  The ubiquity of semilinear parabolic equations has been illustrated in their
numerous applications ranging from physics, biology, to materials and social
sciences. In this paper, we consider a practically desirable property for a
class of semilinear parabolic equations of the abstract form
$u_t=\mathcal{L}u+f[u]$ with $\mathcal{L}$ being a linear dissipative operator
and $f$ being a nonlinear operator in space, namely a time-invariant maximum
bound principle, in the sense that the time-dependent solution $u$ preserves
for all time a uniform pointwise bound in absolute value imposed by its initial
and boundary conditions. We first study an analytical framework for some
sufficient conditions on $\mathcal{L}$ and $f$ that lead to such a maximum
bound principle for the time-continuous dynamic system of infinite or finite
dimensions. Then, we utilize a suitable exponential time differencing approach
with a properly chosen generator of contraction semigroup to develop first- and
second-order accurate temporal discretization schemes, that satisfy the maximum
bound principle unconditionally in the time-discrete setting. Error estimates
of the proposed schemes are derived along with their energy stability.
Extensions to vector- and matrix-valued systems are also discussed. We
demonstrate that the abstract framework and analysis techniques developed here
offer an effective and unified approach to study the maximum bound principle of
the abstract evolution equation that cover a wide variety of well-known models
and their numerical discretization schemes. Some numerical experiments are also
carried out to verify the theoretical results.
