Semilinear Evolution Equations Research Articles

Global well-posedness of some high-order focusing semilinear evolution equations with exponential nonlinearity

Abstract In even space dimensions, the initial value problems for some high-order focusing semilinear evolution equations with exponential nonlinearities are considered. Using the potential well method, global and nonglobal well-posedness in the energy space are obtained.

A Differential Sequence Generated By A Second-Order Recursion Operator

Ponte Academic JournalOct 2017, Volume 73, Issue 10 A Differential Sequence Generated By A Second-Order Recursion OperatorAuthor(s): Adhir MaharajJ. Ponte - Oct 2017 - Volume 73 - Issue 10 doi: 10.21506/j.ponte.2017.10.3 Abstract:The motivation of our investigation is the paper by Petersson {\it et al} \cite{Petersson a} where recursion operators were obtained for a class of semilinear and quasilinear evolution equations. We consider the class of semilinear evolution equations as the basis of our investigation. By setting $a=0, b=0, c=0$ and $u_t=0$ we obtain an ordinary differential equation. We analyse the resulting differential sequence in terms of it's symmetry properties. We also obtain the complete symmetry group for each element of the sequence. Download full text:Check if you have access through your login credentials or your institution Username Password

Elliptic-like regularization of semilinear evolution equations and applications to some hyperbolic problems

Consider in a Hilbert space H the Cauchy problem (P0): u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, where A:D(A)⊂H→H is the generator of a C0-semigroup of contractions and B:H→H is Lipschitzian on bounded sets and monotone. Following the method of artificial viscosity introduced by J.L. Lions, we associate with (P0) the approximate problem (Pε): −εu″(t)+u′(t)+Au(t)+Bu(t)=f(t), 0≤t≤T; u(0)=u0, u(T)=uT, where ε is a positive small parameter. We establish an asymptotic expansion of the solution uε of (Pε), showing that uε corrected by a boundary layer function approximates the solution of (P0) with respect to the sup norm of C([0,T];H). The same asymptotic expansion still holds if B is not necessarily monotone but is Lipschitzian on H. This paper is a significant extension of a previous one by M. Ahsan and G. Moroşanu [2] so that the framework created here allows the treatment of hyperbolic problems (besides parabolic ones). Specifically, our main result is illustrated with the semilinear telegraph system (thus extending a result by N.C. Apreutesei and B. Djafari Rouhani [3]) and the semilinear wave equation.

Square-mean $$\mu $$ μ -pseudo almost periodic and automophic solutions for a class of semilinear integro-differential stochastic evolution equations

In this work, we use the concept of \(\mu \)-square-mean pseudo almost periodic and automorphic processes introduced by Diop et al. (Afr Mat 26(5):779–812, 2015) to discuss the existence and uniqueness of solutions for some semilinear integro-differential stochastic evolution equations. We provide an example to illustrate ours results.

Global rapid-controllability of a class of semilinear evolution equations

In this paper, we are concerned with a class of control systems governed by semilinear evolution equations. Under appropriate conditions of different types, we prove that the control system, with arbitrarily a priori fixed Lipschitz-continuous nonlinearity, is globally rapidly exact-controllable and approximate-controllable, respectively.

A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations

In this paper, we find the critical exponent for global small data solutions to the Cauchy problem in Rn, for dissipative evolution equations with power nonlinearities |u|p or |ut|p,utt+(−Δ)δut+(−Δ)σu={|u|p,|ut|p. Here σ,δ∈N∖{0}, with 2δ≤σ. We show that the critical exponent for each of the two nonlinearities is related to each of the two possible asymptotic profiles of the linear part of the equation, which are described by the diffusion equations: vt+(−Δ)σ−δv=0,wt+(−Δ)δw=0. The nonexistence of global solutions in the critical and subcritical cases is proved by using the test function method (under suitable sign assumptions on the initial data), and lifespan estimates are obtained. By assuming small initial data in Sobolev spaces, we prove the existence of global solutions in the supercritical case, up to some maximum space dimension n̄, and we derive Lq estimates for the solution, for q∈(1,∞). For σ=2δ, the result holds in any space dimension n≥1. The existence result also remains valid if σ and/or δ are fractional.

Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge–Kutta methods for Hamiltonian semilinear evolution equations

We prove that a class of A-stable symplectic Runge–Kutta time semi-discretizations (including the Gauss–Legendre methods) applied to a class of semilinear Hamiltonian partial differential equations (PDEs) that are well posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. Consequently, such time semi-discretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is O(hp)-close to the original energy, where p is the order of the method and h is the time-step size. Examples of such systems are the semilinear wave equation, and the nonlinear Schrödinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace in which the operators occurring in the evolution equation are bounded, and by coupling the number of excited modes and the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form O(exp(–c/h1/(1+q))) with c > 0 and q ⩾ 0; for the semilinear wave equation, q = 1, and for the nonlinear Schrödinger equation, q = 2. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates.

Additive domain decomposition operator splittings—convergence analyses in a dissipative framework

We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions.

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

In this paper, we derive several results related to the long-time behavior of a class of stochastic semilinear evolution equations in a separable Hilbert space H: $$d\mathfrak{u}(t) +[{\rm A}\mathfrak{u}(t)+{\rm B}(\mathfrak{u}(t),\mathfrak{u}(t))] dt = dL(t), \quad\mathfrak{u}(0)=x\in {\rm H}.$$ Here A is a positive self-adjoint operator and B is a bilinear map, and the driving noise L is basically a \({D({\rm A}^{-1/2})}\) -valued levy process satisfying several technical assumptions. By using a density transformation theorem type for levy measure, we first prove a support theorem and an irreducibility property of the Ornstein–Uhlenbeck processes associated to the nonlinear stochastic problem. Second, by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain \({\gamma \in [0,1/4]}\)B is continuous on \({D({\rm A}^\gamma)\times D({\rm A}^\gamma)}\) with values in \({D({\rm A}^ {-1/2})}\). Using a coupling argument, the exponential ergodicity is also proved under the stronger assumption that B is continuous on \({{\rm H} \times{\rm H}}\). While the latter condition is only satisfied by the nonlinearities of GOY and Sabra shell models, the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2D Navier–Stokes, Magnetohydrodynamics equations, the 3D Leray-\({{\varvec{\alpha}}}\) model, the GOY and Sabra shell models.

Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group

In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.

Schauder’s fixed-point theorem in approximate controllability problems

Abstract The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder’s fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.

Weighted pseudo almost automorphic solutions to nonautonomous semilinear evolution equations with delay and $S^{p}$-weighted pseudo almost automorphic coefficients

By some new properties of Stepanov-like weighted pseudo almost automorphic functions established by Chang, Zhang and N'Guerekata recently, we shall deal with weighted pseudo almost automorphic solutions to the nonautonomous semilinear evolution equations with a constant delay: $u'(t)=A(t)u(t)+f(t,u(t-h))$, $ t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $A(t),t\in\mathbb{R}$ generates an exponentially stable evolution family $\{U(t,s)\}$ and $f \colon\mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a $\S^{p}$-weighted pseudo almost automorphic function satisfying some suitable conditions. We obtain our main results by the Leray-Shauder Alternative theorem.

On the Xθ(⋅)-valued function space: Definition, property and applications

This paper deals with the Xθ(⋅)-valued function space. After introducing the notions of bounded topological lattice and Banach space net, we give the definitions of two different Xθ(⋅)-valued spaces: Lp(⋅)(I,Xθ(⋅)) and C−(I,Xθ(⋅)), and make some discussions on their properties, including completeness and imbedding relations. At the end of the paper, a class of abstract semilinear evolution equations with domain-varying nonlinearities are taken into account.

Attraction Property of Admissible Integral Manifolds and Applications to Fisher-Kolmogorov Model

In this paper we investigate the attraction property of an unstable manifold of admissible classes for solutions to the semi-linear evolution equation of the form $u(t) = U(t,s) u(s) + \int_s^t U(t, \xi) f(\xi, u(\xi)) \, d\xi$. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like $L_p$-spaces, the Lorentz spaces $L_{p, q}$ and many other function spaces occurring in interpolation theory. We then apply our abstract results to study Fisher-Kolmogorov model with time-dependent environmental capacity.

A generalisation of a theorem of O. Perron for semilinear evolution equations

AbstractWe generalise a well-known result of O. Perron from the 30s that connects the asymptotic behavior of a linear homogeneous differential equation with the response of the inhomogeneous associated equation to a certain class of inhomogeneities (for this reason, Perron's result is also referred to as “input-output method”, “test function method” or “admissibility”).Our extension is twofold, on the one hand, through the means of a (non)linear evolution family, we deal with the mild solution of a nonautonomous semilinear evolution equation and on the other hand, we collect a very general class of inhomogeneities, eligible for a Perron-type approach in this case.From a technical point of view, the Perron input-output scenario is achieved here by using the Green operator.

High-order splitting schemes for semilinear evolution equations

We first derive necessary and sufficient stiff order conditions, up to order four, for exponential splitting schemes applied to semilinear evolution equations. The main idea is to identify the local splitting error as a sum of quadrature errors. The order conditions of the quadrature rules then yield the stiff order conditions in an explicit fashion, similarly to that of Runge–Kutta schemes. Furthermore, the derived stiff conditions coincide with the classical non-stiff conditions. Secondly, we propose an abstract convergence analysis, where the linear part of the vector field is assumed to generate a group or a semigroup and the nonlinear part is assumed to be smooth and to satisfy a set of compatibility requirements. Concrete applications include nonlinear wave equations and diffusion-reaction processes. The convergence analysis also extends to the case where the nonlinear flows in the exponential splitting scheme are approximated by a sufficiently accurate one-step method.

Global existence and stability for functional evolution equations with state-dependent delay

In this paper we prove some results on the global existence and attractivity of mild solutions for semilinear evolution equations with state-dependent delay in Banach space.

Short-time asymptotic expansions of semilinear evolution equations

We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.

DEGENERATE DISTRIBUTED CONTROL SYSTEMS WITH FRACTIONAL TIME DERIVATIVE

The existence of a unique strong solution for the Cauchy problem to semilinear nondegenerate fractional differential equation and for the generalized Showalter–Sidorov problem to semilinear fractional differential equation with degenerate operator at the Caputo derivative in Banach spaces is proved. These results are used for search of solution existence conditions for a class of optimal control problems to a system described by the degenerate semilinear fractional evolution equation. Abstract results are applied to the research of an optimal control problem solvability for the equations system of Kelvin–Voigt fractional viscoelastic fluids.

Approximate controllability of semilinear evolution equations of fractional order with nonlocal and impulsive conditions via an approximating technique

This paper is concerned with the approximate controllability of the semilinear fractional evolution equations with nonlocal and impulsive conditions. Our main results are obtained by utilizing the technique of approximate solution and the theory of fixed point. In addition, the impulsive functions in this paper are supposed to be continuous and the nonlocal item is divided into two cases: Lipschitz continuous and only continuous, which generalizes the previous contributions. Finally two examples are worked out to illustrate our obtained results.