scholarly journals Blow-up for sign-changing solutions of the critical heat equation in domains with a small hole

2016 ◽  
Vol 18 (01) ◽  
pp. 1550017 ◽  
Author(s):  
Isabella Ianni ◽  
Monica Musso ◽  
Angela Pistoia
Keyword(s):  
Heat Equation ◽  
Stationary Solution ◽  
Bounded Domain ◽  
Blow Up ◽  
Initial Conditions ◽  
Small Hole ◽  
Initial Value ◽  
Smooth Bounded Domain ◽  
Sign Changing Solutions

We consider the critical heat equation [Formula: see text] in [Formula: see text] where [Formula: see text] is a smooth bounded domain in [Formula: see text] and [Formula: see text] is a ball of [Formula: see text] of center [Formula: see text] and radius [Formula: see text] small. We show that if [Formula: see text] is small enough, then there exists a sign-changing stationary solution [Formula: see text] of (CH) such that the solution of (CH) with initial value [Formula: see text] blows up if [Formula: see text] is sufficiently small. This shows, in particular, that the set of the initial conditions for which the solution of (CH) is global and bounded is not star-shaped.

scholarly journals Single blow-up solutions for a slightly subcritical biharmonic equation

2006 ◽  
Vol 2006 ◽  
pp. 1-20 ◽  
Author(s):  
Khalil El Mehdi
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Critical Exponent ◽  
Critical Point ◽  
Bounded Domain ◽  
Biharmonic Equation ◽  
Blow Up ◽  
Asymptotic Behavior Of Solutions ◽  
Smooth Bounded Domain ◽  
Nondegenerate Critical Point

We consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent (Pε):∆2u=u9−ε,u>0inΩandu=∆u=0on∂Ω, whereΩis a smooth bounded domain inℝ5,ε>0. We study the asymptotic behavior of solutions of (Pε) which are minimizing for the Sobolev quotient asεgoes to zero. We show that such solutions concentrate around a pointx0∈Ωasε→0, moreoverx0is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical pointx0of the Robin's function, there exist solutions of (Pε) concentrating aroundx0asε→0.

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.

scholarly journals Blow up Points and the Morse Indices of Solutions to the Liouville Equation in Two-Dimension

2012 ◽  
Vol 12 (1) ◽  
Author(s):  
Futoshi Takahashi
Keyword(s):  
Bounded Domain ◽  
Liouville Equation ◽  
Morse Index ◽  
Blow Up ◽  
Smooth Bounded Domain ◽  
Two Dimension ◽  
Morse Indices

AbstractWe consider the Liouville equation−Δu = λeu in Ω, u =0 on∂Ω,on a smooth bounded domain Ω in ℝfor m ∈ℕ. We prove that the number of blow up points m is less than or equal to the Morse index of uAs a corollary, we show that if a solution u

PROFILE AND EXISTENCE OF SIGN-CHANGING SOLUTIONS TO AN ELLIPTIC SUBCRITICAL EQUATION

2008 ◽  
Vol 10 (06) ◽  
pp. 1183-1216 ◽  
Author(s):  
MOHAMED BEN AYED ◽  
RABEH GHOUDI
Keyword(s):  
Critical Exponent ◽  
Elliptic Problem ◽  
Blow Up ◽  
Low Energy ◽  
Nonlinear Elliptic Problem ◽  
Smooth Bounded Domain ◽  
Nonlinear Elliptic ◽  
Robin Function ◽  
Sign Changing Solutions ◽  
Two Bubbles

In this paper, we study the nonlinear elliptic problem involving nearly critical exponent (Pε) : Δ2 u = |u|(8/(n-4))-εu, in Ω, Δu = u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝn, n ≥ 5. We characterize the low energy sign-changing solutions (uε) of (Pε). We prove that (uε) are close to two bubbles with different signs and they have to blow up either at two different points with the same speed or at a critical point of the Robin function. Furthermore, we construct families of each kind of these solutions and we prove that the bubble-tower solutions exist in our case.

Classification of Global and Blow-up Sign-changing Solutions of a Semilinear Heat Equation in the Subcritical Fujita Range : Second-order Diffusion

2014 ◽  
Vol 14 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Enzo Mitidieri ◽  
Stanislav I. Pohozaev
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Blow Up ◽  
Second Order ◽  
Seminal Paper ◽  
Semilinear Heat Equation ◽  
Sign Changing Solutions

AbstractIt is well known from the seminal paper by Fujita [22] for 1 < p < puwith arbitrary initial data u

Global Sign-changing Solutions of a Higher Order Semilinear Heat Equation in the Subcritical Fujita Range

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Victor A. Galaktionov ◽  
Enzo Mitidieri ◽  
Stanislav I. Pohozaev
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Initial Function ◽  
Blow Up ◽  
Higher Order ◽  
Semilinear Heat Equation ◽  
Sign Changing Solutions

AbstractA detailed study of two classes of oscillatory global (and blow-up) solutions was began in [20] for the semilinear heat equation in the subcritical Fujita rangewith bounded integrable initial data u(x, 0) = uwith the same initial data u∫ ui.e., as for (0.1), any such arbitrarily small initial function u

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 Deriving The Upper Blow-up Rate Estimate for a Parabolic Problem

2020 ◽  
pp. 200-203
Author(s):  
Maan A. Rasheed
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Bounded Domain ◽  
Neumann Boundary Condition ◽  
Parabolic Problem ◽  
Blow Up ◽  
Neumann Boundary ◽  
Rate Estimate ◽  
Blow Up Rate

In this paper, the blow-up solutions for a parabolic problem, defined in a bounded domain, are studied. Namely, we consider the upper blow-up rate estimate for heat equation with a nonlinear Neumann boundary condition defined on a ball in Rn.

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