DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

ON THE GLOBAL BEHAVIOR OF SOLUTIONS TO
NONLINEAR INVERSE SOURCE PROBLEMS
A. EDEN AND V. K. KALANTAROV
We find conditions on data guaranteeing global nonexistence of solutions to inverse
source problems for nonlinear parabolic and hyperbolic equations. We also establish stability
results on a bounded domain for corresponding problems with the opposite sign
on the power-type nonlinearities.
Copyright © 2006 A. Eden and V. K. Kalantarov. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Inverse problems are most of the time ill-posed. Therefore, the powerful tools and techniques
of the dynamical systems theory usually do not apply. However, in the rare and
fortunatecaseswherethegiveninverseproblemiswell-posedonecanstudythelong
time behavior of solutions. The questions that can be addressed are, but not limited to
(i) the global existence or nonexistence of solutions; (ii) the stability of solutions, (iii) the
regularity of solutions; (iv) the stability of solutions in a wider sense, that is, the existence
of an (exponential) attractor, its finite dimensionality, the structure of the global attractor
and so forth.
In [5, 6], we have tried to address some of these questions for a class of semilinear
parabolic and hyperbolic inverse source problems with integral constraints. In particular,
depending on the sign of the nonlinearity we have established both global nonexistence
results as well as stability results. One of the standard tools for establishing the
global nonexistence of solutions is the concavity argument that was introduced by Levine
[11, 10] and was generalized in [9]. This approach requires the appropriate functional to
satisfy the desired differential inequality, with a judicious choice of the initial data. The
main trust of the work in [5, 6] was to prove global nonexistence results for parabolic
and hyperbolic inverse source problems using the generalized concavity argument given
in [9]. In the same works, when the nonlinearities have the opposite sign, it was shown
that the solutions converge to zero in H 1 -norm when the integral constraint that drives
the system tends to 0.
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 391–402