scholarly journals Blow-up for the Porous Media Equation with Source Term and Positive Initial Energy

2000 ◽  
Vol 247 (1) ◽  
pp. 183-197 ◽  
Author(s):  
Enzo Vitillaro
Keyword(s):  
Porous Media ◽  
Source Term ◽  
Blow Up ◽  
Initial Energy ◽  
Porous Media Equation ◽  
Media Equation ◽  
Positive Initial Energy

Boundary blow-up of a logistic-type porous media equation in a multiply connected domain

2010 ◽  
Vol 140 (1) ◽  
pp. 101-117 ◽  
Author(s):  
Huiling Li ◽  
Peter Y. H. Pang ◽  
Mingxin Wang
Keyword(s):  
Porous Media ◽  
Connected Domain ◽  
Blow Up ◽  
Outer Boundary ◽  
Homogeneous Boundary Condition ◽  
Porous Media Equation ◽  
Multiply Connected Domain ◽  
Media Equation ◽  
Multiply Connected ◽  
Limiting Behaviour

We study positive solutions to the porous media equation of degenerate logistic type −Δu = a(x)u1/m – b(x)f(u), m > 1, which blow up at the inner boundary and satisfy a homogeneous boundary condition at the outer boundary of a multiply connected domain. In particular, we investigate uniqueness and blow-up rate near the inner boundary for such solutions. We also look at the limiting behaviour as m ↘ 1 for the special case where b(x) > 0 and f(u) = up/m with p > m.

scholarly journals Blow-up results for a quasilinear von Karman equation of memory type

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Mi Jin Lee ◽  
Jum-Ran Kang
Keyword(s):  
Blow Up ◽  
Suitable Condition ◽  
Nondecreasing Function ◽  
Initial Energy ◽  
Nonincreasing Function ◽  
Positive Initial Energy ◽  
Von Karman ◽  
Von Karman Equation ◽  
Memory Type ◽  
Von Kármán

Abstract In this paper, we consider the blow-up result of solution for a quasilinear von Karman equation of memory type with nonpositive initial energy as well as positive initial energy. For nonincreasing function $g>0$ g > 0 and nondecreasing function f, we prove a finite time blow-up result under suitable condition on the initial data.

scholarly journals Global existence and nonexistence for a class of finitely degenerate coupled parabolic systems with high initial energy level

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yuxuan Chen ◽  
Jiangbo Han
Keyword(s):  
Energy Level ◽  
Global Existence ◽  
Blow Up ◽  
Parabolic Systems ◽  
Initial Energy ◽  
Blowup Time ◽  
Positive Initial Energy ◽  
Existence And Nonexistence ◽  
Nonexistence Of Solutions ◽  
Coupled Parabolic Systems

<p style='text-indent:20px;'>In this paper, we consider a class of finitely degenerate coupled parabolic systems. At high initial energy level <inline-formula><tex-math id="M1">\begin{document}$ J(u_{0})&gt;d $\end{document}</tex-math></inline-formula>, we present a new sufficient condition to describe the global existence and nonexistence of solutions for problem (1)-(4) respectively. Moreover, by applying the Levine's concavity method, we give some affirmative answers to finite time blow up of solutions at arbitrary positive initial energy <inline-formula><tex-math id="M2">\begin{document}$ J(u_{0})&gt;0 $\end{document}</tex-math></inline-formula>, including the estimate of upper bound of blowup time.</p>

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