Existence and Uniqueness of Generalized Solutions to a Telegraph Equation with an Integral Boundary Condition via Galerkin′s Method

Theorem 3.1. Assume that φ ∈ W^1,2(Ω), ψ ∈ L₂(Ω),f ∈ L₂(Q), K(x, y) ∈ C(Ω × Ω), and

then the generalized solution of problem (1.6)–(1.8), if it exists, is unique.

Proof. Suppose that there exist two different generalized solutions u₁ and u₂ for the problem (1.6)–(1.8); then obviously their difference u = u₁ − u₂ is a generalized solution of the problem (1.6)–(1.8) with homogeneous equation and homogeneous initial and non local conditions, that is, f = φ = ψ = 0. We will prove that u = 0 in Q. Let $$, and denote Q^τ = {(x, t); 0 < x < 1,0 < t ≤ τ ≤ T}. Consider the function

The identity (2.8) becomes Substituting v into (3.3), integrating by parts, and then using the fact that v_t(x, t) = −u(x, t), it follows that Using the assumption on the function K, we get where C₁ = max (2ck₁, 2a)/min (c, 1, b). Now, applying Cauchy Schwarz inequality to the first term in the right-hand side of (3.5), then ɛ-Cauchy inequality with ɛ = 1, yields Using the trace inequality, we obtain Considering the function it is easy to see that v(x, t) = w(x, τ) − w(x, t), ∇v(x, 0) = ∇w(x, τ), and (∇v(x,t))² ≤ 2(∇w(x,τ))² + 2(∇w(x,t))². Consequently, substituting w in (3.7), we get Seting it follows that Since τ is arbitrary chosen, let 1 − τC₂ > 0; then (3.11) becomes Applying Gronwall Lemma, we get Consequently, we obtain u(x, τ) = 0, for all x ∈ Ω and τ ∈   ]0, 1/C₂[.

If T ≤ 1/C₂, then u = 0 in Q. In the case where T ≥ 1/C₃, we see that $$, where n₀ = [C₂T] + 1,[C₂T] is the entire part of C₂T; then repeating the preceding reasoning for τ ∈   ](n − 1)/C₂, n/C₂[, we get u(x, τ) = 0, for all τ ∈   ](n − 1)/C₂,   n/C₂[, and then u(x, t) = 0 in Q. Thus, the uniqueness is proved.

Remark 3.2. It should be noted that we can prove the uniqueness theorem by using a priori estimates, so we establish that the generalized solution, if it exists, satisfies the inequality

Theorem 4.1. Assume that the assumptions of Theorem 4.1 hold; then the non-local problem (1.6)–(1.8) has a unique solution u ∈ W^1,2(Q).

Proof. In order to prove the existence of the generalized solution we apply Galerkin’s method. Let {w_k(x)} be a fundamental system in W^1,2(Ω), such that $$. Now we will try to find an approximate solution of the problem (1.6)–(1.8) in the form

The approximates of the functions φ(x) and ψ(x) are denoted, respectively, by Substituting the approximation solution in (1.6), multiplying both sides by w_i(x), and then integrating according to x on Ω, we get Substituting (4.1) in (4.3), we obtain This implies that Denote Then (4.5) becomes We obtain a system of differential equations of second order according to the variable t with smooth coefficients and the initial conditions Consequently, we get a Cauchy problem of linear differential equations with smooth coefficients that is uniquely solvable. So it has a unique solution u⁽ⁿ⁾ satisfying (4.3).

Lemma 4.2. The sequence (u⁽ⁿ⁾) is bounded.

Proof. Multiplying (4.4) by $$ and summing over i from 1 to n, we get

Integrating by parts the left-hand side of (4.9) over t on (0, τ) yields Substituting the four last identities into (4.9), we get Applying Cauchy Schwarz inequality, ϵ-Cauchy inequality, the hypothesis on the operator K, and the fact that $$ to the second term in the right hand side of (4.11), we get We may obtain similar inequalities for the third and the fourth terms in the right-hand side of (4.11). Now if we apply Cauchy Schwarz inequality to the last term in (4.11) we get Using the result inequalities in (4.11) and then regrouping the same terms yield Choose ϵ such that (1 − ɛ) > 0 and $$ and let then (4.14) becomes Now, we apply Gronwall Lemma to get Since φ⁽ⁿ⁾ and ψ⁽ⁿ⁾ are the orthogonal projections of φ and ψ, respectively, from the Pythagorean relation we have that $$ and $$. Integrating (4.17) according to τ on [0, T] yields Consequently, the sequence {u⁽ⁿ⁾} is bounded.

Remark 4.3. We have proved that the sequence {u⁽ⁿ⁾} is bounded, so we can extract a subsequence which we denote by $$ that is weakly convergent; then we prove that its limit is the desired solution of the problem (1.6)–(1.8).

Lemma 4.4. The limit of the subsequence $$ is the solution of the problem (1.6)–(1.8).

Proof. For this we prove that the limit of the subsequence $$ satisfies the identity (2.8) for all functions $$. Since the set S_n$$ is such that $$ is dense in $$, it suffices to prove (2.8) for v ∈ S_n. Multiplying (4.3) by the function v_i(t) ∈ W^1,2(0, T),  v_i(T) = 0, and then taking the sum from i = 0 to n, we obtain

Integrating by parts on [0, T], we get Denote the weak limit of the subsequence $$ by u. When k tends to infinity, we get Indeed, using Cauchy Schwarz inequality then ɛ-Cauchy inequality, we obtain and then by passing to the limit in (4.20) we get that the limit u satisfies(2.8).

Example 4.5. Now we present an example to demonstrate the applications of the results established in the earlier sections. We consider the following:

with the initial conditions and the nonlocal boundary condition

The results of the earlier sections guarantee the existence and uniqueness of a solution. It may be noted that u(x, t) = sinxsint is the unique solution of (4.23)–(4.25).