Georgian Math. J. 19 (2012), 575 – 586
DOI 10.1515 / gmj-2012-0028 © de Gruyter 2012
Global nonexistence and stability of solutions of
inverse problems for a class of Petrovsky systems
Mohammad Shahrouzi and Faramarz Tahamtani
Abstract. In this work, we find conditions on data guaranteeing the global nonexistence
of solutions to inverse source problems for a class of Petrovsky systems. We also estab-
lish asymptotic stability results for the corresponding problems with the opposite sign of
power-type nonlinearities and the integral constraint vanishing as time tends to infinity.
Keywords. Inverse problem, global nonexistence, asymptotic stability.
2010 Mathematics Subject Classification. 35B40, 35B44, 65N21.
1 Introduction and preliminaries
There are numerous papers devoted to the study of stability and global nonex-
istence results for direct problems and the existence, uniqueness of solutions of
inverse problems for various evolutionary partial differential equations (see [2,6,
7, 11, 13–15]). But less is known about the global nonexistence for solutions of
hyperbolic and parabolic inverse problems. The interested reader is referred to the
papers [4, 5].
One of the standard tools for establishing the global nonexistence of solutions is
the concavity argument that was introduced by Levine [9,10] and was generalized
in [8]. In [5], Eden and Kalantarov applied the modified concavity method to the
problem
u
t
u juj
p
u C b.x;t;u; r u/ D F.t/!.x/; x 2 ;t>0;
u.x;t/ D 0; x 2 @;t>0;
u.x;0/ D u
0
.x/; x 2 ;
Z
u.x;t/!.x/dx D 1; t>0;
and established global nonexistence results as well as stability results depending
on the sign of nonlinearity.
In this work, by modifying the methods in [5], we study the global in time be-
havior of solutions to an inverse problem for a class of Petrovsky systems. More-
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