Differential and Integral Equations

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

Julio D. Rossi and Philippe Souplet

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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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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