Blow-up solution and stability to an inverse problem for a pseudo-parabolic equation
© Yaman; licensee Springer 2012
Received: 20 February 2012
Accepted: 14 November 2012
Published: 28 November 2012
We consider a two-fold problem for an inverse problem of pseudo-parabolic equations with a nonlinear term. Sufficient conditions for a blow-up solution are derived and a stability result is established.
Additional information about the solution to the inverse problem is given in the form of the integral overdetermination condition (4). From the physical point of view, this condition may be interpreted as measurements of the temperature by a device averaging over the domain Ω .
This type of equations arises from a variety of mathematical models in engineering and physical sciences; for example, inverse scattering problems in quantum physics, an inverse problem of interest in geophysics .
Existence and uniqueness of solutions to an inverse problem for parabolic and pseudo-parabolic equations are studied in [3–6]. Stability of solutions is investigated by several authors [1, 7]; but less is known about blow-up solutions. Eden and Kalantarov  studied the same problem without a strong damping term . Meyvaci  established a blow-up result for the pseudo-parabolic equation , where is a given integer and is a number.
where conditions (2), (3) and (A1) are used. Substituting (7) into (1), problem (1)-(3) yields a direct problem given by .
2 Blow-up result
Firstly, let us note the following lemma known as ‘generalized concavity lemma’ or ‘Ladyzhenskaya-Kalantarov lemma’. It is an important tool to obtain the blow-up solutions to parabolic- and hyperbolic-type equations.
Here, and .
Proof See . □
where and . Then the solution of the problem (1)-(4) with the weight function blows up in a finite time.
So, inequality (8) is satisfied with , , . Thus, the desired result is obtained by applying Lemma 1. □
3 Stability of problem
with a constant , where is constant in (6).
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