On initial inverse problem for nonlinear couple heat with Kirchhoff type

The main objective of the paper is to study the final model for the Kirchhoff-type parabolic system. Such type problems have many applications in physical and biological phenomena. Under some smoothness of the final Cauchy data, we prove that the problem has a unique mild solution. The main tool is Banach’s fixed point theorem. We also consider the non-well-posed problem in the Hadamard sense. Finally, we apply truncation method to regularize our problem. The paper is motivated by the work of Tuan, Nam, and Nhat [Comput. Math. Appl. 77(1):15–33, 2019].

On a class of stochastic partial differential equations with multiple invariant measures

In this work we investigate the long-time behavior for Markov processes obtained as the unique mild solution to stochastic partial differential equations in a Hilbert space. We analyze the existence and characterization of invariant measures as well as convergence of transition probabilities. While in the existing literature typically uniqueness of invariant measures is studied, we focus on the case where the uniqueness of invariant measures fails to hold. Namely, introducing a generalized dissipativity condition combined with a decomposition of the Hilbert space, we prove the existence of multiple limiting distributions in dependence of the initial state of the process and study the convergence of transition probabilities in the Wasserstein 2-distance. Finally, we apply our results to Lévy driven Ornstein–Uhlenbeck processes, the Heath–Jarrow–Morton–Musiela equation as well as to stochastic partial differential equations with delay.

An inverse source problem for the stochastic wave equation

<p style='text-indent:20px;'>This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.</p>

Construction of a unique mild solution of one-dimensional Keller-Segel systems with uniformly elliptic operators having variable coefficients

A one-dimensional Keller-Segel system which is defined through uniformly elliptic operators having variable coefficients is considered. In the main theorems, the local existence and uniqueness of the mild solution of the system are proved. The main method to construct the mild solution is an argument of successive approximations by means of strongly continuous semi-groups.

Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations

We investigate mild solutions of the fractional order nonhomogeneous Cauchy problemDtαu(t)=Au(t)+f(t), t>0, where0<α<1.WhenAis the generator of aC0-semigroup(T(t))t≥0on a Banach spaceX, we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary conditionu(0)=u(1)admits a unique mild solution for eachf∈C([0,1];X)if and only if the operatorI-Sα(1)is invertible. Here, we use the representationSα(t)x=∫0∞‍Φα(s)T(stα)x ds, t>0in whichΦαis a Wright type function. For the first order case, that is,α=1, the corresponding result was proved by Prüss in 1984. In caseXis a Banach lattice and the semigroup(T(t))t≥0is positive, we obtain existence of solutions of the semilinear problemDtαu(t)=Au(t)+f(t,u(t)),t>0,0<α<1.

Existence results for semilinear neutral functional differential equations involving evolution operators in Fréchet spaces

The existence of a unique mild solution on a semiinfinite interval for a first order semilinear neutral functional differential equations involving evolution operators in Fréchet spaces is investigating using a nonlinear alternative of Leray–Schauder type for contractive maps, combined with semigroup theory.

SOLUTION PROPERTIES OF SOME CLASSES OF OPERATOR EQUATIONS IN HILBERT SPACES

We study properties of solutions of the operator equation [Formula: see text], [Formula: see text], where [Formula: see text] a closable linear operator on a Hilbert space [Formula: see text], such that there exists a self-adjoint operator [Formula: see text] on [Formula: see text], with the resolution of identity E(·), which commutes with [Formula: see text]. We are interested in the question of regular admissibility of the subspace [Formula: see text], i.e. when for every [Formula: see text] there exists a unique (mild) solution u in [Formula: see text] of this equation. We introduce the notion of equation spectrum Σ associated with Eq. (*), and prove that if Λ ⊂ ℝ is a compact subset such that Λ ⋂ Σ = ∅, then [Formula: see text] is regularly admissible. If Λ ⊂ ℝ is an arbitrary Borel subset such that Λ ⋂ Σ = ∅, then, in general, [Formula: see text] needs not be regularly admissible, but we derive necessary and sufficient conditions, in terms of some inequalities, for the regular admissibility of [Formula: see text]. Our results are generalizations of the well-known spectral mapping theorem of Gearhart-Herbst-Howland-Prüss [4], [5], [6], [9], as well as of the recent results of Cioranescu-Lizama [3], Schüler [10] and Vu [11], [12].

EXISTENCE, UNIQUENESS AND REGULARITY w.r.t. THE INITIAL CONDITION OF MILD SOLUTIONS OF SPDEs DRIVEN BY POISSON NOISE

In this paper we investigate stochastic partial differential equations in a separable Hilbert space driven by a compensated Poisson random measure. Our interest is directed towards the existence and uniqueness of mild solutions and their regularity w.r.t. the inital condition. We show the existence of a unique mild solution and prove the Gâteaux differentiability of the mild solution w.r.t. the initial condition. As a consequence, we obtain a gradient estimate for the Gâteaux derivative of the mild solution and for the resolvent associated to the mild solution.