Semilinear Equations Research Articles

Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form

In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u=be−ax2, a<0, a,b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u=be−ax22, respectively, for the third-order PDE, u=AeiΦ, where the amplitude b and the phase function a are complex-valued functions, A>0, and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.

Controllability of semilinear noninstantaneous impulsive neutral stochastic differential equations via Atangana-Baleanu Caputo fractional derivative

This study mainly concerns the controllability of semilinear noninstantaneous impulsive neutral stochastic differential equations via the Atangana-Baleanu (AB) Caputo fractional derivative (FD). The essential findings are created using methods and concepts from semigroup theory, stochastic theory, fractional calculus, K-set contraction, and measure of noncompactness. Finally, an example is provided to demonstrate the applications of the key findings.

Exponential dichotomy and invariant manifolds of semi-linear differential equations on the line

In this paper we investigate the homogeneous linear differential equation vi(t) = A(t)v(t) and the semi-linear differential equation vi(t) = A(t)v(t) + g(t, v(t)) in Banach space X, in which A : R → L(X) is a strongly continuous function, g : R × X → X is continuous and satisfies ϕ-Lipschitz condition. The first we characterize the exponential dichotomy of the associated evolution family with the homogeneous linear differential equation by space pair (E, E∞), this is a Perron type result. Applying the achieved results, we establish the robustness of exponential dichotomy. The next we show the existence of stable and unstable manifolds for the semi-linear differential equation and prove that each a fiber of these manifolds is differentiable submanifold of class C1. Mathematics Subject Classification (2010): 34C45, 34D09, 34D10. Received 14 June 2021; Accepted 09 September 2022

Topics in elliptic problems: from semilinear equations to shape optimization

In this paper, which corresponds to an updated version of the author's Habilitation lecture in Mathematics, we do an overview of several topics in elliptic problems. We review some old and new results regarding the Lane-Emden equation, both under Dirichlet and Neumann boundary conditions, then focus on sign-changing solutions for Lane-Emden systems. We also survey some results regarding fully nontrivial solutions to gradient elliptic systems with mixed cooperative and competitive interactions. We conclude by exhibiting results on optimal partition problems, with cost functions either related to Dirichlet eigenvalues or to the Yamabe equation. Several open problems are referred along the text.

Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.

Lower and upper bounds for the explosion times of a system of semilinear SPDEs

In this paper, we obtain lower and upper bounds for the blow-up times for a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of the explosive times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We provide lower and upper bounds for the probability of finite-time blow-up solution as well. The above obtained results are also extended for semilinear SPDEs forced by two-dimensional Brownian motions.

Recovering coefficients in a system of semilinear Helmholtz equations from internal data

We study an inverse problem for a coupled system of semilinear Helmholtz equations where we are interested in reconstructing multiple coefficients in the system from internal data measured in applications such as thermoacoustic imaging. The system serves as a simplified model of the second harmonic generation process in a heterogeneous medium. We derive results on the uniqueness and stability of the inverse problem in the case of small boundary data based on the technique of first- and higher-order linearization. Numerical simulations are provided to illustrate the quality of reconstructions that can be expected from noisy data.

Unbounded Sturm attractors for quasilinear parabolic equations

AbstractWe analyse the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e. blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincaré projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called equilibria at infinity and their corresponding heteroclinics. Moreover, the diffusion and reaction compete for the dimensionality of the induced dynamics at infinity. If the equilibria are hyperbolic, we explicitly prove the occurrence of heteroclinics between bounded equilibria and/or equilibria at infinity. These unbounded global attractors describe the space of admissible initial data at event horizons of certain black holes.

Exact convergence rate of solutions for a semilinear heat equation with a critical and a supercritical exponent revisited

Exact convergence rate of solutions for a semilinear heat equation with a critical and a supercritical exponent revisited

The One Step Malliavin scheme: new discretization of BSDEs implemented with deep learning regressions

Abstract A novel discretization is presented for decoupled forward–backward stochastic differential equations (FBSDE) with differentiable coefficients, simultaneously solving the BSDE and its Malliavin sensitivity problem. The control process is estimated by the corresponding linear BSDE driving the trajectories of the Malliavin derivatives of the solution pair, which implies the need to provide accurate $\varGamma $ estimates. The approximation is based on a merged formulation given by the Feynman–Kac formulae and the Malliavin chain rule. The continuous time dynamics is discretized with a theta-scheme. In order to allow for an efficient numerical solution of the arising semidiscrete conditional expectations in possibly high dimensions, it is fundamental that the chosen approach admits to differentiable estimates. Two fully-implementable schemes are considered: the BCOS method as a reference in the one-dimensional framework and neural network Monte Carlo regressions in case of high-dimensional problems, similarly to the recently emerging class of Deep BSDE methods (Han et al. (2018 Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510); Huré et al. (2020 Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comp., 89, 1547–1579)). An error analysis is carried out to show $\mathbb{L}^2$ convergence of order $1/2$, under standard Lipschitz assumptions and additive noise in the forward diffusion. Numerical experiments are provided for a range of different semilinear equations up to $50$ dimensions, demonstrating that the proposed scheme yields a significant improvement in the control estimations.

On asymptotic properties of solutions to σ-evolution equations with general double damping

In this paper, we would like to consider the Cauchy problem for semi-linear σ-evolution equations with double structural damping for any σ⩾1. The main purposes of the present work are to not only study the asymptotic profiles of solutions to the corresponding linear equations but also describe large-time behaviors of globally obtained solutions to the semi-linear equations. We want to emphasize that the new contribution is to find out the sharp interplay of “parabolic like models” corresponding to σ1∈[0,σ/2) and “σ-evolution like models” corresponding to σ2∈(σ/2,σ], which together appear in an equation. In this connection, we understand clearly how each damping term influences the asymptotic properties of solutions.

An asymptotic expansion of the solution of a semi-linear partial differential equation implied by a nonlinear Feynman–Kac formula

This paper introduces an asymptotic expansion for the smooth solution of a semi-linear partial differential equation. Our scheme is based on Itô’s formula, Taylor’s expansion, nonlinear Feynman–Kac formula and some algebras.

Existence, uniqueness and regularity for a semilinear stochastic subdiffusion with integrated multiplicative noise

We investigate a semilinear stochastic time-space fractional subdiffusion equation driven by fractionally integrated multiplicative noise. The equation involves the ψ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\psi $$\\end{document}-Caputo derivative of order α∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1)$$\\end{document} and the spectral fractional Laplacian of order β∈(12,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in (\\frac{1}{2},1]$$\\end{document}. The existence and uniqueness of the mild solution are proved in a suitable Banach space by using the Banach contraction mapping principle. The spatial and temporal regularities of the mild solution are established in terms of the smoothing properties of the solution operators.

Solution theory to semilinear stochastic equations of Schrödinger type on curved spaces I: operators with uniformly bounded coefficients

We study the Cauchy problem for Schrödinger type stochastic semilinear partial differential equations with uniformly bounded variable coefficients, depending on the space variables. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.

Existence and blow-up of solutions for a class of semilinear pseudo-parabolic equations with cone degenerate viscoelastic term

Existence and blow-up of solutions for a class of semilinear pseudo-parabolic equations with cone degenerate viscoelastic term

A priori estimates and Liouville type theorems for semilinear equations and systems with fractional Laplacian

In this paper, we firstly obtain a priori estimates of nonnegative solutions to fractional Laplacian equations and systems with critical Sobolev exponent. Secondly, by deriving the universal estimates of solutions, we establish the connections between Liouville type theorems and local properties of nonnegative solutions to fractional Laplacian equations and fractional Lane-Emden systems with subcritical Sobolev exponent. Our main results improve the work of Chen et al. (2016) [8] and Polácik et al. (2007) [26]. Specially, for fractional Lane-Emden systems, our results seem to be the first results on universal estimates to cover the full subcritical range. Furthermore, the novelty of our method lies in a completely new blow-up analysis coupled with a truncating technique and the construction of barrier function which is sufficiently different from the blow-up method used in [8,26].

On a controllability problem for a feedback control system governed by a semilinear differential equation and a sweeping process

We consider a controllability problem for a feedback control system governed by a semilinear differential equation and a sweeping process in Hilbert spaces. We define the operator whose fixed points are generating solutions of the problem. By applying the methods of the fixed point theory for condensing maps we study the properties of this operator, in particular, we prove that under certain conditions it is condensing w.r.t. an appropriate measure of noncompactness. This allows to present the general controllability principle.

Determining a nonlinear hyperbolic system with unknown sources and nonlinearity

AbstractThis paper is devoted to some inverse boundary problems associated with a time‐dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. It is shown in several generic scenarios that one can uniquely determine the nonlinearity and/or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and an approximation property with higher order linearization for the semilinear hyperbolic equation.

The move from Fujita type exponent to a shift of it for a class of semilinear evolution equations with time-dependent damping

The move from Fujita type exponent to a shift of it for a class of semilinear evolution equations with time-dependent damping

Necessary and sufficient condition for existence for a case of eigenvalues of multiplicity two

We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/09/abstr.html