Viscosity solution theory of a class of nonlinear degenerate parabolic equations. I: Uniqueness and existence of viscosity solutions
For the nonlinear degenerate parabolic equations \[ u_t+F(Du,D^2u)= 0,\;(x,t)\in \mathbb{R}^n\times (0,T), \] with \(F\) satisfying the following assumptions: \(F(q,X)\in C(\mathbb{R}^n\times S^n)\), \(F(q,X+Y)\leq F(q,X)\), for all \(Y\geq 0\), \(F(q,X)\geq -A|q|^{p-2}|X|_1- B|q|^{p-1}\) for all \(A,B\geq 0\) and \(p>2\), where \(S^n\) denotes the space of \(n\times n\) symmetric matrices, the author establishes the comparison principle, regularity, existence, and some geometric properties of the interface for the viscosity solutions. The evolution \(p\)-Laplacian equation and its anisotropic version are typical examples for this class of equations. The approximation method and Perron's method developed by Ishii are used to construct the viscosity solution to the Cauchy problem, and the \(W^{1,1,\infty}\) estimates for initial functions in \(W^{2,\infty}\) are obtained by constructing the global obstacle functions.
scientific article; zbMATH DE number 1063725
scientific article