Logarithmic Source Research Articles

Polynomial decay rate for a new class of viscoelastic Kirchhoff equation related with Balakrishnan-Taylor dissipation and logarithmic source terms

In this paper, a polynomial decay rate of Kirchhoff’s nonlinear viscoelastic viscoelastic equation solution related with Balakrishnan-Taylor dissipation solution and logarithmic source terms is obtained, where we obtain the result of energy decay of solutions without impose the usual relationship between certain relaxation functions and their derivatives. This result generalizes the previous results to an arbitrary decay rate, that is not necessarily of the polynomial or exponential decay.

Some existence results for a new class of elliptic Kirchhoff equation with logarithmic source terms

Some existence results for a new class of elliptic Kirchhoff equation with logarithmic source terms

Global well-posedness of solutions for fourth order dispersive wave equation with nonlinear weak damping, linear strong damping and logarithmic nonlinearity

To understand the characteristics of dynamical behavior especially the kinetic evolution for logarithmic nonlinearity, we aim to study the global well-posedness of nonlinear fourth order wave equations with logarithmic source term, where the dispersive, the nonlinear weak damping and linear strong damping are taken into account. Based on the potential well method, the main ingredient of this paper is to construct several conditions for initial data leading to the solution global existence or infinite time blow up with subcritical initial energy. Moreover, we extend the results to the critical initial energy, which is realized by introducing scaling technique. Finally, surrounding the blow up at arbitrarily high initial energy, we compare and discuss the research program stemming from two different strategies. In the first one, it is assumed that the blow up result is bound to the original logarithmic source by weakening the dispersive–dissipative structure, while the second one is based on the nonlinear wave equation with complete dispersive–dissipative structure, but the logarithmic source is replaced by an enhanced version. Through these two strategies, we explain the mechanism of these two structures respectively and establish two kinds of sufficient conditions for initial data leading to high energy blow up.

Publisher Correction to: Existence and a general decay result for a plate equation with non-linear damping and a logarithmic source term

Due to an error in the typesetting process, reference [1] is incorrectly published in the original publication of the article.

A well-posedness and exponential decay of solutions for a coupled Lamé system with viscoelastic term and logarithmic source terms

This paper describes a well-posedness and exponential decay of solutions for a coupled Lamé system with viscoelastic term and logarithmic source terms, where an asymptotic stability result of global solution is obtained.

Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term

Abstract The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three different initial energy levels, i.e., subcritical energy E(0) < d, critical initial energy E(0) = d and the arbitrary high initial energy E(0) > 0 (ω = 0). Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, infinite time blow up of the solution with sub-critical initial energy. Then we parallelly extend all the conclusions for the subcritical case to the critical case by scaling technique. Besides, a high energy infinite time blow up result is established.

Kirchhoff-type system with linear weak damping and logarithmic nonlinearities

For the nonlinear Kirchhoff-type wave system with logarithmic nonlinearities and weak dissipation the global well-posedness of initial boundary value problem is analyzed. Focusing on the interplay between Kirchhoff terms and logarithmic sources, we investigate the Kirchhoff system controlled by logarithmic forces thus amplifying the difficulties in blow up analysis which is the primary scenario of interest. By employing potential well method and concavity method, we obtain several results related to the sufficient conditions posed on subcritical initial energy and critical initial energy, which is used to classify initial data for global existence and finite time blow up. Finally, via careful analysis involving the unstable invariant set under supercritical initial energy, we are able to show an affirmative result that the solution blows up in finite time when initial data satisfy some suitable assumptions.

Qualitative analysis of solutions for a class of Kirchhoff equation with linear strong damping term, nonlinear weak damping term and power-type logarithmic source term

In this paper, we study the initial boundary value problem for Kirchhoff equation with linear strong damping term, nonlinear weak damping term and power-type logarithmic source term at three different initial energy levels, i.e. subcritical energy E(0)<d, critical initial energy E(0)=d and the arbitrary high energy E(0)>0. By potential well method, we prove global existence, finite time blow up and asymptotic behavior of solutions in cases of E(0)<d and E(0)=d. We also prove the finite time blow up of solutions to the problem with linear weak and strong damping term in the case of E(0)>0.

Classical Solution of Non-stationary Heat Equation with Nonlinear Logarithmic Source Using Generalized Functional Separable Method (GFSM)

Classical Solution of Non-stationary Heat Equation with Nonlinear Logarithmic Source Using Generalized Functional Separable Method (GFSM)

Quenching of the solution to the discrete heat equation with logarithmic type sources on graphs

ABSTRACTIn the current paper, we mainly consider the following discrete heat equation on graphs with logarithmic type sources: First, the local existence and uniqueness of the above problem are obtained via the Banach fixed point theorem. By the comparison principle, the quenching behavior of the above problem and the blow-up of its time-derivatives of its solution at finite time under some suitable conditions are also given. Moreover, we also obtain the existence of the critical exponent , when , the above problem admits a global solution by the implicit function theorem. On the other hand, as , its solution will quench at finite time. Finally, a numerical experiment on the graph with five vertexes is used to explain the theoretical results.

General Stability and Exponential Growth for a Class of Semi-linear Wave Equations with Logarithmic Source and Memory Terms

In this work we investigate asymptotic stability and instability at infinity of solutions to a logarithmic wave equation $$\begin{aligned} u_{tt}-\Delta u + u + (g\,*\, \Delta u)(t)+ h(u_{t})u_{t}+|u|^{2}u=u\log |u|^{k}, \end{aligned}$$ in an open bounded domain $$\Omega \subseteq \mathbb {R}^3$$ whith $$h(s)=k_{0}+k_{1}|s|^{m-1}.$$ We prove a general stability of solutions which improves and extends some previous studies such as the one by Hu et al. (Appl Math Optim, https://doi.org/10.1007/s00245-017-9423-3 ) in the case $$g=0$$ and in presence of linear frictional damping $$u_{t}$$ when the cubic term $$|u|^2u$$ is replaced with u. In the case $$k_{1}=0,$$ we also prove that the solutions will grow up as an exponential function. Our result shows that the memory kernel g dose not need to satisfy some restrictive conditions to cause the unboundedness of solutions.

Global Solution and Blow-up for a Class of p-Laplacian Evolution Equations with Logarithmic Nonlinearity

The main goal of this work is to study an initial boundary value problem for a quasilinear parabolic equation with logarithmic source term. By using the potential well method and a logarithmic Sobolev inequality, we obtain results of existence or nonexistence of global weak solutions. In addition, we also provided sufficient conditions for the large time decay of global weak solutions and for the finite time blow-up of weak solutions.

The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term

In this paper, we consider a plate equation with a logarithmic nonlinearity in the presence of a frictional damping. We prove, for a suitably chosen initial data, a global existence as well as a decay result.

Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term

In this paper, we consider a plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping. Using the Galaerkin method, we establish the existence of solutions of the problem and we prove an explicit and general decay rate result, using the multiplier method and some properties of the convex functions. Our result is obtained without imposing any restrictive growth assumption on the damping term.

Instability of positive periodic solutions for semilinear pseudo-parabolic equations with logarithmic nonlinearity

In this paper, we consider the periodic problem for semilinear heat equation and pseudo-parabolic equation with logarithmic source. After establishing the existence of positive periodic solutions, we discuss the instability of such solutions.

Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

ABSTRACTIn this article, we prove optimal convergence rates results for regularization methods for solving linear ill-posed operator equations in Hilbert spaces. The results generalizes existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions.

About a deficit in low-order convergence rates on the example of autoconvolution

We revisit in -spaces the autoconvolution equation with solutions which are real-valued or complex-valued functions defined on a finite real interval, say . Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity, deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low-order convergence rates for regularized solutions of nonlinear ill-posed operator equations with solutions in a Hilbert space setting. So, for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex -functions with a fixed and uniformly bounded modulus function.

Symmetry preserving numerical schemes for partial differential equations and their numerical tests

The method of equivariant moving frames is used to construct symmetry preserving finite difference schemes of partial differential equations invariant under finite-dimensional symmetry groups. Invariant numerical schemes for a heat equation with logarithmic source and the spherical Burgers' equation are obtained. Numerical tests show how invariant schemes can be more accurate than standard discretizations.

Logarithmic convergence rate of Levenberg–Marquardt method with application to an inverse potential problem

Abstract We prove logarithmic convergence rate of the Levenberg–Marquardt method in a Hilbert space if a logarithmic source condition is satisfied. This method is applied to an inverse potential problem. Numerical implementations demonstrate the convergence rate.

A UNIFIED APPROACH FOR REGULARIZING DISCRETIZED LINEAR ILL‐POSED PROBLEMS

In this paper we deal with regularization approaches for discretized linear ill‐posed problems in Hilbert spaces. As opposite to other contributions concerning this topic the smoothness of the unknown solution is measured with so‐called approximative source conditions. This idea allows us to generalize known convergence rates results to arbitrary classes of smoothness conditions including logarithmic and general source conditions. The considerations include an a‐posteriori parameter choice strategy for the regularization parameter and the discretization level. Results of one numerical example are presented.