Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

Abstract

Abstract. We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over R d and in L p -spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous secondorder elliptic operator with unbounded coefficients defined in I × R d , (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.

Existence results for Isaacs equations with local conditions and related semilinear Cauchy problems

“…During the peer revision process we have also discovered that the same set of conditions (Assumption 1) is largely covered by the recent result of Addona et al [2,Theorem 3.6] and proved by exploiting the fixed point approach. But, they have used slightly different technique which operates on the solution defined on the small time interval (T − δ, T ] and they have not proved global uniform convergence to the fixed point.…”

Equivalence of Sobolev norms with respect to weighted Gaussian measures

“…In this section we provide other important results which involve vector-valued functions, and which have been already proved in the scalar case, both in finite dimension for more general operators (see [2,5,14]) and when X is a Hilbert space (see [4,Section 4]). We begin by proving a Logarithmic Sobolev Inequality for functions F ∈ C 1 b (X; V ) which generalize the scalar case.…”

“…In the former we provide the vector-valued Poincaré inequalities (1.1) and (1.2), in the latter we give the equivalence of the norms • k,p and • p,D k H . We remark that the proof of (1.1) relies on duality arguments: when p ≥ 2 we use the fact that, for any F ∈ W 1,p (X, ν), the function F * := |F | p−2 F ∈ W 1,p ′ (X, ν) and an explicit formula for D H F * is available (see Lemma 2.16), while when p ∈ [1,2) we employ the duality between L p (X, ν; H ⊗ V ) and L p ′ (X, ν; H ⊗ V ). Regarding the equivalence of • k,p and • p,D k H , we adapt to our situation the idea of [19,Corollary 3.2], which we explain when k = 2.…”