Preamble

This work, following the developments in [7], considers the mixed problem for a nonlinear partial differential equation in the domain $Q = {0 < t < T, 0 < x < \pi}$. We seek a function $u(t, x)$ satisfying the equation:
$$\rho(x) \frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = f(t, x, u, u_t, u_x) \quad \text{(1)}$$
subject to the initial conditions:
$$u(0, x) = \phi_0(x), \quad u_t(0, x) = \phi_1(x) \quad \text{(2)}$$
and the boundary conditions:
$$u(t, 0) = u(t, \pi) = 0 \quad \text{(3)}$$
where $\rho(x) \geq \rho_0 > 0$ for $x \in [0, \pi]$. We assume that the eigenfunctions $\psi_k(x)$ of the corresponding linear operator satisfy the normalization condition $\int_0^\pi \rho(x) \psi_k(x) \psi_i(x) dx = \delta_{ki}$.

1. Construction of the Approximate Solution

We seek an approximate solution $u_n(t, x)$ using the Galerkin method in the form:
$$u_n(t, x) = \sum_{k=1}^n u_k^{(n)}(t) \psi_k(x) \quad \text{(4)}$$
Substituting this expression into equation (1) and projecting onto the basis ${\psi_k(x)}$, we obtain a system of ordinary differential equations for the coefficients $u_k^{(n)}(t)$:
$$\frac{d^2 u_k^{(n)}}{dt^2} + \lambda_k u_k^{(n)} = \int_0^\pi f(t, x, u_n, \dots) \psi_k(x) dx \quad \text{(5)}$$
with initial conditions:
$$u_k^{(n)}(0) = \int_0^\pi \rho(x) \phi_0(x) \psi_k(x) dx, \quad \dot{u}_k^{(n)}(0) = \int_0^\pi \rho(x) \phi_1(x) \psi_k(x) dx \quad \text{(6)}$$
This system can be written in vector form as:
$$\ddot{u}^{(n)}(t) + A_n u^{(n)}(t) = f^{(n)}(t, u^{(n)}(t), \dots) \quad \text{(7)}$$
$$u^{(n)}(0) = \phi_0^{(n)}, \quad \dot{u}^{(n)}(0) = \phi_1^{(n)} \quad \text{(8)}$$
where $A_n$ is a diagonal matrix of eigenvalues $\lambda_k$. The solution to this system on the interval $[0, T]$ can be represented by the integral equation:
$$u^{(n)}(t) = \cos(\sqrt{A_n} t) \phi_0^{(n)} + (\sqrt{A_n})^{-1} \sin(\sqrt{A_n} t) \phi_1^{(n)} + \int_0^t (\sqrt{A_n})^{-1} \sin(\sqrt{A_n}(t-\tau)) f^{(n)}(\tau, u^{(n)}(\tau), \dots) d\tau \quad \text{(9)}$$

2. Convergence and Stability

To establish the existence of a solution on $[0, T]$, we impose growth conditions on the nonlinear term $f$. Specifically, we assume:
$$|f(t, x, u, u_t, u_x)| \leq a_1(t, x)|u| + a_2(t)|u_t| + a_3(t)|u_x| + a_4(t, x) \quad \text{(10)}$$
Using Gronwall-type inequalities and energy estimates, we demonstrate that the sequence of approximate solutions $u_n(t, x)$ is bounded in the appropriate Sobolev spaces. For the norm $|u^{(n)}(t)|^2 + |\dot{u}^{(n)}(t)|^2$, we derive the following estimate:
$$|u^{(n)}(t)|^2 + |\dot{u}^{(n)}(t)|^2 \leq C \left( |\phi_0^{(n)}|^2 + |\phi_1^{(n)}|^2 + \int_0^t \beta_n(\tau) d\tau \right) \exp\left( \int_0^t \alpha_n(\tau) d\tau \right) \quad \text{(11)}$$
where $\alpha_n(t)$ and $\beta_n(t)$ are determined by the coefficients in (10).

Furthermore, we assume a Lipschitz condition for the uniqueness of the solution:
$$|f(t, u_1, u_2, u_3) - f(t, v_1, v_2, v_3)| \leq \sum_{i=1}^3 \kappa_i(t, K) |u_i - v_i| \quad \text{(12)}$$
Under these conditions, the sequence $u_n(t, x)$ converges to a function $u(t, x)$ that serves as a generalized solution to the original problem (1)–(3).

3. Regularity of the Solution

If the initial data $\phi_0(x)$ and $\phi_1(x)$ possess higher regularity, specifically $\phi_0 \in W_2^{(2)}$ and $\phi_1 \in W_2^{(1)}$, and satisfy the compatibility conditions $\phi_0(0) = \phi_0(\pi) = 0$, then the solution $u(t, x)$ belongs to the space $W_2^{(2)}(Q)$. In this case, the second-order derivatives exist in the sense of $L_2$, and the equation (1) is satisfied almost everywhere.

The proof utilizes the properties of the operator $B_n = \sqrt{A_n}$ and the convergence of the Galerkin approximations in stronger norms. We show that:
$$|B_n^2 u^{(n)}(t)|^2 + |B_n \dot{u}^{(n)}(t)|^2 \leq C_T \quad \text{(14)}$$
which ensures that the limit function $u(t, x)$ has the required square-integrable second derivatives.

Conclusion

The results obtained demonstrate the global existence and uniqueness of the solution to the nonlinear mixed problem. The method of semi-discretization (Galerkin method) provides not only a theoretical existence proof but also a basis for numerical implementation. The stability of the solution with respect to initial data and the right-hand side $f$ follows from the derived energy inequalities. These findings extend classical results for linear hyperbolic equations to a broad class of nonlinearities relevant in mathematical physics.