Control of Weakly Blowing up Semilinear Heat Equations

  • E. Fernández-Cara
  • E. Zuazua
Part of the NATO Science Series book series (ASIC, volume 569)


In these notes we consider a semilinear heat equation in a bounded domain of ℝ d , with control on a subdomain and homogeneous Dirichlet boundary conditions. We consider nonlinearities for which, in the absence of control, blow up arises.

We prove that when the nonlinearity grows at infinity fast enough, due to the local (in space) nature of the blow up phenomena, the control may not avoid the blow up to occur for suitable initial data. This is done by means of localized energy estimates.

However, we also show that when the nonlinearity is weak enough, and provided the system admits a globally defined solution (for some initial data and control), the choice of a suitable control guarantees the global existence of solutions and moreover that the solution may be driven in any finite time to the globally defined solution. In order for this to be true we require the nonlinearity f to satisfy at infinity the growth condition
$$ \frac{{f(s)}} {{\left| s \right|\log ^{3/2} (1 + \left| s \right|)}} \to 0 as \left| s \right| \to \infty . $$
This is done by means of a fixed point argument and a careful analysis of the control of linearized heat equations relying on global Carleman estimates. The problem of controlling the blow up in this sense remains open for nonlinearities growing at infinity like f(s) ~ |s| log p (1 + |p|) with 3/2 ≤ p ≤ 2.


Heat Equation Global Existence Adjoint System Exact Controllability Approximate Controllability 
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© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • E. Fernández-Cara
    • 1
  • E. Zuazua
    • 2
  1. 1.Departamento de Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevillaSpain
  2. 2.Departamento de Matemática AplicadaUniversidad ComplutenseMadridSpain

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