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.

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

In this paper, we consider a plate equation with nonlinear damping and logarithmic source term. By the contraction mapping principle, we establish the local existence. The global existence and decay estimate of the solution at subcritical initial energy are obtained. We also prove that the solution with negative initial energy blows up in finite time under suitable conditions. Moreover, we also give the blow-up in finite time of solution at the arbitrarily high initial energy for linear damping (i.e. \begin{document}$ m = 2 $\end{document} ).

https://www.aimsciences.org/article/exportPdf?id=10a9e564-6a01-4c86-a724-800b08396287

The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term

&lt;p style='text-indent:20px;'&gt;This article is a comparative study on an initial-boundary value problem for a class of semilinear pseudo-parabolic equations with the fractional Caputo derivative, also called the fractional Sobolev-Galpern type equations. The purpose of this work is to reveal the influence of the degree of the source nonlinearity on the well-posedness of the solution. By considering four different types of nonlinearities, we derive the global well-posedness of mild solutions to the problem corresponding to the four cases of the nonlinear source terms. For the advection source function case, we apply a nontrivial limit technique for singular integral and some appropriate choices of weighted Banach space to prove the global existence result. For the gradient nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to show that the solution either exists globally in time or blows up at finite time. For the polynomial form nonlinearity, by assuming the smallness of the initial data we derive the global well-posed results. And for the case of exponential nonlinearity in two-dimensional space, we derive the global well-posedness by additionally using an Orlicz space.&lt;/p&gt; 

https://www.aimsciences.org/article/exportPdf?id=02ab2374-661b-44ec-a318-b6d1d4eff50d

Global well-posedness for fractional Sobolev-Galpern type equations

In this paper, we are concerned with the decay rate of the solution of a viscoelastic plate equation with infinite memory and logarithmic nonlinearity. We establish an explicit and general decay rate results with imposing a minimal condition on the relaxation function. In fact, we assume that the relaxation function h satisfies $$ h^{\prime}(t)\le-\xi(t) H\bigl(h(t)\bigr),\quad t\geq0, $$ where the functions ξ and H satisfy some conditions. Our proof is based on the multiplier method, convex properties, logarithmic inequalities, and some properties of integro-differential equations. Moreover, we drop the boundedness assumption on the history data, usually made in the literature. In fact, our results generalize, extend, and improve earlier results in the literature.

/pdf/stability-result-of-a-viscoelastic-plate-equation-with-past-2w1sgwkwqr.pdf

Stability result of a viscoelastic plate equation with past history and a logarithmic nonlinearity

In this paper we consider the initial boundary value problem for a viscoelastic wave equation with strong damping and logarithmic nonlinearity of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \int ^{t}_{0} g(t-s) \Delta u(x,s)\,ds - \Delta u_{t} (x,t) = \bigl\vert u(x,t) \bigr\vert ^{p-2} u(x,t) \ln \bigl\vert u(x,t) \bigr\vert $$ in a bounded domain $\varOmega \subset {\mathbb{R}}^{n} $, where g is a nonincreasing positive function. Firstly, we prove the existence and uniqueness of local weak solutions by using Faedo–Galerkin’s method and contraction mapping principle. Then we establish a finite time blow-up result for the solution with positive initial energy as well as nonpositive initial energy.

/pdf/blow-up-phenomena-for-a-viscoelastic-wave-equation-with-2p287usodd.pdf

Blow-up phenomena for a viscoelastic wave equation with strong damping and logarithmic nonlinearity

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.

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

Abstract In this paper, the initial boundary value problem for a nonlocal semilinear pseudo-parabolic equation is investigated, which was introduced to model phenomena in population dynamics and biological sciences where the total mass of a chemical or an organism is conserved. The existence, uniqueness and asymptotic behavior of the global solution and the blowup phenomena of solution with subcritical initial energy are established. Then these results are extended parallelly to the critical initial energy. Further the blowup phenomena of solution with supercritical initial energy is proved, but the existence, uniqueness and asymptotic behavior of the global solution with supercritical initial energy are still open.

/pdf/global-existence-and-finite-time-blowup-for-a-nonlocal-37no1cq0hu.pdf

Runzhang Xu

Papers

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

Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation