scholarly journals Existence of a stable blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term

2019 ◽  
Vol 371 (8) ◽  
pp. 5899-5972 ◽  
Author(s):  
Slim Tayachi ◽  
Hatem Zaag
Keyword(s):  
Heat Equation ◽  
Critical Power ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Gradient Term

Global blow-up of a nonlinear heat equation

1986 ◽  
Vol 104 (1-2) ◽  
pp. 161-167 ◽  
Author(s):  
A. A. Lacey
Keyword(s):  
Heat Equation ◽  
Parabolic Equations ◽  
Finite Time ◽  
Positive Measure ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Semilinear Parabolic Equations ◽  
Semilinear Parabolic

SynopsisSolutions to semilinear parabolic equations of the form ut = Δu + f(u), x in Ω, which blow up at some finite time t* are investigated for “slowly growing” functions f. For nonlinearities such as f(s) = (2 +s)(ln(2 +s))1+b with 0 < b < l,u becomes infinite throughout Ω as t→t* −. It is alsofound that for marginally more quickly growing functions, e.g. f(s) = (2 + s)(ln(2 +s))2, u is unbounded on some subset of Ω which has positive measure, and is unbounded throughout Ω if Ω is a small enough region.

Local well‐posedness and blow‐up for an inhomogeneous nonlinear heat equation

2020 ◽  
Vol 43 (8) ◽  
pp. 5264-5272
Author(s):  
Rasha Alessa ◽  
Aisha Alshehri ◽  
Haya Altamimi ◽  
Mohamed Majdoub
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Well Posedness

The Cauchy problem for a system of nonlinear heat equations in two space dimensions

2015 ◽  
Vol 08 (01) ◽  
pp. 1550004
Author(s):  
Amel Chouichi ◽  
Sarah Otsmane
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Dimensional Space ◽  
Local Existence ◽  
Energy Functional ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Uniqueness Of Solutions ◽  
Semilinear Heat Equation ◽  
Singular Initial Data

This paper is devoted to system of semilinear heat equations with exponential-growth nonlinearity in two-dimensional space which is the analogue of the scalar model problem studied in [S. Ibrahim, R. Jrad, M. Majdoub and T. Saanouni, Local well posedness of a 2D semilinear heat equation, Bull. Belg. Math. Soc.21 (2014) 1–17]. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space (H1× H1)(ℝ2). The uniqueness part is nontrivial although it follows Brezis–Cazenave's proof [H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math.68 (1996) 73–90] in the case of monomial nonlinearity in dimension d ≥ 3. Next, we show that in the defocusing case our solution is bounded, and therefore exists globally in time. Finally, for this system, we treat the question of blow-up in finite time under the negativity condition on the energy functional. The technique to be used is adapted from [Bull. Belg. Math. Soc. 21 (2014) 1–17].

scholarly journals Blow up of solutions of semilinear heat equations in general domains

2015 ◽  
Vol 17 (02) ◽  
pp. 1350042 ◽  
Author(s):  
Valeria Marino ◽  
Filomena Pacella ◽  
Berardino Sciunzi
Keyword(s):  
Boundary Condition ◽  
Initial Data ◽  
Heat Equation ◽  
Dirichlet Boundary ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Initial Value ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

Consider the nonlinear heat equation vt - Δv = |v|p-1v in a bounded smooth domain Ω ⊂ ℝn with n > 2 and Dirichlet boundary condition. Given up a sign-changing stationary classical solution fulfilling suitable assumptions, we prove that the solution with initial value ϑup blows up in finite time if |ϑ - 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent [Formula: see text]. Since for ϑ = 1 the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped with respect to the origin. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.

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