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Carathéodory solutions of hyperbolic functional differential inequalities with first order derivatives. (English) Zbl 1165.35416

Summary: We consider the Darboux problem for a functional differential equation: \begin{align*} &\frac{\partial^2u}{\partial x \partial y}(x,y) =f\bigg(x,y,u_{(x,y)},u(x,y),\frac{\partial u}{\partial x}(x,y), \frac{\partial u}{\partial y}(x,y)\bigg) \quad \text{a.e. in } [0,a]\times[0,b],\\ &u(x,y)=\psi(x,y) \quad \text{on } [-a_{0},a]\times[-b_{0},b]\setminus(0,a]\times(0,b], \end{align*} where the function \(u_{(x,y)}:[-a_{0},0]\times[-b_{0},0] \to {\mathbb R}^k\) is defined by \(u_{(x,y)}(s,t)=u({s+x},{t+y})\) for \((s,t)\in [-a_{0},0]\times[-b_{0},0]\). We give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35R10 Partial functional-differential equations
35R45 Partial differential inequalities and systems of partial differential inequalities
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