## Differential and Integral Equations

### Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level

#### Abstract

For a class of semilinear parabolic equations on a bounded domain $\Omega$, we analyze the behavior of the solutions when the initial data varies in the phase space $H^1_0(\Omega)$. We obtain both global solutions and finite time blow-up solutions. Our main tools are the comparison principle and variational methods. Particular attention is paid to initial data at high energy level; to this end, a basic new idea is to exploit the weak dissipativity (respectively antidissipativity) of the semiflow inside (respectively outside) the Nehari manifold.

#### Article information

Source
Differential Integral Equations, Volume 18, Number 9 (2005), 961-990.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356060117

Mathematical Reviews number (MathSciNet)
MR2162421

Zentralblatt MATH identifier
1212.35248

Subjects
Primary: 35K55: Nonlinear parabolic equations
Secondary: 35B40: Asymptotic behavior of solutions

#### Citation

Gazzola, Filippo; Weth, Tobias. Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level. Differential Integral Equations 18 (2005), no. 9, 961--990. https://projecteuclid.org/euclid.die/1356060117