Appl Math Optim 46:97–105 (2002)
2002 Springer-Verlag New York Inc.
Null Controllability for the Dissipative Semilinear Heat Equation
and Daniel Tataru
Facultatea de Mathematica,
Department of Mathematics, University of California,
Berkeley, CA 94720, USA
Abstract. We consider the exact null controllability problem for the semi-
linear heat equation with dissipative nonlinearity in a bounded domain of R
main result of the article asserts that if the nonlinearity is even mildly superlinear,
then global null controllability in an arbitrarily short time fails; instead we pro-
vide sharp estimates for the controllability time in terms of the size of the initial
Key Words. Null controllability, Heat equation, Nonlinear parabolic equation.
AMS Classiﬁcation. 93B05, 35K05, 35K60.
Let be a bounded domain in R
with smooth boundary ∂. We consider the internal
exact null controllability problem for a semilinear heat equation with Dirichlet boundary
− y + f (x, y) = m(x)u in × (0, T ),
y = 0in∂ × (0, T ),
y(0) = y
Here T > 0 and m is the characteristic function of a nonempty open subset ω ⊂ . The
function f is locally lipschitz and satisﬁes f (x, 0) = 0.