The Cauchy problem for the fast p-Laplacian evolution equation. Characterization of the global Harnack principle and fine asymptotic behaviour

Functional inequalities and applications to doubly nonlinear diffusion equations

“…with C 1 = 2 q C M . The above inequality implies (18), which by Proposition 2.1 completes the proof of (HP) with w 1 (x) = h(|x|) and w 2 (x) = |x| q h(|x|). Bounds on the constant result from the estimates on c P provided in Theorem 1.…”

“…equation ( 1) with m = 1 and 1 < p < 2. Only few results are available in this range, see [3,18]. Therefore, in a forthcoming paper, we will give a more detailed analysis for a natural class of initial data.…”

“…Since the first term on the right-hand side above can be estimated due to (18) and then by properties of the gradient ( 14), we get…”

Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights

“…There is a long list of literature devoted to studying the smoothness of solutions to unweighted quasilinear elliptic and parabolic equations, see e.g. [2,14,17,18,[26][27][28][29][30][31][33][34][35][36]38,56,57,73] and the references therein. Besides its own mathematical interest, this nonlinear evolution model is also of great relevance to applications in nonlinear porous media flow, image processing and game theory etc, see e.g.…”