Semilinear Equations Research Articles

L2-convergence of Yosida approximation for semi-linear backward stochastic differential equation with jumps in infinite dimension

PurposeThe main motivation of this paper is to present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established.Design/methodology/approachThe authors establish a result concerning the L2-convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation.Findings The authors carry out a convergence rate of Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension.Originality/valueIn this paper, the authors present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established.

On a reaction diffusion problem with a moving impulse on boundary

Abstract We study an asymptotic problem of a semilinear partial differential equation (PDE) with Neumann boundary condition, periodic coefficients and highly oscillating drift and nonlinear terms. Our analysis focuses on the double limiting behavior of the PDE-solution perturbed by ε (viscosity parameter) and δ (scaling coefficient) both tending to zero. To do so, we state basic properties of the large deviations principle (LDP) and we express the logarithmic asymptotic of the PDE-solution. Particularly, we provide it for the case when ε converges more quickly than δ.

Existence of blowup solutions to the semilinear heat equation with double power nonlinearity

Existence of blowup solutions to the semilinear heat equation with double power nonlinearity

Continuity and pullback attractors for a semilinear heat equation on time-varying domains

We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term f(cdot ) to satisfy int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{L^{2}},ds<infty for all tin mathbb{R}, we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in H^{1} topology and the usual (L^{2},L^{2}) pullback mathscr{D}_{lambda}-attractor indeed can attract in the H^{1}-norm, provided that int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{H^{-1}(mathcal{O}_{s})},ds< infty and fin L^{2}_{mathrm{loc}}(mathbb{R},L^{2}(mathcal{O}_{s})).

On regularity and asymptotic stability for semilinear nonlocal pseudo-parabolic equations

We deal with a class of nonlocal pseudo-parabolic equations involving strong nonlinearities. The questions on existence, regularity and stability of solutions are addressed by using local estimates, fixed point arguments, and the relation between the Hilbert scales and fractional Sobolev spaces.

Boundary controllability of Riemann–Liouville fractional semilinear equations

We study the boundary regional controllability of a class of Riemann–Liouville fractional semilinear sub-diffusion systems with boundary Neumann conditions. The result is obtained by using semi-group theory, the fractional Hilbert uniqueness method, and Schauder’s fixed point theorem. Conditions on the order of the derivative, internal region, and on the nonlinear part are obtained. Furthermore, we present appropriate sufficient conditions for the considered fractional system to be regionally controllable and, therefore, boundary regionally controllable. An example of a population density system with diffusion is given to illustrate the obtained theoretical results. Numerical simulations show that the proposed method provides satisfying results regarding two cases of the control operator.

Application of branching processes to solve boundary value problems for the semi-linear Helmholtz equation

This work devoted to construct a probabilistic model for solving boundary value problems for the semi-linear Helmholtz equation. Probabilistic representations of problem solving in the form of the mathematical expectation of some random variable are obtained. In accordance with probabilistic representation, branching processes are constructed and modeling formulas are specified for each of the branching processes. It has been proven that the constructed branching processes degenerate with probability one and the average number of particles of each n-th generation is less than or equal to one. For the considered problems, unbiased estimates were constructed on the trajectories of the corresponding random processes. Unbiased estimates which constructed on the trajectories of a branching process with a limited average number of branches, easily modeled in computer and has limited variance. Based on the proposed estimates, computational experiments were carried out. The results of the computational experiment show that using the constructed algorithm it is possible to solve semi-linear problems that occur in practice.

Existence of positive solutions for generalized quasilinear Schrödinger equations with Sobolev critical growth

In this paper, we are devoted to establishing that the existence of positive solutions for a class of generalized quasilinear elliptic equations in R N with Sobolev critical growth, which have appeared from plasma physics, as well as high-power ultrashort laser in matter. To begin, by changing the variable, quasilinear equations are transformed into semilinear equations. The positive solutions to semilinear equations are then presented using the Mountain Pass Theorem for locally Lipschitz functionals and the Concentration-Compactness Principle. Finally, an inverse translation reveals the presence of positive solutions to the original quasilinear equations.

Breakdown of $ C^{\infty} $-smoothing effects of solutions to the semilinear heat equation in the whole space

Breakdown of $ C^{\infty} $-smoothing effects of solutions to the semilinear heat equation in the whole space

Gradient blowup profile for the semilinear heat equation

Gradient blowup profile for the semilinear heat equation

Global boundary null-controllability of one-dimensional semilinear heat equations

Global boundary null-controllability of one-dimensional semilinear heat equations

Singular solutions of a fractional Dirichlet problem in a punctured domain

Let D be a bounded regular domain in Rn (n ? 3) containing 0, 0 &lt; ? &lt; 2, and ? &lt; 1. We take up in this article the existence and asymptotic behavior of a positive continuous solution for the following semi-linear fractional differential equation (??|D)?2 u = a(x)u?(x) in D\{0}, with the boundary Dirichlet conditions lim |x|?0 |x|n??u(x) = 0 and lim x??D ?(x)2??u(x) = 0, where (??|D)?/2 is the fractional Laplace associated to the subordinate killed Brownian motion process in D and ?(x) = dist ( x, ?D) denotes the Euclidean distance between x and ?D. The function a is a positive continuous function in D\{0}, which may be singular at x = 0 and/or at the boundary ?D satisfying some appropriate assumption related to Karamata class. More precisely, we shall prove the existence and global asymptotic behavior of a positive continuous solution on ?D\{0}. We will use some potential theory arguments and Karamata regular variation theory tools.

Initial boundary value problem for semilinear hyperbolic equations with weakly damping term on conical manifolds with critical and supcritical initial data

Initial boundary value problem for semilinear hyperbolic equations with weakly damping term on conical manifolds with critical and supcritical initial data

Boundedness of semilinear Duffing equations with Liouvillean frequency

Boundedness of semilinear Duffing equations with Liouvillean frequency

On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart

On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart

Time two-grid technique combined with temporal second order difference method for semilinear fractional diffusion-wave equations

Time two-grid technique combined with temporal second order difference method for semilinear fractional diffusion-wave equations

Existence and controllability results of semilinear Sobolev type difference equation

Existence and controllability results of semilinear Sobolev type difference equation

Dynamics of locally monotone stochastic evolution equations

This paper is concerned with the dynamics of the locally monotone stochastic evolution equations in Bochner spaces. We first prove the well-posedness of the abstract stochastic evolution equations with locally Lipschitz nonlinear diffusion terms. Then we establish a mean random dynamical system and prove the existence and uniqueness of weak pullback mean random attractors generated by the mean random dynamical system. Furthermore, we prove the existence of invariant measures for the locally monotone model. As applications, the related dynamics for several stochastic models such as stochastic semilinear equations, stochastic Burgers equation and stochastic 2D Navier-Stokes equations are established.

Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O(λN−γ) in space and O(τγ) in time, where γ∈(0,1] from the assumption ∥Aγ−12Q12∥L2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings.

Finite-Approximate Controllability of ν-Caputo Fractional Systems

This paper introduces a methodology for examining finite-approximate controllability in Hilbert spaces for linear/semilinear ν-Caputo fractional evolution equations. A novel criterion for achieving finite-approximate controllability in linear ν-Caputo fractional evolution equations is established, utilizing resolvent-like operators. Additionally, we identify a control strategy that not only satisfies the approximative controllability property but also ensures exact finite-dimensional controllability. Leveraging the approximative controllability of the corresponding linear ν-Caputo fractional evolution system, we establish sufficient conditions for achieving finite-approximative controllability in the semilinear ν-Caputo fractional evolution equation. These findings extend and build upon recent advancements in this field. The paper also explores applications to ν-Caputo fractional heat equations.