scholarly journals Blow-up for a semilinear heat equation with Fujita’s critical exponent on locally finite graphs

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Yiting Wu
Keyword(s):  
Heat Equation ◽  
Critical Exponent ◽  
Blow Up ◽  
Locally Finite ◽  
Finite Graphs ◽  
Semilinear Heat Equation ◽  
Locally Finite Graphs

Blow-up for quasilinear heat equations with critical Fujita's exponents

1994 ◽  
Vol 124 (3) ◽  
pp. 517-525 ◽  
Author(s):  
Victor A. Galaktionov
Keyword(s):  
Heat Equation ◽  
Critical Exponent ◽  
Critical Case ◽  
Source Term ◽  
Blow Up ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
The Cauchy Problem ◽  
Fixed Constant ◽  
Image Position

We consider the Cauchy problem for the quasilinear heat equationwhere σ > 0 is a fixed constant, with the critical exponent in the source term β = βc = σ + 1 + 2/N. It is well-known that if β ∈(1,βc) then any non-negative weak solution u(x, t)≢0 blows up in a finite time. For the semilinear heat equation (HE) with σ = 0, the above result was proved by H. Fujita [4].In the present paper we prove that u ≢ 0 blows up in the critical case β = σ + 1 + 2/N with σ > 0. A similar result is valid for the equation with gradient-dependent diffusivitywith σ > 0, and the critical exponent β = σ + 1 + (σ + 2)/N.

scholarly journals The edmonds—Gallai decomposition for matchings in locally finite graphs

COMBINATORICA ◽  
1982 ◽  
Vol 2 (3) ◽  
pp. 229-235 ◽  
Author(s):  
François Bry ◽  
Michel Las Vergnas
Keyword(s):  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs

scholarly journals Topological Infinite Gammoids, and a New Menger-Type Theorem for Infinite Graphs

10.37236/6083 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Johannes Carmesin
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Infinite Graphs ◽  
Disjoint Paths ◽  
Natural Topology ◽  
Locally Finite ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Vertex Sets ◽  
Special Case

Answering a question of Diestel, we develop a topological notion of gammoids in infinite graphs which, unlike traditional infinite gammoids, always define a matroid.As our main tool, we prove for any infinite graph $G$ with vertex-sets $A$ and $B$, if every finite subset of $A$ is linked to $B$ by disjoint paths, then the whole of $A$ can be linked to the closure of $B$ by disjoint paths or rays in a natural topology on $G$ and its ends.This latter theorem implies the topological Menger theorem of Diestel for locally finite graphs. It also implies a special case of the infinite Menger theorem of Aharoni and Berger.

scholarly journals The Structure of Locally Finite Two-Connected Graphs

10.37236/1211 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Carl Droms ◽  
Brigitte Servatius ◽  
Herman Servatius
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Finite ◽  
Connected Graphs ◽  
Finite Graphs ◽  
Locally Finite Graphs ◽  
Necessary And Sufficient ◽  
Block Tree ◽  
Labeled Tree

We expand on Tutte's theory of $3$-blocks for $2$-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the $3$-block tree of a $2$-connected graph.

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