# The zero set of a solution of a parabolic equation.

@article{Angenent1988TheZS,
title={The zero set of a solution of a parabolic equation.},
author={Sigurd B. Angenent},
journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
year={1988},
volume={1988},
pages={79 - 96}
}
• S. Angenent
• Published 1988
• Mathematics
• Journal für die reine und angewandte Mathematik (Crelles Journal)
On etudie l'ensemble nul d'une solution u(t,x) de l'equation u t =a(x,t)u xx +b(x,t)u x +C(x,t)u, sous des hypotheses tres generales sur les coefficients a, b, et c
445 Citations
Entire solutions and a Liouville theorem for a class of parabolic equations on the real line
We consider a class of semilinear heat equations on R, including in particular the Fujita equation ut = uxx + |u|p−1u, x ∈ R, t ∈ R, where p > 1. We first give a simple proof and an extension of a
Asymptotic behavior of solutions of semilinear heat equations on S1
We study the dynamical behavior of the initial value problem for the equation u t = u xx + f(u, u x ), x ∈ S 1 =R/Z, t > 0. One of our main results states that any C 1-bounded solution approaches
On the quenching behavior of the solution of a semilinear parabolic equation
where /? > 0, E > 0, 0 6 u0 0 there exists an E(B) > 0 such that u quenches at finite time T if E > E(B). We say that a is a quenching point for u if there exists a sequence {(x,, t,)} with x, + a
Large-time behavior of solutions of parabolic equations on the real line with convergent initial data
• Mathematics
Nonlinearity
• 2018
We consider the semilinear parabolic equation on the real line, where f is a locally Lipschitz function on We prove that if a solution u of this equation is bounded and its initial value has distinct
Solution of a nonlinear heat equation with arbitrarily given blow-up points
We consider the equation where I ⊂ ℝ u is scalar-valued and p > 1. It has been proven that if u(t) blows up at time T, the blow-up points are finite in number and located in I°.
A Note on the Dynamics of Piecewise-Autonomous Bistable Parabolic Equations
• Mathematics
• 2000
For a family of piecewise-autonomous one-dimensional bistable parabolic equations, with vanishing diffusion and Neumann boundary conditions, we determine the number and Morse indices of their
Infinite Time Blow-Up of Solutions to a Nonlinear Parabolic Problem
Abstract We investigate the blow-up of solutions of nonuniformly parabolic equations. It will be shown that the gradient of the solution tends to infinity as time goes to infinity, even though
Slow decay of solutions in a semilinear dissipative parabolic equation
This paper is concerned with a Cauchy problem (P) { u t = u x x − | u | p − 1 u in R x ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) in R , , where p > 1 and u0 e L∞(R). A solution u of (P) is said to decay
Quantitative unique continuation for a parabolic equation
• Physics, Mathematics
• 2017
We address the quantitative uniqueness properties of the solutions of the parabolic equation $\partial_t u - \Delta u = w_j (x,t) \partial_j u + v(x,t) u$ where $v$ and $w$ are bounded. We prove
ORDER STRUCTURES AND THE HEAT
• 1997
In this paper we shall study the evolution of the zero set of the solution of the heat equation perturbed by a potential c and Neumann boundary conditions in one and two dimensions. 1. Introduction

## References

SHOWING 1-10 OF 12 REFERENCES
Numbers of zeros on invariant manifolds in reaction-diffusion equations
• Mathematics
• 1986
On considere l'equation a une dimension u t =u xx +f(x,u), t>0, 0<x<1 avec les conditions aux limites de Dirichlet u(t,0)=(t,1)=0, ou f:[0,1]×R→R est BC 1 ∩C K , K<1
The Morse-Smale property for a semi-linear parabolic equation
On etudie la dynamique d'equations aux derivees partielles paraboliques semi-lineaires du type suivant: u t =u xx +f(x,u) 0 0, u(0,t)=u(1,t)=0, t>0, u(x,0)=φ(x)
Simplicity of zeros in scalar parabolic equations
• Mathematics
• 1986
Soit le systeme u t =u xx +f(x,u), x∈(0,1), u(t,0)=u(t,1)=0, f(x,0)=0, u(0,x)=u 0 (x). Soit Σ:={t>0/u(t,.) a seulement des zeros simples}; soit Z(u(t,.)) le nombre de zeros de u(0,.). On suppose
The heat equation shrinks embedded plane curves to round points
Soit C(•,0):S 1 →R 2 une courbe lisse plongee dans le plan. Alors C:S 1 ×[0,T)→R 2 existe en satisfaisant δC/δt=K•N, ou K est la courbure de C, et N est son vecteur unite normal entrant. C(•,t) est
Maximum principles in differential equations
• Mathematics
• 1967
The One-Dimensional Maximum Principle.- Elliptic Equations.- Parabolic Equations.- Hyperbolic Equations.- Bibliography.- Index.
Connecting orbits in scalar reaction diffusion equations
• Physics
• 1988
We consider the flow of a one-dimensional reaction diffusion equation $${u_t} = {u_{xx}} + f(u),x \in (0,1)$$ (1.1) with Dirichlet boundary conditions $$u(t,0) = u(t,1) = 0{\text{ }}$$
Zero numbers on invariant manifolds in scalar reaction diffusion equations
• Nonlinear Analysis TMA
• 1986
Convergence of Solutions of one dimensional parabolic equations
• J. Math. Kyoto Univ
• 1978