Gradient estimates for a degenerate parabolic equation with gradient absorption and applications

Abstract We consider a quasilinear degenerate parabolic equation driven by the orthotropic p-Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.

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Hölder gradient estimates for a class of singular or degenerate parabolic equations

Advances in Nonlinear Analysis ◽

10.1515/anona-2016-0197 ◽

2017 ◽

Vol 8 (1) ◽

pp. 845-867 ◽

Cited By ~ 4

Author(s):

Cyril Imbert ◽

Tianling Jin ◽

Luis Silvestre

Keyword(s):

Parabolic Equation ◽

Parabolic Equations ◽

Degenerate Parabolic Equation ◽

Divergence Form ◽

Gradient Estimates ◽

Degenerate Parabolic Equations ◽

Spatial Gradients ◽

Degenerate Parabolic ◽

Laplacian Equations ◽

Hölder Estimates

Abstract We prove interior Hölder estimates for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation u_{t}=\lvert\nabla u\rvert^{\kappa}\operatorname{div}(\lvert\nabla u\rvert^{p-% 2}\nabla u), where {p\in(1,\infty)} and {\kappa\in(1-p,\infty)} . This includes the from {L^{\infty}} to {C^{1,\alpha}} regularity for parabolic p-Laplacian equations in both divergence form with {\kappa=0} , and non-divergence form with {\kappa=2-p} .

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Null controllability of a semilinear degenerate parabolic equation with a gradient term

Boundary Value Problems ◽

10.1186/s13661-020-01351-2 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Fengdan Xu ◽

Qian Zhou ◽

Yuanyuan Nie

Keyword(s):

Parabolic Equation ◽

Degenerate Parabolic Equation ◽

Null Controllability ◽

Gradient Term ◽

Degenerate Parabolic

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On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02550-w ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Sujun Weng

Keyword(s):

Parabolic Equation ◽

Newtonian Fluid ◽

Weak Solutions ◽

Mixed Type ◽

Degenerate Parabolic Equation ◽

Type Equation ◽

Existence Of Weak Solutions ◽

Degenerate Parabolic ◽

The Stability ◽

Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

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