Gradient estimates for degenerate diffusion equations. I

1982 ◽  
Vol 259 (1) ◽  
pp. 53-70 ◽  
Author(s):  
Nicholas D. Alikakos ◽  
Rouben Rostamian
Keyword(s):  
Diffusion Equations ◽  
Gradient Estimates ◽  
Degenerate Diffusion ◽  
Degenerate Diffusion Equations

Gradient estimates for degenerate diffusion equations II

1982 ◽  
Vol 91 (3-4) ◽  
pp. 335-346 ◽  
Author(s):  
Nicholas D. Alikakos ◽  
Rouben Rostamian
Keyword(s):  
Initial Data ◽  
Lower Bounds ◽  
Space Time ◽  
Diffusion Equations ◽  
Upper And Lower Bounds ◽  
Gradient Estimates ◽  
Degenerate Diffusion ◽  
Lipschitz Continuous ◽  
Degenerate Diffusion Equations

SynopsisWe establish upper and lower bounds for various norms of solutions and their gradients for the equation ut = div (|∇u|m−1 ∇u) in ℝN in terms of the norms of the initial data. Based on the L∞ estimate of ∇u, we conclude that u(x, t) is Lipschitz continuous in space-time, for all t>0, whenever u(x,0) is in L1(ℝN).

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