Uniform convergence of operator semigroups without time regularity

When we are interested in the long-term behaviour of solutions to linear evolution equations, a large variety of techniques from the theory of $$C_0$$ C 0 -semigroups is at our disposal. However, if we consider for instance parabolic equations with unbounded coefficients on $${\mathbb {R}}^d$$ R d , the solution semigroup will not be strongly continuous, in general. For such semigroups many tools that can be used to investigate the asymptotic behaviour of $$C_0$$ C 0 -semigroups are not available anymore and, hence, much less is known about their long-time behaviour. Motivated by this observation, we prove new characterisations of the operator norm convergence of general semigroup representations—without any time regularity assumptions—by adapting the concept of the “semigroup at infinity”, recently introduced by M. Haase and the second named author. Besides its independence of time regularity, our approach also allows us to treat the discrete-time case (i.e. powers of a single operator) and even more abstract semigroup representations within the same unified setting. As an application of our results, we prove a convergence theorem for solutions to systems of parabolic equations with the aforementioned properties.

Generation results for vector-valued elliptic operators with unbounded coefficients in $$L^p$$ spaces

We consider a class of vector-valued elliptic operators with unbounded coefficients, coupled up to the first order, in the Lebesgue space $$L^p({\mathbb {R}}^d;{\mathbb {R}}^m)$$ L p ( R d ; R m ) with $$p \in (1,\infty )$$ p ∈ ( 1 , ∞ ) . Sufficient conditions to prove generation results of an analytic $$C_0$$ C 0 -semigroup $${\varvec{T}}(t)$$ T ( t ) , together with a characterization of the domain of its generator, are given. Some results related to the hypercontractivity and the ultraboundedness of the semigroup are also established.

Correctness conditions for high-order differential equations with unbounded coefficients

We give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

The solvability conditions for the second order nonlinear differential equation with unbounded coefficients in L2(R)

The article deals with the existence of a generalized solution for the second order nonlinear differential equation in an unbounded domain. Intermediate and lower coefficients of the equation depends on the required function and considered smooth. The novelty of the work is that we prove the solvability of a nonlinear singular equation with the leading coefficient not separated from zero. In contrast to the works considered earlier, the leading coefficient of the equation can tend to zero, while the intermediate coefficient tends to infinity and does not depend on the growth of the lower coefficient. The result obtained formulated in terms of the coefficients of the equation themselves; there are no conditions on any derivatives of these coefficients.