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Blow-up solution and stability to an inverse problem for a pseudo-parabolic equation
Journal of Inequalities and Applications volume 2012, Article number: 274 (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.
Let us consider the following inverse problem for a pseudo-parabolic equation:
where is a bounded domain with a sufficiently smooth boundary ∂ Ω, p and a are positive constants, and are given functions satisfying
with weight function , and a constant
The inverse problem consists of finding a pair of functions satisfying (1)-(4) when
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.
Here, we used the following notations:
are the arithmetic-geometric inequality and Young’s inequality for respectively;
with , and the Poincare-Friedrich inequality
where is the first eigenvalue of the eigenvalue problem
Multiplying both sides of (1) by ω and integrating the resulting equation over Ω lead to the following relation:
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.
Lemma 1 Let , and . Suppose that a positive, twice differentiable function satisfies the inequality
then goes to infinity as
Here, and .
Proof See . □
Theorem 1 Assume that (A1)-(A3) are satisfied and suppose that the initial condition satisfies the following condition:
where and . Then the solution of the problem (1)-(4) with the weight function blows up in a finite time.
Proof Multiplying (1) by u and integrating over Ω give
Also, multiplying (1) by and integrating over Ω, we obtain
Now, let us consider the following function:
where is a nonnegative parameter to be chosen later. It is clear that
Using the Cauchy-Schwarz inequality, we have
Substituting (11) into (12), we obtain
We take the derivative of (7) with respect to t
Rewrite (16) in view of (17)
After applying the arithmetic-geometric inequality to estimate the terms on the right-hand side of (18), we obtain
For , , the following inequality is satisfied for some :
We apply the Hölder inequality, with , , , to the last term in (18),
It follows from (24) and (25) with
where . Substituting the estimates (19)-(23) and (26) into (18), we write
Since coefficients of the term are greater than those of on the right-hand side of (27), multiplying both sides of (27) by , we get
where . From (15) and (28), we have
where . We choose in the last inequality and multiply both sides of (29) by , which gives
So, inequality (8) is satisfied with , , . Thus, the desired result is obtained by applying Lemma 1. □
3 Stability of problem
In this part, we consider the following inverse source problem:
where is a bounded domain with a sufficiently smooth boundary ∂ Ω and ω, and are given functions, . Assume that ω satisfies the conditions
Theorem 2 Suppose that the conditions (A5) and (A6) are satisfied and assume that φ and are continuous functions defined on which tend to zero as . Then
with a constant , where is constant in (6).
Proof We multiply (31) by ω, integrate over Ω and use (34) to obtain
Inserting (35) into (31), we obtain
Now, let us multiply (36) by and integrate over Ω:
Using Cauchy, Poincare and Young inequalities on the right-hand side of (37), we have
Rewriting (37) with estimates (38)-(43), we obtain the following inequality:
We choose such that and take
So, (44) follows
The last term on the left-hand side of (45) can be written
where . It follows from (45) and (46)
Here, and . After solving first-order differential inequality (47), it follows that
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The author declares that they have no competing interests.
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Yaman, M. Blow-up solution and stability to an inverse problem for a pseudo-parabolic equation. J Inequal Appl 2012, 274 (2012). https://doi.org/10.1186/1029-242X-2012-274
- inverse problem
- pseudo-parabolic equation