Evolution Equations Of Parabolic Type Research Articles

Symmetric-conjugate splitting methods for evolution equations of parabolic type

Symmetric-conjugate splitting methods for evolution equations of parabolic type

Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control

In a separable Hilbert space X, we study the controlled evolution equation u′(t)+Au(t)+p(t)Bu(t)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u'(t)+Au(t)+p(t)Bu(t)=0, \\end{aligned}$$\\end{document}where Age -sigma I (sigma ge 0) is a self-adjoint linear operator, B is a bounded linear operator on X, and pin L^2_{loc}(0,+infty ) is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the jth eigensolution for any jge 1. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed.

Exact controllability to eigensolutions of the bilinear heat equation on compact networks

<p style='text-indent:20px;'>Partial differential equations on networks have been widely investigated in the last decades in view of their application to quantum mechanics (Schrödinger type equations) or to the analysis of flexible structures (wave type equations). Nevertheless, very few results are available for diffusive models despite an increasing demand arising from life sciences such as neurobiology. This paper analyzes the controllability properties of the heat equation on a compact network under the action of a single input bilinear control.</p><p style='text-indent:20px;'>By adapting a recent method due to [F. Alabau-Boussouira, P. Cannarsa, C. Urbani, <i>Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control</i>, arXiv: 1811.08806], an exact controllability result to the eigensolutions of the uncontrolled problem is obtained in this work. A crucial step has been the construction of a suitable biorthogonal family under a non-uniform gap condition of the eigenvalues of the Laplacian on a graph. Application to star graphs and tadpole graphs are included.</p>

Periodic and almost periodic evolution flows and their stability on non-compact Einstein manifolds and applications

Considering the evolution equations of parabolic type on a non-compact Einstein manifold $(\mathcal M,g)$ with negative Ricci curvature tensor, we establish results on the existence, uniqueness of time-periodic (on the time half-axis) and almost periodic

Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families

<p style='text-indent:20px;'>In this paper, we investigate the non-autonomous stochastic evolution equations of parabolic type with nonlinear noise and nonlocal initial conditions in Hilbert spaces, where the operators in linear part depend on time <inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> and generate an noncompact evolution family. New existence result of mild solutions is established under some weaker growth and measure of noncompactness conditions on nonlinear functions and nonlocal term. The discussions are based on Sadovskii's fixed-point theorem as well as the theory of evolution family. At last, as a sample of application, the obtained abstract result is applied to a class of non-autonomous stochastic partial differential equations of parabolic type with nonlocal initial conditions. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions

In this paper, we study the non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions in Hilbert spaces, where the operators in linear part (possibly unbounded) depend on time \begin{document}$ t $\end{document} and generate an evolution family. New existence result of mild solutions is established under more weaker conditions by introducing a new Green's function. The discussions are based on Schauder's fixed-point theorem as well as the theory of evolution family. At last, an example is also given to illustrate the feasibility of our theoretical results. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations

<p style='text-indent:20px;'>In this paper, we are considered with approximate controllability for a class of non-autonomous stochastic evolution equations of parabolic type with discrete nonlocal initial conditions. Some new results about existence of mild solutions as well as approximate controllability are established under more natural conditions on nonlinear functions and control operator by introducing a new Green function and using the theory of evolution family, Schauder fixed point theorem and the resolvent operator condition. At last, as a sample of application, these results are applied to a class of non-autonomous stochastic partial differential equation of parabolic type with discrete nonlocal initial conditions. The results obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations

Abstract The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition or Babuška–Brezzi condition for the DG bilinear form. Then, a nearly best approximation property and a nearly symmetric error estimate are obtained as corollaries. Moreover, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new for the DG time-stepping method; it differs from previous works by which the method is formulated as the one-step method. We apply our abstract results to the finite element approximation of a second-order parabolic equation with space-time variable coefficient functions in a polyhedral domain, and derive the optimal order error estimates in several norms.

Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families

This paper investigates the Cauchy problem to a class of stochastic non-autonomous evolution equations of parabolic type governed by noncompact evolution families in Hilbert spaces. Combining the theory of evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we established some new existence results of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the strong restriction on the constants in the condition of noncompactness measure is completely deleted, and also the condition of uniformly continuity of the nonlinearity is not required. At last, as samples of applications, we consider the Cauchy problem to a class of stochastic non-autonomous partial differential equation of parabolic type.

A PARABOLIC SHAPE OPTIMIZATION PROBLEM

In this article, we discuss some approximation methods for optimal design problems governed by evolution equations of parabolic type. The two investigated approaches are of fixed domain type. We also formulate supplementary questions and problems, related to this subject

Non-autonomous Evolution Equations of Parabolic Type with Non-instantaneous Impulses

In this paper, we study the Cauchy problem to a class of non-autonomous evolution equations of parabolic type with non-instantaneous impulses in Banach spaces, where the operators in linear part (possibly unbounded) depend on time t and generate an evolution family. New existence result of piecewise continuous mild solutions is established under more weaker conditions. At last, as a sample of application, the abstract result is applied to a class of non-autonomous partial differential equation of parabolic type with non-instantaneous impulses. The result obtained in this paper is a supplement to the existing literature and essentially extends some existing results in this area.

Asymptotic convergence of solutions for Laplace reaction–diffusion equations

We study the initial–boundary value problem for a Laplace reaction–diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show asymptotic convergence of bounded global solutions if they exist under the assumption that the reaction function is analytic in neighborhoods of their ω-limit sets. Reduction of degenerate evolution equation to multivalued evolution equation enables us to use the theory of the infinite-dimensional Łojasiewicz–Simon gradient inequality.

Global existence for Laplace reaction-diffusion equations

We study the initial-boundary value problem for a Laplace reaction-diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show global existence under suitable assumptions on the reaction function. We also show that the problem generates a dynamical system in a suitably set universal space and that this dynamical system possesses a Lyapunov function.

Existence results for linear evolution equations of parabolic type

We study a stochastic parabolic evolution equation of the form $ dX+AXdt = F(t)dt+G(t)dW(t)$ in Banach spaces. Existence of mild and strict solutions and their space-time regularity are shown in both the deterministic and stochastic cases. Abstract results are applied to a nonlinear stochastic heat equation.

On a Finite Difference Scheme for an Inverse Integro-Differential Problem Using Semigroup Theory: A Functional Analytic Approach

ABSTRACTIn this article, the problem of reconstructing an unknown memory kernel from an integral overdetermination in an abstract linear (of convolution type) evolution equation of parabolic type is considered. After illustrating some results of the existence and uniqueness of a solution for the differential problem, we study its approximation by Rothe's method. We prove a result of stability and another concerning the order of approximation of the solution in dependence of its regularity. The main tool is a maximal regularity result for solutions to abstract parabolic finite difference schemes. Two model problems to which the results are applicable are illustrated.

Direct meshless kernel techniques for time-dependent equations

We provide a class of positive definite kernels that allow to solve certain evolution equations of parabolic type for scattered initial data by kernel-based interpolation or approximation, avoiding time intergation completely. Some numerical illustrations are given.

Nonlinear Evolution Problems

In this workshop geometric evolution equations of parabolic type, nonlinear hyperbolic equations, and dispersive equations and their interrelations were the subject of 21 talks and several shorter special presentations.

Strong and weak orders in averaging for SPDEs

We show an averaging result for a system of stochastic evolution equations of parabolic type with slow and fast time scales. We derive explicit bounds for the approximation error with respect to the small parameter defining the fast time scale. We prove that the slow component of the solution of the system converges towards the solution of the averaged equation with an order of convergence 1/2 in a strong sense–approximation of trajectories–and 1 in a weak sense–approximation of laws. These orders turn out to be the same as for the SDE case.

Approximating travelling waves by equilibria of non-local equations

We consider an evolution equation of parabolic type in R having a travelling wave solution. We study the effects on the dynamics of an appropriate change of variables which transforms the equation into a non-local evolution one having a travelling wave solution with zero speed of propagation with exactly the same profile as the original one. This procedure allows us to compute simultaneously the travelling wave profile and its propagation speed avoiding moving meshes, as we illustrate with several numerical examples. We analyze the relation of the new equation with the original one in the entire real line. We also analyze the behavior of the non-local problem in a bounded interval with appropriate boundary conditions. We show that it has a unique stationary solution which approaches the traveling wave as the interval gets larger and larger and that is asymptotically stable for large enough intervals.

Optimal Control Problems for Evolution Equations of Parabolic Type with Nonlinear Perturbations

In this paper, we study the optimal control problems governed by the semilinear parabolic type equation in Hilbert spaces. Under Lipschitz continuity condition of the nonlinear term, we can obtain the optimal conditions and maximal principles for a given equation, which are described by the adjoint state corresponding to the given equation without the rigorous conditions for the nonlinear term.