scholarly journals Regional Blow Up in a Semilinear Heat Equation with Convergence to a Hamilton–Jacobi Equation

1993 ◽  
Vol 24 (5) ◽  
pp. 1254-1276 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Juan L. Vazquez
Keyword(s):  
Heat Equation ◽  
Jacobi Equation ◽  
Blow Up ◽  
Semilinear Heat Equation ◽  
Hamilton Jacobi Equation

scholarly journals Blow up and globality of solutions for a nonautonomous semilinear heat equation with Dirichlet condition

2019 ◽  
Vol 53 (1) ◽  
pp. 57-72
Author(s):  
Marcos Josías Ceballos-Lira ◽  
Aroldo Pérez
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Mild Solution ◽  
Blow Up ◽  
Local Existence ◽  
Dirichlet Condition ◽  
Infinite Time ◽  
Diffusion Operator ◽  
Intrinsic Ultracontractivity ◽  
Semilinear Heat Equation

In this paper we prove the local existence of a nonnegative mild solution for a nonautonomous semilinear heat equation with Dirichlet condition, and give sucient conditions for the globality and for the blow up infinite time of the mild solution. Our approach for the global existence goes back to the Weissler's technique and for the nite time blow up we uses the intrinsic ultracontractivity property of the semigroup generated by the diffusion operator.

On blow-up and degeneracy for the semilinear heat equation with source

1990 ◽  
Vol 115 (1-2) ◽  
pp. 19-24 ◽  
Author(s):  
V. A. Galaktionov
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Semilinear Parabolic Equation ◽  
Quasilinear Parabolic Equations ◽  
Open Problems ◽  
Semilinear Heat Equation ◽  
Self Similar Solution ◽  
Self Similar ◽  
Semilinear Parabolic ◽  
Hamilton Jacobi Equation

SynopsisThe asymptotic behaviour of the solution of the semilinear parabolic equation ut = uxx + (1 + u)ln2(l + u) for t > 0, x ∊[−π, π ], ux(t, ± π) = 0 for t > 0 and u(0, x) = u0(x) ≧ 0 in [−π, π], which blows up at a finite time T0, is investigated. It is proved that for some two-parametric set of initial functions u0 the behaviour of u(t, x) near t = T0 is described by the approximate self-similar solution va(t, x) = exp {(T0 −t)−1 cos2 (x/2)} − 1, satisfying the first order nonlinear Hamilton–Jacobi equation vt, = (vx)2 /(1 + v) + (1 + v) ln2 (1 + v). Some open problems of degeneracy near a finite blow-up time for other semilinear or quasilinear parabolic equations with source ut, = Δu + (1 + u) lnβ (1 + u) (β >1), ut, = Δu + uβ(β > l), ut = Δu + eu; ut = ∇. (lnσ(1 + u)∇u)+ (1 + u)lnβ(1 + u) (σ > 0, β > 1) are discussed.

scholarly journals Stability of ODE blow-up for the energy critical semilinear heat equation

2017 ◽  
Vol 355 (1) ◽  
pp. 65-79 ◽  
Author(s):  
Charles Collot ◽  
Frank Merle ◽  
Pierre Raphaël
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Semilinear Heat Equation
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