Semilinear Equations Research Articles

Finite-time blowup and existence of global solutions for a logarithmic semilinear hyperbolic equation

In this paper we consider the semilinear wave equation with the product of logarithmic and polynomial nonlinearities and establish the global existence and finite-time blowup of solutions by using the potential well method.

Existence study of semilinear fractional differential equations and inclusions for multi–term problem under Riemann–Liouville operators

In the present paper, we study the existence and uniqueness of the solution for multi–term fractional boundary value problem under Riemann–Liouville fractional operators by using Banach's fixed point theorem. Afterward, we investigate some existence results for a semilinear fractional differential inclusions multi–term problem by using some notions and properties on set‐valued maps together with classical fixed point theorems due to Leray–Schauder. The cases when the set‐valued function has convex as well as nonconvex values are considered. Finally, some examples are given to illustrate our main results.

Liouville Type Theorems Involving the Fractional Laplacian on the Upper Half Euclidean Space

In this paper, we mainly establish Liouville-type theorems for the elliptic semi-linear equations involving the fractional Laplacian on the upper half of Euclidean space. We employ a direct approach by studying an equivalent integral equation instead of using the conventional extension method. Applying the method of moving planes in integral forms, we prove the non-existence of positive solutions under very weak conditions. We also extend the results to a more general equation.

Weak solutions of impulsive pseudoparabolic equations with an infinitesimal transition layer

We study the multi-dimensional initial–boundary value problem for the quasilinear pseudoparabolic equation with a regular nonlinear minor term, which models a non-instantaneous impulsive impact. The minor term depends on a small parameter ɛ>0 and, as ɛ→0, converges weakly⋆ to the expression incorporating the Dirac delta function, which, in turn, models an instantaneous impulsive impact. We prove that the infinitesimal transition layer, associated with the Dirac delta function, is formed as ɛ→0, and that the family of weak solutions of the original problem converges to the weak solution of a two-scale microscopic–macroscopic model. This model consists of two equations and the set of initial, boundary, and matching conditions, so that the ‘outer’ macroscopic solution beyond the transition layer is governed by the quasilinear homogeneous pseudoparabolic equation at the macroscopic (‘slow’) timescale, while the transition layer solution is defined at the microscopic level and obeys the semilinear pseudoparabolic equation at the microscopic (‘fast’) timescale. The latter equation inherits the full information about the profile of the original non-instantaneous impulsive impact.

Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel

We consider the semilinear diffusion equation ∂ t u = Au + |u| α u in the half-space R N + := R N −1 × (0, +∞), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ′ , 0) = 0 for x ′ ∈ R N −1. We prove that if the symbol of the operator A is of order a|ξ| β near the origin ξ = 0, for some β ∈ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 β/(N + 1). Notice that in the case of the half-space, the exponent β/(N + 1) is smaller than the so-called Fujita exponent β/N in R N. As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of R N , which are odd in the x N direction (and thus sign changing).

Nonlinear modulational dynamics of spectrally stable Lugiato–Lefever periodic waves

We consider the nonlinear stability of spectrally stable periodic waves in the Lugiato– Lefever equation (LLE), a damped nonlinear Schrödinger equation with forcing that arises in nonlinear optics. So far, nonlinear stability of such solutions has only been established against co-periodic perturbations by exploiting the existence of a spectral gap. In this paper, we consider perturbations which are localized, i.e., integrable on the line. Such localized perturbations naturally yield the absence of a spectral gap, so we must rely on a substantially different method with origins in the stability analysis of periodic waves in reaction–diffusion systems. The relevant linear estimates have been obtained in recent work by the first three authors through a delicate decomposition of the associated linearized solution operator. Since its most critical part just decays diffusively, the nonlinear iteration can only be closed if one allows for a spatio-temporal phase modulation. However, the modulated perturbation satisfies a quasilinear equation yielding an apparent loss of regularity. To overcome this obstacle, we incorporate tame estimates on the unmodulated perturbation, which satisfies a semilinear equation in which no derivatives are lost, yet where decay is too slow to close an independent iteration scheme. We obtain nonlinear stability of periodic steady waves in the LLE against localized perturbations with precisely the same decay rates as predicted by the linear theory.

Semilinear elliptic equations on rough domains

The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the usual arguments based on maximum principles do not apply. A key result is an application of Kato's inequality to prove a comparison theorem for eigenfunctions that only requires interior regularity and avoids the use of the Hopf boundary maximum principle. We demonstrate the theory on an abstract degenerate logistic equation by proving the existence, uniqueness and stability of non-trivial positive solutions. Examples of operators include the Dirichlet Laplacian on arbitrary bounded domains, a simplified construction of the Robin Laplacian on arbitrary domains with boundary of finite measure and general elliptic operators in divergence form.

Bounds for blow-up solutions of a semilinear pseudo-parabolic equation with a memory term and logarithmic nonlinearity in variable space

In this article, we investigate the initial boundary value problem for a pseudo-parabolic equation under the influence of a linear memory term and a logarithmic nonlinear source term \[ u_{t}-\Delta u_{t}+\int _{0}^{t}g( t-s) \Delta u( x,s) \mathrm {d}s-\Delta u\]\[=|u|^{p(\cdot ) -2}u\ln (|u|), \]with a Dirichlet boundary condition. Under appropriate assumptions about the relaxation function $g$, the initial data $u_{0}$ and the function exponent $p$, we not only set the lower bounds for the blow-up time of the solution when blow-up occurs, but also by assuming that the initial energy is negative, we give a new blow-up criterion and an upper bound for the blow-up time of the solution.

Error of the Galerkin scheme for a semilinear subdiffusion equation with time-dependent coefficients and nonsmooth data

We investigate the error of the (semidiscrete) Galerkin method applied to a semilinear subdiffusion equation in the presence of nonsmooth initial data. The diffusion coefficient is allowed to depend on time. We use the energy method to find optimal bounds on the error under weak natural assumptions on the diffusivity. First, we prove the result for the linear problem and then use the “frozen nonlinearity” technique coupled with various generalizations of Grönwall inequality to carry the result to the semilinear case. The paper ends with numerical illustrations supporting the theoretical results.

Existence of solutions to a class of quasilinear elliptic equations

This paper is concerned with the existence and nonexistence of nontrivial solutions for a class of quasilinear elliptic equations in RN involving the p-Laplacian. By utilizing a change of variables, the quasilinear equations are reduced to semilinear equations, whose associated functionals satisfy the geometric hypotheses of the mountain pass theorem. We establish the existence of nontrivial solutions via Mountain-Pass theorem. Furthermore, by using a Pohozaev type identity we prove the nonexistence of nontrivial solution under certain conditions.

Stabilization of non-autonomous infinite-dimensional systems depending on a parameter

ABSTRACT In this paper, we investigate the practical stabilization of non-autonomous infinite-dimensional systems depending on a parameter. By the Lyapunov method, several sufficient conditions are provided to obtain the practical stabilization of two classes of non-autonomous semilinear evolution equations in Hilbert spaces. We propose some classes of memoryless state linear and nonlinear feedback controllers. Numerical examples are given to illustrate the validity of the proposed theoretical results.

Existence and uniqueness of periodic solution to second‐order impulsive differential equations

In this paper, existence and uniqueness of periodic solutions of second‐order semilinear impulsive differential equations are obtained. Some sufficient conditions that guarantee the existence of periodic solutions to second‐order linear nonhomogeneous impulsive differential equations are obtained. In addition, by using the appropriate fixed theorem and the theory of coincidence degree, some conditions ensuring the existence and uniqueness of periodic solution to second‐order semilinear impulsive differential systems are derived. Finally, some examples are provided to illustrate the availability of our results.

Existence of Atangana-Baleanu fractional neutral Volterra integro-differential equations with non-instantaneous impulses

The main focus of this manuscript is centered around Atangana-Baleanu semilinear fractional integrodifferential equations accompanied by non-instantaneous impulses. The outcomes are demonstrated using the Darbo fixed point theorem along with its results when the measure of non-compactness collaborates. Eventually, a demonstration example is proposed.

Sharp patterns for some semilinear nonlocal dispersal equations

Sharp patterns for some semilinear nonlocal dispersal equations

An addendum to “Mild solutions to semilinear stochastic partial differential equations with locally monotone coefficients”

In this addendum we provide an existence and uniqueness result for mild solutions to semilinear stochastic partial differential equations driven by Wiener processes and Poisson random measures in the framework of the semigroup approach with locally monotone coefficients, where the semigroup is allowed to be pseudo-contractive. This improves an earlier paper of the author, where the equation was only driven by Wiener processes, and where the semigroup was only allowed to be a semigroup of contractions.

Existence and compactness of solution of semilinear integro-differential equations with finite delay

UDC 517.9 We present some existence and uniqueness results for a class of functional integro-differential evolution equations generated by the resolvent operator for which the semigroup is not necessarily compact. It is proved that the set of solutions is compact. Our approach is based on fixed point theory. Finally, some examples are given to illustrate the results.

Concavity of solutions to semilinear equations in dimension two

AbstractWe consider the Dirichlet problem for a class of semilinear equations on two‐ dimensional convex domains. We give a sufficient condition for the solution to be concave. Our condition uses comparison with ellipses, and is motivated by an idea of Kosmodem'yanskii. We also prove a result on propagation of concavity of solutions from the boundary, which holds in all dimensions.

Existence and uniqueness of the mild solution of an abstract nonlocal semilinear fractional integrodifferential equation

The purpose of this paper is to study the existence and uniqueness of mild solutions to a semilinear Cauchy problem for an abstract nonlocal fractional integrodifferential equation, which has a distinctive nonlinear term. Two sufficient conditions on the existence of mild solutions will be displayed. Continuous dependence of solutions on initial conditions and local [Formula: see text]-approximate mild solutions will be discussed. An example will be given to elucidate the main results. The results obtained are based upon the method of semigroups, contraction mapping principle, and Krasnoselskii’s fixed point theorem.

Existence and partial approximate controllability of nonlinear Riemann–Liouville fractional systems of higher order

This article studies the existence and partial approximate controllability of higher order nonlocal semilinear fractional differential equations with Riemann–Liouville derivatives avoiding Lipschitz assumptions of nonlinear operator and nonlocal functions. To derive the existence result, we make approximate systems corresponding to the original system. For this, we construct the mild solutions in terms of fractional resolvent. Then, we prove the partial approximate controllability of the nonlinear system by using the obtained existence result. Finally, we give an example to illustrate the established theory.

Functional Integro-Differential Equations with State-Dependent Delay and Non-Instantaneous Impulsions: Existence and Qualitative Results

This paper addresses some existence, attractivity and controllability results for semilinear integrodifferential equations having non-instantaneous impulsions on an infinite interval via resolvent operators in case of neutral and state-dependent delay problems. Our criteria were obtained by applying a Darbo’s fixed-point theorem combined with measures of noncompactness. The obtained result is illustrated by an example at the end.