## Differential and Integral Equations

### Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system

#### Abstract

We study simultaneous and nonsimultaneous blow-up for solutions of the following system $$\left\{\begin{array}{l} u_t = \Delta u + u^r + v^p, \\ v_t = \Delta v + v^s + u^q, \end{array} \quad\mbox{ in } \Omega \times (0,T), \right.$$ with Dirichlet boundary conditions. We show that, in the range of exponents where either component may blow up alone, there also exist initial data for which both components blow up simultaneously. The proof is based on a continuity argument, which requires upper and lower blow-up estimates, independent of initial data, and continuous dependence of the existence time. In turn, we prove a result of continuous dependence of the existence time, under the assumption of uniform upper blow-up estimates, in the framework of general abstract semiflows.

#### Article information

Source
Differential Integral Equations, Volume 18, Number 4 (2005), 405-418.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356060194

Mathematical Reviews number (MathSciNet)
MR2122706

Zentralblatt MATH identifier
1212.35219

Subjects
Primary: 35K50
Secondary: 35B40: Asymptotic behavior of solutions 35K55: Nonlinear parabolic equations

#### Citation

Rossi, Julio D.; Souplet, Philippe. Coexistence of simultaneous and nonsimultaneous blow-up in a semilinear parabolic system. Differential Integral Equations 18 (2005), no. 4, 405--418. https://projecteuclid.org/euclid.die/1356060194