse source problem (see also Tahamtani and Shahrouzi 5 ). Shahrouzi 6 studied the following damped viscoelastic inverse problem:…”

On behavior of solutions to a class of nonlinear hyperbolic inverse source problem

“…Therefore, we have Proof of Theorem 2.2 Inspired by the idea in [24], we define F (t) = ME(t) + N (ψ 1 (t) + ξψ 2 (t)), (3.11) where…”

“…For more information about inverse problems, the interested reader is referred to the papers [1,6,11,12,15,20,23] Our objective in the present work is to extend the results of [3,24] by considering problem (1.1)- (1.4) in the presence of m-Laplacian operator and boundary conditions. Motivated by the aforementioned works, our result here is twofold: First, we consider a = 1 and show that if we take initial data and parameters in the appropriately domain, then solutions of (1.1)-(1.4) are asymptotically stable when φ(t) tends to zero as time goes to infinity.…”

Long-time Behaviour of Solution to Inverse Problem for Higher-order Parabolic Equation

“…There are some papers devoted to the study of existence and uniqueness of solutions of inverse problems for various parabolic type equations with unknown source functions [1,3,5]. In [6,7,8,9,10] authors studied global nonexistence and blow-up solution to fourth-order equations.…”