A simple algorithm for solving Cauchy problem of nonlinear heat equation without initial value

2015 ◽  
Vol 80 ◽  
pp. 562-569 ◽  
Author(s):  
Chein-Shan Liu ◽  
Chih-Wen Chang
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Simple Algorithm ◽  
Nonlinear Heat Equation ◽  
Initial Value

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.

Asymptotics for a class of heat equations with inhomogeneous nonlinearity

Analysis ◽  
2018 ◽  
Vol 38 (1) ◽  
pp. 21-36
Author(s):  
Tarek Saanouni ◽  
Radhia Ghanmi
Keyword(s):  
Heat Equation ◽  
Initial Value Problem ◽  
Energy Space ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Well Posedness ◽  
Initial Value ◽  
Power Nonlinearity

AbstractThe initial value problem for an inhomogeneous nonlinear heat equation with a pure power nonlinearity is investigated. In the radial energy space, global and non-global well-posedness are discussed.

On the Cauchy problem for the nonlinear heat equation

2019 ◽  
Author(s):  
Elena Nikolova ◽  
Mirko Tarulli ◽  
George Venkov
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Nonlinear Heat Equation ◽  
The Cauchy Problem

On the Cauchy problem for a generalized nonlinear heat equation

2018 ◽  
Vol 25 (2) ◽  
pp. 169-180
Author(s):  
Franka Baaske ◽  
Hans-Jürgen Schmeißer
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Space Variable ◽  
Strong Solutions ◽  
Nonlinear Heat Equation ◽  
Well Posedness ◽  
Weierstrass Semigroup ◽  
Smooth Functions ◽  
The Cauchy Problem ◽  
Generalized Heat Equation

Abstract The paper is concerned with the Cauchy problem for a nonlinear generalized heat equation which is related to the generalized Gauss–Weierstrass semigroup via Duhamel’s principle. For the initial data we assume that they belong to some fractional Sobolev spaces. We study the existence and uniqueness of mild and strong solutions which are local in time. Moreover, they are smooth functions and belong to Lebesgue spaces with respect to the space variable. We use both fixed point arguments and mapping properties of the generalized Gauss–Weierstrass semigroup. Finally, we study the well-posedness of the problem.

scholarly journals Sign-changing solutions of the nonlinear heat equation with persistent singularities

2020 ◽  
Vol 26 ◽  
pp. 126
Author(s):  
Thierry Cazenave ◽  
Flávio Dickstein ◽  
Ivan Naumkin ◽  
Fred B. Weissler
Keyword(s):  
Heat Equation ◽  
Nonlinear Heat Equation ◽  
Initial Value ◽  
Time Solution ◽  
Self Similar ◽  
Sign Changing Solutions

We study the existence of sign-changing solutions to the nonlinear heat equation ∂tu = Δu + |u|αu on ℝN, N ≥ 3, with 2/N−2 <α<α0, where α0=4/N−4+2√N−1 ∈ (2/N−2,4/N−2), which are singular at x = 0 on an interval of time. In particular, for certain μ > 0 that can be arbitrarily large, we prove that for any u0 ∈ Lloc∞(ℝN\{0}) which is bounded at infinity and equals μ|x|−2/α in a neighborhood of 0, there exists a local (in time) solution u of the nonlinear heat equation with initial value u0, which is sign-changing, bounded at infinity and has the singularity β|x|−2/α at the origin in the sense that for t > 0, |x|2/αu(t,x) → β as |x|→ 0, where β=2/α(N−2−2/α). These solutions in general are neither stationary nor self-similar.

scholarly journals Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value

2020 ◽  
Vol 142 (5) ◽  
pp. 1439-1495
Author(s):  
Thierry Cazenave ◽  
Flávio Dickstein ◽  
Ivan Naumkin ◽  
Fred B. Weissler
Keyword(s):  
Heat Equation ◽  
Nonlinear Heat Equation ◽  
Initial Value ◽  
Self Similar ◽  
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