scholarly journals Quenching for a Non-Newtonian Filtration Equation with a Singular Boundary Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Xiliu Li ◽  
Chunlai Mu ◽  
Qingna Zhang ◽  
Shouming Zhou
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Finite Time ◽  
Singular Boundary ◽  
Quenching Rate ◽  
Filtration Equation ◽  
Laplacian Equation ◽  
Singular Boundary Condition

This paper deals with a nonlinearp-Laplacian equation with singular boundary conditions. Under proper conditions, the solution of this equation quenches in finite time and the only quenching point thatisx=1are obtained. Moreover, the quenching rate of this equation is established. Finally, we give an example of an application of our results.

scholarly journals Simultaneous and Non-Simultaneous Quenching for a System of Multi-Dimensional Semi-Linear Heat Equations

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2075
Author(s):  
Ratinan Boonklurb ◽  
Tawikan Treeyaprasert ◽  
Aong-art Wanna
Keyword(s):  
Boundary Conditions ◽  
Finite Time ◽  
Blow Up ◽  
Initial Conditions ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Heat Equations ◽  
Quenching Rate ◽  
Linear Heat ◽  
Quenching Set

This article deals with finite-time quenching for the system of coupled semi-linear heat equations ut=uxx+f(v) and vt=vxx+g(u), for (x,t)∈(0,1)×(0,T), where f and g are given functions. The system has the homogeneous Neumann boundary conditions and the bounded nonnegative initial conditions that are compatible with the boundary conditions. The existence result is established by using the method of upper and lower solutions. We obtain sufficient conditions for finite time quenching of solutions. The quenching set is also provided. From the quenching set, it implies that the quenching solution has asymmetric profile. We prove the blow-up of time-derivatives when quenching occurs. We also find the criteria to identify simultaneous and non-simultaneous quenching of solutions. For non-simultaneous quenching, the corresponding quenching rate of solutions is given.

The solvability of an elliptic system under a singular boundary condition

2006 ◽  
Vol 136 (3) ◽  
pp. 509-546 ◽  
Author(s):  
J. García-Melián ◽  
J. Sabina de Lis ◽  
R. Letelier-Albornoz
Keyword(s):  
Boundary Condition ◽  
Elliptic System ◽  
Positive Solutions ◽  
Detailed Account ◽  
Singular Boundary ◽  
Dimensional Problem ◽  
One Dimensional ◽  
All Solutions ◽  
The One ◽  
Singular Boundary Condition

In this work we are considering both the one-dimensional and the radially symmetric versions of the elliptic system Δu = vp, Δv = uq in Ω, where p, q > 0, under the boundary condition u|∂Ω = +∞, v|∂Ω = +∞. It is shown that no positive solutions exist when pq ≤ 1, while we provide a detailed account of the set of (infinitely many) positive solutions if pq > 1. The behaviour near the boundary of all solutions is also elucidated, and symmetric solutions (u, v) are completely characterized in terms of their minima (u(0), v(0)). Non-symmetric solutions are also deeply studied in the one-dimensional problem.

scholarly journals Quenching for Porous Medium Equations

2021 ◽  
Vol 53 ◽  
Author(s):  
Burhan Selcuk
Keyword(s):  
Porous Medium ◽  
Lower Bound ◽  
Boundary Conditions ◽  
Finite Time ◽  
Initial Function ◽  
Local Existence ◽  
Singular Boundary ◽  
Porous Medium Equations ◽  
Quenching Time

This paper studies the following two porous medium equations with singular boundary conditions. First, we obtain that finite time quenching on the boundary, as well as kt blows up at the same finite time and lower bound estimates of the quenching time of the equation kt = (kn)xx + (1 − k)−α, (x,t) ∈ (0,L) × (0,T) with (kn)x (0,t) = 0, (kn)x (L,t) = (1 − k(L,t))−β, t ∈ (0,T) and initial function k(x,0) = k0 (x), x ∈ [0, L] where n > 1, α and β and positive constants. Second, we obtain that finite time queching on the boundary, as well as kt blows up at the same finite time and a local existence resultbythehelpofsteadystateoftheequationkt =(kn)xx,(x,t)∈(0,L)×(0,T)with (kn)x (0,t) = (1 − k(0,t))−α, (kn)x (L,t) = (1 − k(L,t))−β, t ∈ (0,T) and initial function k (x, 0) = k0 (x), x ∈ [0, L] where n > 1, α and β and positive constants.

Boundary behavior of large solutions to a class of Hessian equations

2020 ◽  
pp. 1-16
Author(s):  
Ling Mi ◽  
Chuan Chen
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Nonlinear Term ◽  
Boundary Behavior ◽  
Singular Boundary ◽  
Structure Condition ◽  
Hessian Equation ◽  
Large Solutions ◽  
Hessian Equations ◽  
Singular Boundary Condition

In this paper, we consider the m-Hessian equation S m [ D 2 u ] = b ( x ) f ( u ) > 0 in Ω, subject to the singular boundary condition u = ∞ on ∂ Ω. We give estimates of the asymptotic behavior of such solutions near ∂ Ω when the nonlinear term f satisfies a new structure condition.

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