Regularity and geometric character of solution of a degenerate parabolic equation

Abstract

Abstract This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation $$u_{t}=\Delta {}u^{m}$$ u t = Δ u m . Our main objective is to improve the H $$\ddot{o}$$ o ¨ lder estimate obtained by pioneers and then, to show the geometric characteristic of free boundary of degenerate parabolic equation. To be exact, for the weak solution u(x, t), the present work will show that: 1. The function $$\phi =(u(x,t))^{\beta }\in {}C^{1}(\mathbb {R}^{n})$$ ϕ = ( u ( x , t ) ) β ∈ C 1 ( R n ) for given $$t>0$$ t > 0 if $$\beta $$ β is large sufficiently; 2. The surface $$\phi =\phi (x,t)$$ ϕ = ϕ ( x , t ) is tangent to $$\mathbb {R}^{n}$$ R n at the boundary of the positivity set of u(x, t); 3. The function $$\phi (x,t)$$ ϕ ( x , t ) is a classical solution to another degenerate parabolic equation. Moreover, some explicit derivative estimates and expressions about the speed of propagation of u(x, t) and the continuous dependence on the nonlinearity of the equation are obtained.

Keywords

• Degenerate parabolic equation
• Regularity
• Geometric character
• H $$\ddot{o}$$ o ¨ lder Estimate