Abstract
We study the Dirichlet problem for viscous Hamilton–Jacobi equations. Despite this type of equations seems to be uniformly elliptic, loss of boundary conditions may occur because of the strong nonlinearity of the first-order part and therefore the Dirichlet boundary condition has to be understood in the sense of viscosity solutions theory. Under natural assumptions on the initial and boundary data, we prove a Strong Comparison Result which allows us to obtain the existence and the uniqueness of a continuous solution which is defined globally in time.
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