Nonlinear Diffusion Equations

An Alternative Variational Framework for Image Denoising

“…In this section, we present the definition of weak solution to the evolution problem (27)- (29), propose an approximating evolution equation, establish existence result for the solution of the approximate evolution equation, and, then, by logical mathematical manipulation and passing to the limits, present a proof of the existence and uniqueness of the solution of the evolution problem (27)- (29). We refer to the works in [25][26][27][28] as the motivation for our definition of the weak solution to the evolution problem (27)- (29).…”

On a Nonlinear Heat Equation with Degeneracy on the Boundary

“…(1.1) becomes a linear parabolic equation, we would like to suggest that, for linear equations, any boundedness estimate is equivalent to a stability result (i.e., control of differences of solutions in terms of differences of data), but this is not the truth for nonlinear equations generally. One can see the well-known monographs or textbooks [1][2][3][4][5][6][7] and the references therein. However, in some special case, if we add some restrictions to k (u,x,t) , the character may be still true.…”

On the n-Dimensional Porous Medium Diffusion Equation and Global Actions of the Symmetry Group

“…, x n ) ∈ R n and t ∈ R. When m > 1, this equation models slow diffusion phenomena so this condition is often assumed (cf. [17]). However, for reasons that will become evident below, we will allow m ∈ R\{0, 1}.…”