Aleksandrov and Bony maximum principles for parabolic equations

1991 ◽  
pp. 121-148
Keyword(s):  
Parabolic Equations ◽  
Maximum Principles

scholarly journals Strong Maximum Principles for Anisotropic Elliptic and Parabolic Equations

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Jérôme Vétois
Keyword(s):  
Parabolic Equations ◽  
Maximum Principles ◽  
Nonnegative Solutions ◽  
Elliptic And Parabolic Equations

AbstractWe investigate vanishing properties of nonnegative solutions of anisotropic elliptic and parabolic equations. We describe the optimal vanishing sets, and we establish strong maximum principles.

scholarly journals Liouville Theorems for Fractional Parabolic Equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenxiong Chen ◽  
Leyun Wu
Keyword(s):  
Half Space ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Maximum Principles ◽  
Qualitative Properties ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Narrow Region ◽  
Liouville Type Theorems ◽  
Whole Space

Abstract In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 u\to 0 at infinity to a polynomial growth on 𝑢 by constructing proper auxiliary functions. Then we derive monotonicity for the solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and obtain some new connections between the nonexistence of solutions in a half space R + n × R \mathbb{R}_{+}^{n}\times\mathbb{R} and in the whole space R n - 1 × R \mathbb{R}^{n-1}\times\mathbb{R} and therefore prove the corresponding Liouville type theorems. To overcome the difficulty caused by the nonlocality of the fractional Laplacian, we introduce several new ideas which will become useful tools in investigating qualitative properties of solutions for a variety of nonlocal parabolic problems.

scholarly journals Maximum principles for nonlocal parabolic Waldenfels operators

2019 ◽  
Vol 09 (03) ◽  
pp. 1950015 ◽  
Author(s):  
Qiao Huang ◽  
Jinqiao Duan ◽  
Jiang-Lun Wu
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Stochastic Differential Equations ◽  
Markov Processes ◽  
Parabolic Equations ◽  
Lévy Processes ◽  
Stochastic Dynamics ◽  
Levy Processes ◽  
Maximum Principles

As a class of Lévy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under Lévy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for ‘parabolic’ equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with [Formula: see text]-stable Lévy processes are presented to illustrate the maximum principles.

scholarly journals Maximum principles for implicit parabolic equations

1973 ◽  
Vol 49 (10) ◽  
pp. 785-788 ◽  
Author(s):  
Norio Yoshida
Keyword(s):  
Parabolic Equations ◽  
Maximum Principles

scholarly journals Maximum principles for parabolic equations

1994 ◽  
Vol 56 (1) ◽  
pp. 41-52 ◽  
Author(s):  
Giovanni Porru ◽  
Salvatore Serra
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Smooth Function ◽  
Suitable Condition ◽  
Maximum Principles ◽  
Spatial Gradient ◽  
Parabolic Boundary ◽  
Maximum Value

AbstractLet u(x, t) be a smooth function in the domain Q = Ω × (0, L), Ω in n, let Du be the spatial gradient of u(x, t) and let ∇u = (Du, u1). If u(x, t) satisfies the parabolic equation F(u, Du, D2u) = ut, we define w(x, t) by g(w) = │∇u│−1G(∇u) (g is positive and decreasing, G is concave and homogeneous of degree one) and we prove that w(x, t) attains its maximum value on the parabolic boundary of Q. If u(x, t) satisfies the equation Δu + 2h(q2) uiujuij = ut(q2 = │Du│2, 1 + 2q2h(q2) > 0) we prove that qf (u) takes its maximum value on the parabolic boundary of Q provided f satisfies a suitable condition. If u(x, t) satisfies the parabolic equation aij (Du)uij − b(x, t, u, Du) = ut (b is concave with respect to (x, t, u)) we define C(x, y, t, τ) = u(z, θ) − αu(x, t) − βu(y, τ) (0 < α, 0 < β, α + β = 1, z αx +y, θ = αt + βτ) and we prove that if C(x, y, t, r) ≤0 when x, y, z ∈ Ω2 and one of t, τ = 0, and when t, τ ∈ (0, L], and one of x, y, z, ∈ ∂Ω, then it is C(x, y, t, τ) ≤0 everywhere.

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.

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