Preamble

This section considers the existence and uniqueness of solutions for a class of partial differential equations with functional arguments. Specifically, we investigate the following equation:

$$\begin{aligned} U_{xt} = f(t, x, U(t, x), U(t-\tau, x), U_t(t, x), U_t(t-\tau, x), U_x(t, x), U_x(t-\tau, x)) \end{aligned}$$

where the delay argument is defined by $\tau = \tau(t, x, U(t, x), U_x(t, x), U_t(t, x))$. The domain of interest is defined for $t \in [t_0 - \tau_0, t_0 + h]$ and $x \in \Omega$. We assume that the delay satisfies the condition $\tau(t, x, U, U_x, U_t) < t$, ensuring that the functional dependence remains within the past state of the system. This work builds upon the foundational results established in \cite{1, 2} and further developed in \cite{3, 4}.

1. Existence and Uniqueness Conditions

To establish the existence of a solution $U(t, x)$ for equation (1), we impose the following conditions:

a) The initial function $\phi(t, x)$ is defined and sufficiently smooth on the initial set $[t_0 - \tau_0, t_0] \times \Omega$. We assume that $\phi$ and its derivatives $\phi_t, \phi_x, \phi_{tx}$ are continuous and bounded.

b) The function $f$ satisfies a Lipschitz condition with respect to its functional arguments. Specifically, there exists a constant $L_f > 0$ such that for any two sets of arguments, the difference in $f$ is bounded by the sum of the absolute differences of its components.

c) The delay function $\tau$ is also Lipschitz continuous with constant $L_\tau > 0$. We assume that for $t \in [t_0, t_0 + h]$, the solution remains within a bounded region where the partial derivatives of $f$ and $\tau$ are controlled.

2. Operator Formulation and Convergence

The problem can be reformulated as a fixed-point problem for an operator $T$ acting on a function space $V$. We define the operator $T(U(t, x))$ based on the integral form of the differential equation:

$$\begin{aligned} T(U(t, x)) = \phi(t_0, x) + \int_{t_0}^t \int_{x_0}^x f(\xi, \eta, U, U_\tau, U_t, U_{t\tau}, U_x, U_{x\tau}) \, d\eta \, d\xi \end{aligned}$$

To prove the existence of a unique solution, we demonstrate that $T$ is a contraction mapping in a suitably chosen Banach space. We define the norm on this space as:

$$\begin{aligned} \rho(V, W) = \sup |V(t, x) - W(t, x)| + \sup |V_t(t, x) - W_t(t, x)| + \sup |V_x(t, x) - W_x(t, x)| \end{aligned}$$

By applying the Lipschitz conditions on $f$ and $\tau$, we derive an estimate for $\rho(T(V), T(W))$. Let $B_1$ and $B_2$ be bounds such that $|f| \leq B_1$ and the partial derivatives of $f$ are bounded by $B_2$. After a series of estimations over the domain $ABC$ (as illustrated in [FIGURE:1]), we obtain:

$$\begin{aligned} \rho(T(V), T(W)) \leq L_f \cdot h \cdot [2 + (B_1 + B_1 a_1 + B_2 a_1) L_\tau] \cdot \rho(V, W) \end{aligned}$$

For the operator to be a contraction, we require the coefficient $\alpha = L_f \cdot h \cdot [2 + (B_1 + B_1 a_1 + B_2 a_1) L_\tau]$ to be less than 1. This condition is satisfied by choosing a sufficiently small time interval $h$. Under these constraints, the Banach fixed-point theorem guarantees the existence of a unique solution $U(t, x)$ for the given initial conditions.

References

1. Ivanov, V. R. (1960). On certain properties of differential equations with delay. Matematicheskii Sbornik, 39(1-2), 29–36.
2. El'sgol'ts, L. E. (1965). Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis.
3. Kamenskii, G. A. (1966). Boundary value problems for equations with deviating arguments. Differentsial'nye Uravneniya, 2(12), 61–64.
4. Myshkis, A. D. (1955). Linear Differential Equations with Retarded Argument. Moscow-Leningrad.
5. Edelstein, I. V. (1965). Methods of Mathematical Physics. Moscow.