𝔖 Bobbio Scriptorium
✦   LIBER   ✦

The Blow-Up Rate for a Strongly Coupled System of Semilinear Heat Equations with Nonlinear Boundary Conditions

✍ Scribed by Chunlai Mu; Shaoyong Lai

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
106 KB
Volume
254
Category
Article
ISSN
0022-247X

No coin nor oath required. For personal study only.

✦ Synopsis

The paper deals with the blow-up rate of positive solutions to the system l 11 l 12 l 21 l 22 Ε½ . u s u q u Β¨, Β¨s Β¨q u Β¨with boundary conditions u 1, t s t x x t x x x Ε½ p 11 p 12 .Ε½ . Ε½ . Ε½ p 21 p 22 .Ε½ . u Β¨1, t and Β¨1, t s u Β¨1, t . Under some assumptions on the x Ε½ .

Ε½ . Ε½ . matrices L s l and P s p and on the initial data u , Β¨, the solution u, Γ― j i j 0 0 Ε½ . Ε½ blows up at finite time T, and we prove that max u x, t resp.

x g w0, 1x

Ž .. Ž . ␣ 1 r2 Ž Ž . ␣ 2 r2 . max ¨x, t goes to infinity as T y t resp. T y t , where ␣ -0 x g w0, 1x i Ž . Ž . t Ž . t are the solutions of P y Id ␣ , ␣ s y1, y1 .

πŸ“œ SIMILAR VOLUMES

Blow-up analysis for a system of heat eq
Blow-up analysis for a system of heat equations coupled via nonlinear boundary conditions
✍ Xianfa Song πŸ“‚ Article πŸ“… 2007 πŸ› John Wiley and Sons 🌐 English βš– 128 KB πŸ‘ 1 views

## Abstract In this paper, we study a system of heat equations $u\_t=\Delta u, \, v\_t=\Delta v\,{\rm in}\,\Omega\times(0,T)$ coupled __via__ nonlinear boundary conditions Here __p__, __q__>0. We prove that the solutions always blow up in finite time for non‐trivial and non‐negative initial value

The Blow-up Rate for a System of Heat Eq
The Blow-up Rate for a System of Heat Equations with Non-trivial Coupling at the Boundary
✍ Julio D. Rossi πŸ“‚ Article πŸ“… 1997 πŸ› John Wiley and Sons 🌐 English βš– 222 KB πŸ‘ 2 views

Under some natural hypothesis on the matrix P"(p GH ) that guarrantee the blow-up of the solution at time ΒΉ, and some assumptions of the initial data u G , we find that if "" x """1 then u G (x , t)goestoinfinitylike(ΒΉ!t) G /2 , where the G (0 are the solutions of (P!Id) ( , )R"(!1, !1)R. As a corol

Note on Critical Exponents for a System
Note on Critical Exponents for a System of Heat Equations Coupled in the Boundary Conditions
✍ Shu Wang; Chunhong Xie; Mingxin Wang πŸ“‚ Article πŸ“… 1998 πŸ› Elsevier Science 🌐 English βš– 137 KB

This note establishes the blow up estimates near the blow up time for a system of heat equations coupled in the boundary conditions. Under certain assumptions, the exact rate of blow up is established. We also prove that the only solution with vanishing initial values when pq G 1 is the trivial one.