A Class of Globally Controllable Semilinear Heat Equations with Superlinear Terms1

It is well known nowadays that a rather general semilinear parabolic equation, governed in a bounded domain by boundary or locally distributed controls, can be made globally approximately controllable in L p-type spaces, provided the nonlinearity is globally Lipschitz. However, the question whether this property can possibly take place in the superlinear case in several space dimensions seems to be open. The goal of this article is to describe a special class of semilinear heat equations with superlinear terms for which this property indeed holds. Our technique combines the asymptotic method of A. Y. Khapalov 1995, J. Math. Anal. Appl., 194, 858]882. with certain a priori estimates for the corresponding truncated spectral problem.