Research Article | Open Access
A Boundary Value Problem with Multivariables Integral Type Condition for Parabolic Equations
We study a boundary value problem with multivariables integral type condition for a class of parabolic equations. We prove the existence, uniqueness, and continuous dependence of the solution upon the data in the functional wieghted Sobolev spaces. Results are obtained by using a functional analysis method based on two-sided a priori estimates and on the density of the range of the linear operator generated by the considered problem.
Certain problems of modern physics and technology can be effectively described in terms of nonlocal problems with integral conditions for partial differential equations.These nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. Motivated by this, we consider in the rectangular domain the following nonclassical boundary value problem of finding a solution such that where the function and its derivative are bounded on the interval
Mathematical modelling of different phenomena leads to problems with nonlocal or integral boundary conditions. Such a condition occurs in the case when one measures an averaged value of some parameter inside the domain. This amounts to the specification of the energy or mass contained in a portion of the conductor or porous medium as a function of time. This problems arise in plasma physics, heat conduction, biology and demography, as well as modelling of technological process, see, for example, [1–5]. Boundary-value problems for parabolic equations with integral boundary condition are investigated by Batten , Bouziani and Benouar , Cannon [8, 9], Cannon, Perez Esteva and van der Hoek , Ionkin , Kamynin , Shi and Shillor , Shi , Marhoune and Bouzit , Marhoune and Hameida , Yurchuk , and many references therein. The problem with one-variable (resp., two-variable) boundary integral type condition is studied in  and by Marhoune and Latrous  (resp., in Marhoune ).
Mention that in the cited paper , the author proved the existence, uniqueness, and continuous dependence of a stronge solution in weighted Sobolev spaces to the problem
under the following conditions:
This last integral condition in the form
arises, for example, in biochemistry in which is a constant, and in this case is known as the conservation of protein . Further, in , the author studied a similar problem with the weak integral condition
The same problem with the new integral condition
was investigated in . The present paper is an extension in the same direction. By constructing a suitable multiplicator, we will try to establish existence and uniqueness of solution of problem (1.1)–(1.5). Note that the multivariables integral type condition (1.5) is considerably much weaker and better than that used in . In fact, some physical problems have motivated specialists to consider nonlocal integral condition (1.5), which tells us the integral total effect of the solution over several independent portions , and of interval at certain time that give this effect over the entire or part of this interval.
We associate with (1.1)–(1.5) the operator defined from into where is the Banach space of functions satisfying (1.4) and (1.5), with the finite norm and is the Hilbert space of vector-valued functions obtained by completion of the space with respect to the norm
Using the energy inequalities method proposed in , we establish two-sided a priori estimates. Then, we prove that the operator is a linear homeomorphism between the spaces and .
2. Two-Sided A Priori Estimates
Theorem 2.1. For any function one has the a priori estimate where the constant is independent of . In fact,
Theorem 2.2. For any function , one has the a priori estimate with the constant and is such that
Before proving this theorem, we need the following lemma.
Lemma 2.3 (see ). For one has
Proof of Theorem 2.2. Define We consider for the quadratic formula with the constant satisfying (2.5), obtained by multiplying (1.1) by , by integrating over , where , with , and by taking the real part. Integrating by parts in (2.8) by report to with the use of boundary conditions (1.4) and (1.5), we obtain On the other hand, by using the elementary inequalities we get Again, integrating by parts the second, third, fourth, and fifth terms of the right-hand side of the inequality (2.10) by report to and taking into account the initial condition (1.3) and (2.5) gives Using (2.11) in (2.10), we get By using the -inequalities on the first integral in the left-hand side of (2.12) and Lemma 2.3, we obtain Now, from (1.1), we have Combining inequalities (2.13) and (2.14), we get As the right-hand side of (2.15) is independent of , by replacing the left-hand side by its upper bound with respect to in the interval , we obtain the desired inequality.
3. Solvability of the Problem
From estimates (2.1) and (2.3), it follows that the operator is continuous and its range is closed in Therefore, the inverse operator exists and is continuous from the closed subspace onto , which means that is an homeomorphism from onto . To obtain the uniqueness of solution, it remains to show that . The proof is based on the following lemma.
Lemma 3.1. Let If for and some , one has where then
Proof. From (3.2) we have
Now, for given , we introduce the function
Integrating by parts with respect to , we obtain
which implies that
Then, from (3.4), we obtain
If we introduce the smoothing operators with respect to , and then these operators provide the solutions of the respective problems:
and also have the following properties: for any , the functions and are in such that and Morever, commutes with so and for
Putting in (3.8), where the constant satisfies , and using (3.11), we obtain
Integrating by parts each term in the right-hand side of (3.12) and taking the real parts yield Using -inequalities, we obtain Combining (3.13) and (3.15), we get From (3.16), we deduce that Then, for we obtain We conclude that , hence , which ends the proof of the the lemma.
Theorem 3.2. The range of coincides with .
Proof. Since is a Hilbert space, we have if and only if the relation
for arbitrary and , implies that and
Putting in (3.19), we conclude from Lemma 3.1 that , where then .
Taking in (3.19) yields The range of the operator is everywhere dense in Hilbert space with the norm hence,
The authors would like to thank the referee for helpful suggestions and comments.
- A. A. Samarskiĭ, “Some problems of the theory of differential equations,” Differentsial'nye Uravneniya, vol. 16, no. 11, pp. 1221–1228, 1980.
- A. L. Marhoune, “A three-point boundary value problem with an integral two-space-variables condition for parabolic equations,” Computers & Mathematics with Applications, vol. 53, no. 6, pp. 940–947, 2007.
- R. E. Ewing and T. Lin, “A class of parameter estimation techniques for fluid flow in porous media,” Advances in Water Resources, vol. 14, no. 2, pp. 89–97, 1991.
- P. Shi, “Weak solution to an evolution problem with a nonlocal constraint,” SIAM Journal on Mathematical Analysis, vol. 24, no. 1, pp. 46–58, 1993.
- A. V. Kartynnik, “A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic equations,” Differential Equations, vol. 26, no. 9, pp. 1160–1166, 1990.
- G. W. Batten Jr., “Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations,” Mathematics of Computation, vol. 17, pp. 405–413, 1963.
- A. Bouziani and N.-E. Benouar, “Mixed problem with integral conditions for a third order parabolic equation,” Kobe Journal of Mathematics, vol. 15, no. 1, pp. 47–58, 1998.
- J. R. Cannon, “The solution of the heat equation subject to the specification of energy,” Quarterly of Applied Mathematics, vol. 21, pp. 155–160, 1963.
- J. R. Cannon, The One-Dimensional Heat Equation, vol. 23 of Encyclopedia of Mathematics and Its Applications, Addison-Wesley, Mento Park, Calif, USA, 1984.
- J. R. Cannon, S. Pérez Esteva, and J. van der Hoek, “A Galerkin procedure for the diffusion equation subject to the specification of mass,” SIAM Journal on Numerical Analysis, vol. 24, no. 3, pp. 499–515, 1987.
- N. I. Ionkin, “The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition,” Differentsial'nye Uravneniya, vol. 13, no. 2, pp. 294–304, 1977.
- L. I. Kamynin, “A boundary-value problem in the theory of heat conduction with non-classical boundary conditions,” U.S.S.R. Computational Mathematics and Mathematical Physicsi, vol. 4, pp. 33–59, 1964.
- P. Shi and M. Shillor, “Design of contact patterns in one dimentional thermoelasticity,” in Theoretical Aspects of Industrial Design, SIAM, Philadelphia, Pa, USA, 1992.
- A. L. Marhoune and M. Bouzit, “High order differential equations with integral boundary condition,” Far East Journal of Mathematical Sciences, vol. 18, no. 3, pp. 341–350, 2005.
- A. L. Marhoune and A. Hameida, “Mixed problem with an integral space variable condition for a third order parabolic equation of mixed type,” Far East Journal of Mathematical Sciences, vol. 29, no. 2, pp. 409–418, 2008.
- N. I. Yurchuk, “Mixed problem with an integral condition for some parabolic equations,” Differential Equations, vol. 22, pp. 1457–1463, 1986.
- A. L. Marhoune and C. Latrous, “A strong solution of a high-order mixed type partial differential equation with integral conditions,” Applicable Analysis, vol. 87, no. 6, pp. 625–634, 2008.
- G. Fairweather and R. D. Saylor, “The reformulation and numerical solution of certain nonclassical initial-boundary value problems,” SIAM Journal on Scientific and Statistical Computing, vol. 12, no. 1, pp. 127–144, 1991.
- G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge Press, Cambridge, UK, 1934.
Copyright © 2009 A. L. Marhoune and F. Lakhal. 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.