On the generalized Dirichlet problem for viscous Hamilton--Jacobi equations.
The initial and boundary value problem for viscous Hamilton-Jacobi equation \(u_t-\Delta u + | D u| ^p=0, \quad u(x,0)=u_0(x), \quad x \in \Omega, u(x,t)=\phi(x,t),\) where \(\Omega\) is an subset of \(\mathbb R^N\) and \(p>0,\) \(u(x,t)=\phi(x,t)\) on \(\partial \Omega \times (0,\infty)\), \(u_0(x)\) and \(\phi\) are continuons functions satistying the compatiblity conditions, is studied. The authors revisit the results of \textit{M. Fila} and \textit{G. M. Lieberman} [Differ. Integral Equ., 7, No. 3--4, 811--821 (1994; Zbl 0811.35059)], \textit{P. Suplet} [SIAM J. Control Optim. 24, 335-353 (1986)]. It is shown under suitable and natural assumptions on the initial and boundary data the solution of the Dirichlet problem exists for all time, provided that the Dirichlet boundary that condition is understood in the generalized sense of viscosity solutions. This generalized notion of boundary conditions allows loss of boundary conditions, and therefore the authors show that, for \(p > 2,\) such loss of boundary data can actually happen, while the Dirichlet problem can be solved in a classical way for \(1 < p \leq 2\) and even for \(p > 0.\) The authors indicate why a blow up of the gradient or at least some related property is necessary to have such loss of boundary data, explaining the phenomena observed in the above mentioned works.
scientific article; zbMATH DE number 2066879
scientific article