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.


Introduction
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 Ω 0, 1 × 0, T , the following nonclassical boundary value problem of finding a solution u x, t such that where the function a t and its derivative are bounded on the interval 0, T : δ u x, t dx 0, 0 < α < β < γ < δ < 1, α 1 − δ γ − β, t ∈ 0, T .

1.5
Here, we assume that the known function ϕ satisfies the conditions given in 1.4 and 1.5 , that is, When considering the classical solution of the problem 1.1 -1.5 , along with 1.5 , there should be the fulfilled conditions:

1.7
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 Yurchuk 16 , and many references therein. The problem with onevariable resp., two-variable boundary integral type condition is studied in 5 and by Marhoune and Latrous 17 resp., in Marhoune 2 . Mention that in the cited paper 16 This last integral condition in the form The same problem with the new integral condition was investigated in 2 . 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 2 . 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 u over several independent portions 0, α , β, γ , and δ, 1 of interval I 0, 1 at certain time t that give this effect over the entire or part of this interval. We associate with 1.1 -1.5 the operator L L, l , defined from E into F, where E is the Banach space of functions u ∈ L 2 Ω , satisfying 1.4 and 1.5 , with the finite norm

Journal of Applied Mathematics and Stochastic Analysis
and F is the Hilbert space of vector-valued functions F f, ϕ obtained by completion of the space L 2 Ω × W 2 2 0, 1 with respect to the norm

1.15
Using the energy inequalities method proposed in 16 , we establish two-sided a priori estimates. Then, we prove that the operator L is a linear homeomorphism between the spaces E and F.

Two-Sided A Priori Estimates
Theorem 2.1. For any function u ∈ E, one has the a priori estimate where the constant c 4 is independent of u. In fact, c 4 2 max 1, c 2 1 .
Proof. Using 1.1 and initial condition 1.3 , we obtain

2.2
Combining the inequalities in 2.2 , we obtain 2.1 for u ∈ E. and c is such that Before proving this theorem, we need the following lemma.

2.6
Proof of Theorem 2.2. Define

2.7
We consider for u ∈ E the quadratic formula

2.9
On the other hand, by using the elementary inequalities we get

2.10
Journal of Applied Mathematics and Stochastic Analysis 7 Again, integrating by parts the second, third, fourth, and fifth terms of the right-hand side of the inequality 2.10 by report to t and taking into account the initial condition 1.3 and 2.5 gives

2.11
Using 2.11 in 2.10 , we get

2.15
As the right-hand side of 2.15 is independent of τ, by replacing the left-hand side by its upper bound with respect to τ in the interval 0, T , we obtain the desired inequality.
Journal of Applied Mathematics and Stochastic Analysis 9

Solvability of the Problem
From estimates 2.1 and 2.3 , it follows that the operator L : E → F is continuous and its range is closed in F. Therefore, the inverse operator L −1 exists and is continuous from the closed subspace R L onto E, which means that L is an homeomorphism from E onto R L .
To obtain the uniqueness of solution, it remains to show that R L F. The proof is based on the following lemma.
If for u ∈ D 0 L and some w ∈ L 2 Ω , one has Now, for given w x, t , we introduce the function v x, t

3.5
Integrating by parts with respect to ξ, we obtain Then, from 3.4 , we obtain If we introduce the smoothing operators with respect to t 16 , I −1 ε I ε∂/∂t −1 and I −1 ε * , then these operators provide the solutions of the respective problems: and also have the following properties: for any g ∈ L 2 0, T , the functions g ε I −1 ε g and g * ε I −1 ε * g are in W 1 2 0, T such that g ε t t 0 0 and g * ε t t T 0. Morever, I −1 ε commutes with ∂/∂t, so T 0 |g ε − g| 2 dt → 0 and T 0 |g * ε − g| 2 dt → 0 for ε → 0. Putting u t 0 exp cτ v * ε x, τ dτ in 3.8 , where the constant c satisfies cc 0 − c 3 − εc 2 3 /c 0 ≥ 0, 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 2 Re We conclude that v 0, hence w 0, which ends the proof of the the lemma. for arbitrary u ∈ E and f, ϕ ∈ F, implies that f 0 and ϕ 0.