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On the uniqueness of viscosity solutions for first order partial differential-functional equations. (English) Zbl 0804.35138

Viscosity solutions for the differential-functional equation \[ D_ t z(t,x) + H \bigl( t,x,z_{\langle t,x \rangle},\;D_ xz(t,x) \bigr) = 0 \quad \text{in } \quad \Theta, \qquad z(t,x) = g(t,x) \quad \text{in } \quad \Theta_ 0 \cup \partial_ 0 \Theta \] where \(\Theta_ 0 = ( - \tau_ 0,0] \times \Omega_ \tau\), \(\partial_ 0 \Omega = \Omega_ \tau \backslash \Omega\), \(\Theta = (0,T) \times \Omega\) are considered. Here, \(\Omega_ \tau\) is a \(\tau\)-neighbourhood of \(\Omega\). Let \(u_ g\) denote the viscosity solution of this system with initial-boundary function \(g\) and let \(E = \Theta \cup \Theta_ 0 \cup \partial_ 0 \Theta\) with the sup norm. Then the following uniqueness result is proved: \[ \bigl \| (u_ f - u_ g)^ + \bigr \|_ E \leq \exp (CT) \bigl \| (f-g)^ + \bigr \|_{\Theta_ 0 \cup \partial_ 0 \Theta} \] for some constant \(C\), under a simple continuity condition on \(H\). A similar result for the corresponding boundary value problem \[ H \bigl( x,z_{\langle x \rangle}, Dz(x) \bigr) = 0 \quad \text{in} \quad \Omega, \qquad z(x) = g(x) \quad \text{in} \quad \partial_ 0 \Theta \] is also proved.

MSC:

35R10 Partial functional-differential equations
35F30 Boundary value problems for nonlinear first-order PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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