scholarly journals Classical solvability of multidimensional two-phase Stefan problem for degenerate parabolic equations and Schauder’s estimates for a degenerate parabolic problem with dynamic boundary conditions

2014 ◽  
Vol 22 (2) ◽  
pp. 185-237 ◽  
Author(s):  
S. P. Degtyarev
Keyword(s):  
Boundary Conditions ◽  
Stefan Problem ◽  
Parabolic Equations ◽  
Parabolic Problem ◽  
Degenerate Parabolic Equations ◽  
Dynamic Boundary Conditions ◽  
Two Phase ◽  
Classical Solvability ◽  
Degenerate Parabolic Problem ◽  
Degenerate Parabolic

scholarly journals Blow-up results of the positive solution for a class of degenerate parabolic equations

2021 ◽  
Vol 19 (1) ◽  
pp. 773-781
Author(s):  
Chenyu Dong ◽  
Juntang Ding
Keyword(s):  
Positive Solution ◽  
Parabolic Equations ◽  
Blow Up ◽  
Degenerate Parabolic Equations ◽  
Maximum Principles ◽  
First Order ◽  
Degenerate Parabolic Problem ◽  
Inequality Technique ◽  
Degenerate Parabolic ◽  
Upper Estimate

Abstract This paper is devoted to discussing the blow-up problem of the positive solution of the following degenerate parabolic equations: ( r ( u ) ) t = div ( ∣ ∇ u ∣ p ∇ u ) + f ( x , t , u , ∣ ∇ u ∣ 2 ) , ( x , t ) ∈ D × ( 0 , T ∗ ) , ∂ u ∂ ν + σ u = 0 , ( x , t ) ∈ ∂ D × ( 0 , T ∗ ) , u ( x , 0 ) = u 0 ( x ) , x ∈ D ¯ . \left\{\begin{array}{ll}{(r\left(u))}_{t}={\rm{div}}(| \nabla u{| }^{p}\nabla u)+f\left(x,t,u,| \nabla u\hspace{-0.25em}{| }^{2}),& \left(x,t)\in D\times \left(0,{T}^{\ast }),\\ \frac{\partial u}{\partial \nu }+\sigma u=0,& \left(x,t)\in \partial D\times \left(0,{T}^{\ast }),\\ u\left(x,0)={u}_{0}\left(x),& x\in \overline{D}.\end{array}\right. Here p > 0 p\gt 0 , the spatial region D ⊂ R n ( n ≥ 2 ) D\subset {{\mathbb{R}}}^{n}\hspace{0.33em}\left(n\ge 2) is bounded, and its boundary ∂ D \partial D is smooth. We give the conditions that cause the positive solution of this degenerate parabolic problem to blow up. At the same time, for the positive blow-up solution of this problem, we also obtain an upper bound of the blow-up time and an upper estimate of the blow-up rate. We mainly carry out our research by means of maximum principles and first-order differential inequality technique.

The two-phase Stefan problem for parabolic equations

2019 ◽  
Vol 2019 (4) ◽  
pp. 54-64
Author(s):  
A.N. Elmurodov
Keyword(s):  
Stefan Problem ◽  
Parabolic Equations ◽  
Two Phase
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