Well-posedness and gradient blow-up estimate near the boundary for a Hamilton–Jacobi equation with degenerate diffusion

Boundedness of global solutions of a p-Laplacian evolution equation with a nonlinear gradient term

“…Similar gradient estimates in any space dimension are already obtained in [4] using a more technical Bernstein-type argument.…”

“…This, in turn requires good convergence properties and estimates of regularized solutions. For this, we heavily rely on results from our previous work [4] (which concerned the higher dimensional problem as well) and an extension up to the boundary of a result of DiBenedetto-Friedman on the regularity of the derivative of weak solutions of a degenerate parabolic problem (see Proposition 2.1). Let us mention some results concerning related equations possessing solutions with unbounded gradient.…”

“…Section 4 is devoted to the proof of Theorem 1.1. It is known (see e.g., [4]) that there exists T max = T max (u 0 ) ∈ (0, ∞] such that for each T ∈ (0, T max ), (1.1) admits a unique solution u such that u ∈ L ∞ ((0, T ); W 1,∞ (0, 1)). In the rest of this paper, the maximal weak solution of problem (1.1) will refer to this solution.…”

“…The uniqueness of the solution of (1.1) implies thatũ = u and that the whole sequence converges to u. We recall that u ε is bounded in L ∞ ([0,τ ]; W 1,∞ (0, 1)) (see Step 3 of the proof of Theorem 1.1 in [4]). The boundary regularity result of Proposition 2.1 implies that the convergence of {(u ε ) x } to u x holds in C loc ((0,τ ) × [0, 1]) (that is up to the boundary).…”

Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation

“…Due to its mathematical relevance as a simple model for an equation with nonlinear dependence on the gradient of the solution, (1.8) has been studied from numerous points of view and with different qualitative results in mind: existence and uniqueness of classical solutions ( [16]); existence and nonexistence of global, classical solutions and gradient blow-up and related phenomena ( [31], [32], [34], [23]; see also [26] and the references therein for a broader context); global existence of viscosity solutions, assuming boundary conditions in the viscosity sense ( [4]); and, closest to our work, regarding LOBC, [24]. Some of the previous results have been extended to more general equations, still in the second-order setting: e.g., to degenerate equations in [1], [2]; and, by the authors, to fully nonlinear, uniformly parabolic equations in [25], which the present work closely parallels.…”