The asymptotic behavior of solutions to an inverse problem for differential operator equations

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

“…They proved the global nonexistence of solutions with sufficiently large initial energy by using the concavity arguments [15,16]. Also Guvenilir and Kalantarov established the same results for an inverse problem for differential operator equations in [12]. Gozukizil and Yaman in [13,14] used contraction mapping theorem and proved the existence and unique solvability of parabolic and hyperbolic inverse source problems.…”

“…In the presence of unknown function .f .t/ ¤ 0/ and in contrast with the extensive literature on asymptotic stability and global nonexistence results for direct problems, there are few stability and blow-up results for hyperbolic and parabolic inverse problems (see [10][11][12][13][14]). In [11], Eden and Kalantarov considered the problem: u tt u juj p u C a.x, t, u, ru/ D f .t/!.x/, x 2 , t > 0, u.x, t/ D 0, x 2 , t > 0, u.x, 0/ D u 0 .x/, u t .x, 0/ D u 1…”

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

“…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.…”

Global nonexistence and stability of solutions of inverse problems for a class of Petrovsky systems

“…There are numerous papers devoted to the study of stability and global nonexistence results for direct problems and the existence, uniqueness of solutions of inverse problems for various evolutionary partial differential equations (see [2,6,7,11,[13][14][15]). But less is known about the global nonexistence for solutions of hyperbolic and parabolic inverse problems.…”