Propagation of sound in a viscous gas and other similar processes of the same nature can be described by the model hyperbolic equation of the third order, which includes a mixed derivative with respect to spatial and time variables

where η is a positive constant, and ηΔ_xu_t represents low viscosity.

Many important physical phenomena can be modeled with the use of Equation 1 and its generalizations. These are, in particular, processes that occur in viscous media (propagation of disturbances in viscoelastic and viscous-plastic rods, movement of a viscous compressible fluid, sound propagation in a viscous gas), wave processes in different media, acoustic waves in environments where wave propagation disrupts the state of thermodynamic and mechanical equilibrium, liquid filtration processes in porous media, heat transfer in a heterogeneous environment, moisture transfer in soils, and longitudinal vibrations in a homogenous bar with viscosity. The term Δ_xu_t indicates that the level of stress is proportional to the level of strains and to the strain rate [1–5].

Due to its wide range of applications, different problems for Equation 1 were investigated by many authors. For example, the unique solvability of the direct initial-boundary value problems for Equation 1 and its nonlinear generalizations with power nonlinearities have been studied in other research [1, 2, 4–11].

The inverse problems, with the integral overdetermination conditions, of identifying of the coefficients in the right-hand side function of hyperbolic equations without damping or for other types of equations have been investigated in many studies [12–18]. Their unique solvability has been solved with the use of the methods such as integral equations, the Green function, regularization, and the Shauder principle [14] and successive approximations [18]. The unique solvability of a two-dimensional inverse problem for the linear third-order hyperbolic equation with constant coefficients and with the unknown time-dependent lower coefficient has been proved in Mehraliyev et al. [19].

The main objective of this study is to determine the sufficient conditions for the existence and uniqueness of the solution to the inverse problem for the third-order semilinear hyperbolic equation with an unknown time- dependent function on its right-hand side. The unknown function is determined from the equation, subject to initial, boundary, and integral type overdetermination conditions. To prove the main results of the study, we use the properties of the solution for the corresponding initial-boundary value problem and the method of successive approximations. These results are new for semilinear n-dimensional third-order hyperbolic equations with non-constant coefficients and an unknown function on their right-hand side. The unique solvability of the initial-boundary value problem has been proved using of the method of Galerkin approximations and the methods of monotonicity and compactness.

Let Ω ⊂ ℝⁿ, n ∈ ℕ, be a bounded domain with the smooth boundary ∂Ω ∈ C¹ and 0 < T < ∞. Denote Q_τ = Ω × (0, τ), τ ∈ (0, T]; Q_{t1, t2} = Ω × (t₁, t₂), t₁, t₂ ∈ (0, T]. In this study, we consider the following inverse problem: find the sufficient conditions for the existence of a pair of functions (u(x, t), g(t)) that satisfies the equation with strong damping (in the sense of Definition 3.1).

and the initial, boundary, and overdetermination conditions

We shall use Lebesgue and Sobolev spaces L^∞(·), L²(·), H¹(·): = W^{1, 2}(·), C^k(·), C([0, T];L²(G)), H01(·):=W01,2(·) (see, e.g., Gajewski et al. [20]).

Suppose that the data of the problem (2–5) satisfy the following conditions.

(H1): aij, bij, aijt, bijt, bij,xi∈C([0,T];L∞(Ω)), a_ij(x, t) = a_ji(x, t), b_ij(x, t) = b_ji(x, t), and

for all ξ ∈ ℝⁿ, almost all x ∈ Ω, all t ∈ [0, T], and i, j = 1, ..., n, where α₀, α₁ and β₀, β₁ are positive constants.

(H2): functions φ₁(x, ξ), φ₂(x, ξ) are measurable with respect to x ∈ Ω for all ξ ∈ ℝ¹ and continuously differentiable concerning ξ ∈ ℝ. Moreover,

for almost all x ∈ Ω and ξ, η ∈ ℝ, where L_i,0, L_i,1 are positive constants.

(H3): f1∈L2(Ω), f2∈C([0,T];L2(Ω)), u0∈H01(Ω),  u1∈H01(Ω).

(H4): E ∈ C²([0, T]), ∫ΩK(x)u0(x)dx=E(0), ∫ΩK(x)u1(x)dx=E′(0).

(H5): K∈H2(Ω)∩H01(Ω).

Denote f~(x,t):=f1(x)g(t)+f2(x,t).

Let γ₀ = γ₀(Ω) be a coefficient in Friedrich's inequality.

Definition 3.1. A function u(x, t) is considered to be a solution of problem 2–4 if u∈C([0,T];H01(Ω)), ut∈L2(0,T;H01(Ω))∩C([0,T];L2(Ω)), utt∈L2(QT), u satisfies (3), and

for all functions v∈L2(0,T;H01(Ω)) and τ ∈ (0, T].

Theorem 3.2. Under the assumptions (H1)–(H3) and g ∈ L²(0, T), a_ijt ≤ 0 for all i, j = 1, 2, …, n, the problem (2–4) has a unique solution.

Proof. First, using Galerkin method, we prove the existence of a solution for the problem. Let {wk}k=1∞,k=1,2,..., be a basis in H01(Ω), orthonormal in L²(Ω). We will consider the sequence of functions

where the set (c1N(t),....,cNN(t)) is a solution of the initial value problem

Here u0N(x)=∑k=1Nu0,kNwk(x), u1N(x)=∑k=1Nu1,kNwk(x) and

The solution of system (8) exists on some interval [0, τ₀] (Carathéodory's Theorem [21, p. 43]). The estimation (13) from below implies that this solution could be extended on [0, T]. Multiplying each equation of (8) on function (ckN(t))′ respectively, summing up for k from 1 to N and integrating to t on interval [0, τ], τ ≤ τ₀, we obtain

After transformations of terms from (9), we get

Note that

then from (10) we obtain

We rewrite the last inequality in the form

where

Then by Grönwall's lemma, from (12), we get

Therefore, from (11) we also get

Multiplying each equation of (8) on function (ckN(t))″ respectively, summing up with respect to k from 1 to N and integrating on interval [0, τ], τ ≤ T, we obtain

After transformations in all terms from (15), we get

where ∑i,j=1nbijt(x,t)ξiξj≤β2||ξ||2, β2>0, α2=maxi,jsupt∈[0,T]|aijt(x,t)|. Taking into account (13), (14), from (16) we obtain

where

The right-hand sides of the estimates (13), (14), and (17) are positive constants, independent of N. Therefore, there exists a subsequence of {uN}N=1∞ (which will be denoted by the same notation), such that as N → ∞

It follows from (18) that uN→u  in  L2(QT), and therefore, φ1(x,uN)→φ1(x,u)weakly in L2(QT) as N → ∞. Besides, u∈C([0,T];H01(Ω)), ut∈C([0,T];L2(Ω)) and φ2(x,utN)→χ weakly in L2(QT).

Equations 8 and 18 imply the equality

for all functions v∈L2(0,T;H01(Ω)) and τ ∈ (0, T].

Let us prove that χ = φ₂(x, u_t).

Note that ||φ1(x,uN+k)-φ1(x,uN)||L2(QT)≤L1,0||uN+k-uN||L2(QT) for all k ∈ ℕ. Due to (18), {u}k=1∞ is fundamental in L2(QT). So, for any ε > 0, there exists such a number N₀ that for all N, k ∈ ℕ, N > N₀ the inequality ||uN+k-uN||L2(QT)≤ε holds; thus, {φ1(x,u)}k=1∞ is also fundamental in L2(QT) and, therefore,

Consider the sequence

where η∈C([0,T];H01(Ω)), ηt∈L2(0,T;H01(Ω))∩C([0,T];L2(Ω)), ηtt∈L2(QT). From (9), it follows that

After substitution (22) in (21), passing to the limit as N → ∞, taking into account (18), (20), and the assumptions of Theorem 3.2, we obtain

Choosing here η=u-ϰw, ϰ>0, w∈L2(0,T;H01(Ω))∩C([0,T];L2(Ω)), wt∈L2(0,T;H01(Ω))∩C([0,T];L2(Ω)),wtt∈L2(QT), dividing the result on ϰ and then tending ϰ → 0 we obtain χ = φ₂(x, u_t). Hence, from (19), it follows (7).

Now we prove the uniqueness of the solution for the problems (2–4). On the contrary, suppose that there exist two solutions u₍₁₎(x, t) and u₍₂₎(x, t) of problems (2–4). Then ũ: = ũ(x, t) = u₍₁₎(x, t)−u₍₂₎(x, t) satisfies the conditions ũ(x, 0) ≡ 0, ũ_t(x, 0) ≡ 0, and the equality

holds for all v∈L2(0,T;H01(Ω)), τ ∈ (0, T].

After choosing v = ũ_t in (23) we get

From (24) by the same way as from (11) we got (12), we find the following estimate

Then from Grönwall's lemma and (25) we obtain ∫Ω(|ũt(x,τ)|2+∑i=1n|ũxi(x,τ)|2)dx≤0 and ∫Qτ∑i=1n|ũxit(x,t)|2dxdt≤0, hence, ũ ≡ 0, and, therefore, u₍₁₎ = u₍₂₎ in Q_T.

Definition 4.1. A pair of functions (u(x, t), g(t)) is a solution to the problem (2–5), if u∈C([0,T];H01(Ω)), ut∈L2(0,T;H01(Ω))∩C([0,T];L2(Ω)), utt∈L2(QT), and g ∈ C([0, T]), and it satisfies (5) and