Differential and Integral Equations

Multiplicity and stability topics in semilinear parabolic equations

Myriam Comte, Alain Haraux, and Petru Mironescu

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Under suitable hypotheses on the nonlinear function $f$, the number of connected components of the complement of the nodal set of $\varphi$ is estimated when $\varphi$ is a solution of the elliptic equation $ -\Delta\varphi +f(\varphi) = 0$ in a bounded, open domain $\Omega$ with Dirichlet homogeneous boundary condition, and in the simplest case a dynamical consequence is derived for the corresponding semilinear heat equation. In addition, for simple domains such as a one-dimensional interval, a rectangle or a ball of arbitrary dimension, we establish the dynamical instability of solutions which do not have a constant sign in all the reasonable-looking cases.

Article information

Differential Integral Equations, Volume 13, Number 7-9 (2000), 801-811.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B35: Stability 35J60: Nonlinear elliptic equations


Comte, Myriam; Haraux, Alain; Mironescu, Petru. Multiplicity and stability topics in semilinear parabolic equations. Differential Integral Equations 13 (2000), no. 7-9, 801--811. https://projecteuclid.org/euclid.die/1356061198

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