Blow-up Behavior Research Articles

Blow-Up Analysis of L2-Norm Solutions for an Elliptic Equation with a Varying Nonlocal Term

This paper is devoted to studying a type of elliptic equation that contains a varying nonlocal term. We provide a detailed analysis of the existence, non-existence, and blow-up behavior of L2-norm solutions for the related equation when the potential function V(x) fulfills an appropriate choice.

Blow-up for a Porous Medium Equation with Local Linear Boundary Dissipation

This article investigates the blow-up behaviors for a porous medium equation with a superlinear source and local linear boundary dissipation. Making use of the concavity method, we establish sufficient conditions to guarantee the occurrence of the finite time blow-up phenomenon. Meanwhile, we show the existence of the finite time blow-up solutions for arbitrarily high initial energy. Finally, we derive the life span bounds (i.e., the lower and upper bounds of the blow-up time).

Global Existence, Blowup, and Asymptotic Behavior for a Kirchhoff-Type Parabolic Problem Involving the Fractional Laplacian with Logarithmic Term

In this paper, we studied a class of semilinear pseudo-parabolic equations of the Kirchhoff type involving the fractional Laplacian with logarithmic nonlinearity: ut+M([u]s2)(−Δ)su+(−Δ)sut=|u|p−2uln|u|,in Ω×(0,T),u(x,0)=u0(x),in Ω,u(x,t)=0,on ∂Ω×(0,T),, where [u]s is the Gagliardo semi-norm of u, (−Δ)s is the fractional Laplacian, s∈(0,1), 2λ<p<2s*=2N/(N−2s), Ω∈RN is a bounded domain with N>2s, and u0 is the initial function. To start with, we combined the potential well theory and Galerkin method to prove the existence of global solutions. Finally, we introduced the concavity method and some special inequalities to discuss the blowup and asymptotic properties of the above problem and obtained the upper and lower bounds on the blowup at the sublevel and initial level.

Global existence, blow-up and dynamical behavior in a nonlocal parabolic problem with variational structure

In this paper, we study an initial boundary value problem for a nonlocal parabolic equation that includes a diffusion term and convex–concave nonlinearities. By establishing the Lq-estimate and analyzing its energy, we prove the existence of global solutions and obtain some blow-up conditions. Given the variational structure of the problem, we utilize the Mountain-pass theorem to demonstrate the existence of nontrivial steady-state solutions. Additionally, we establish the dynamical behavior of global solutions whose trajectory is relatively compact in H01(Ω), and which uniformly converge to some non-zero steady state after a long time, owing to the energy functional satisfying the P.S. condition. Finally, we derive an unstable steady states sequence by using another minimax theorem.

Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source

This manuscript deals with global solution, polynomial stability and blow-up behavior at a finite time for the nonlinear system $$ \left\{ \begin{array}{rcl} & u'' - \Delta_{p} u + \theta + \alpha u' = \left\vert u\right\vert ^{p-2}u\ln \left\vert u\right\vert \\ &\theta' - \Delta \theta = u' \end{array} \right. $$ where $\Delta_{p}$ is the nonlinear $p$-Laplacian operator, $ 2 \leq p < \infty$. Taking into account that the initial data is in a suitable stability set created from the Nehari manifold, the global solution is constructed by means of the Faedo-Galerkin approximations. Polynomial decay is proven for a subcritical level of initial energy. The blow-up behavior is shown on an instability set with negative energy values.

Blow-Up Behavior of $$L^{2}$$-Norm Solutions for Kirchhoff Equation in a Bounded Domain

Blow-Up Behavior of $$L^{2}$$-Norm Solutions for Kirchhoff Equation in a Bounded Domain

The existence and asymptotic behavior of normalized solutions for Kirchhoff equation with singular potential

In this paper, the existence, non-existence and asymptotic behavior of minimizers for a constrained Kirchhoff type minimization problem with singular potential were considered. Using concentration-compactness lemma, the existence of minimizers of this minimization problem was obtained. By truncation function technique, the non-existence of minimizers was got. Compared with the known result for this problem with trapping potential (Guo, Zhang, Zhou, 2018, [4]) that there exists minimizer for the case of critical exponent p=p⁎ and critical parameter β=βp⁎, our problem has no minimizers in the same condition. Therefore, we discuss the blow-up behavior of minimizer for p=p⁎ as β approaches βp⁎ or β=β˜p and p approaches p⁎, and they appear to be new.

Singularities in $L^1$-supercritical Fokker–Planck equations: A qualitative analysis

A class of nonlinear Fokker–Planck equations with superlinear drift is investigated in the L^1 -supercritical regime, which exhibits a finite critical mass. The equations have a formal Wasserstein-like gradient-flow structure with a convex mobility and a free energy functional whose minimising measure has a singular component if above the critical mass. Singularities and concentrations also arise in the evolutionary problem and their finite-time appearance constitutes a primary technical difficulty. This paper aims at a global-in-time qualitative analysis with main focus on the isotropic case, where solutions will be shown to converge to the unique minimiser of the free energy as time tends to infinity. A key step in the analysis consists in properly controlling the singularity profiles during the evolution. Our study covers the three-dimensional Kaniadakis–Quarati model for Bose–Einstein particles, and thus provides a first rigorous result on the continuation beyond blow-up and long-time asymptotic behaviour for this model.

Blow-Up Behavior of Solutions to Critical Hartree Equations on Bounded Domain

Blow-Up Behavior of Solutions to Critical Hartree Equations on Bounded Domain

Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equation

In this work, we consider the following focusing inhomogeneous nonlinear Schrödinger equation i∂tu+Δu+|x|−b|u|pu=0,(t,x)∈R×RNwith 0<b<min{2,N} and 4−2bN<p<4−2bN−2. Assume that u0∈H1(RN) and beyond the ground state threshold, then we prove the following two statements, (1) when 4−2bN<p<min{4N,4−2bN−2}, or p=4N when b∈(0,4N), then the corresponding solution blows up in finite time; (2) when 4N<p<4−2bN−2, we prove the finite or infinite time blow-up. Moreover, we can further obtain a precise lower bound of infinite time blow-up rate, that is supt∈[0,T]‖∇u(t)‖L2≳Tκ,for someκ>0.To our knowledge, the statement (1) establishes the first finite time blow-up result for this equation in the intercritical case when the initial data u0 does not have finite variance and is non-radial. The statement (2) gives the first result for the infinite time blow-up rate for this equation.

Gradient blowup behavior for a viscous Hamilton-Jacobi equation with degenerate gradient nonlinearity

The paper is concerned with gradient blowup behavior for a semilinear parabolic equationut−Δu=δm(x)|∇u|p+ΛinΩ×(0,T) with the zero Dirichlet condition. Two results with p>m+2 and m⩾0 are established. One of which is that any gradient blowup solution follows a global ODE type behavior, with domination of normal derivatives over the tangential derivatives. The other is about time-increasing solutions. Zhang and Hu (2010) [50] obtained the precise gradient blowup rate in one-dimensional case, but the higher dimensional case was left as an open problem. Here we solve this problem by establishing the gradient blowup rate, for any small γ>0,C(T−t)−m+1p−m−2⩽‖∇u‖∞⩽Cγ(T−t)−m+1p−m−2−γ for suitable ranges of p and m, which extends the result of (Attouchi and Souplet (2020) [3]) to the case m≠0. As an important by-product which is of independent interests itself, the gradient estimate near boundary for the corresponding elliptic equation is derived under weaker assumptions on the inhomogeneous term.

Large time behaviour of semilinear stochastic partial differential equations perturbed by a mixture of Brownian and fractional Brownian motions

We study the trajectorywise blowup behaviour of a semilinear partial differential equation that is driven by a mixture of multiplicative Brownian and fractional Brownian motion, modelling different types of random perturbations. The linear operator is supposed to have an eigenfunction of constant sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters of the equation, on the occurrence of a blowup in finite time of the solution. We give estimates for the probability of finite time blowup and of blowup before a given fixed time. Essential tools are the mild and weak form of an associated random partial differential equation.

Upper and lower blow-up rate estimates of a semilinear heat equation with a nonlinear boundary condition

In this paper, we consider the blow-up solutions of a semi-linear heat equation with a Neumann boundary condition, where the nonlinear terms, appear in the differential equation and in the boundary conditions, are of exponential types. We study the effect of the nonlinear terms on the last blow-up behavior. Precisely, the upper (lower) blow-up rate estimates are established by using integral equation method and maximum principle techniques. The results show that the presentence of the reaction and boundary terms has important effects on the upper (lower) blow-up rate estimates forms.

Existence and blow-up behavior of positive solutions to integral equations on bounded domains

Existence and blow-up behavior of positive solutions to integral equations on bounded domains

Classification of the Blow-Up Behavior for a Semilinear Wave Equation with Nonconstant Degenerate Coefficients

We consider a nonlinear wave equation with nonconstant coefficients. In particular, the coefficient in front of the second-order space derivative is degenerate. We give the blow-up behavior and the regularity of the blow-up set. Partial results are given at the origin, where the degeneracy occurs. Some nontrivial obstacles, due to the nonconstant speed of propagation, have to be surmounted.

Sign-changing blow-up for the Yamabe equation at the lowest energy level

We investigate the blow-up behavior of sequences of sign-changing solutions for the Yamabe equation on a Riemannian manifold (M,g) of positive Yamabe type. For each dimension n≥11, we describe the value of the minimal energy threshold at which blow-up occurs. In dimensions 11≤n≤24, where the set of positive solutions is known to be compact, we show that the set of sign-changing solutions is not compact and that blow-up already occurs at the lowest possible energy level. We prove this result by constructing a smooth, non-locally conformally flat metric on space forms Sn/Γ, Γ≠{1}, whose Yamabe equation admits a family of sign-changing blowing-up solutions. As a counterpart of this result, we also prove a sharp compactness result for sign-changing solutions at the lowest energy level, in small dimensions or under strong geometric assumptions.

On the Riccati dynamics of 2D EulerPoisson equations with attractive forcing

The Euler–Poisson (EP) system describes the dynamic behaviour of many important physical flows. In this work, a Riccati system that governs pressureless two-dimensional EP equations is studied. The evolution of divergence is governed by the Riccati type equation with several nonlinear/nonlocal terms. Among these, the vorticity accelerates divergence while others further amplify the blow-up behaviour of a flow. The growth of these blow-up amplifying terms are related to the Riesz transform of density, which lacks a uniform bound makes it difficult to study global solutions of the multi-dimensional EP system. We show that the Riccati system can afford to have global solutions, as long as the growth rate of blow-up amplifying terms is not higher than exponential, and admits global smooth solutions for a large set of initial configurations. To show this, we construct an auxiliary system in 3D space and find an invariant space of the system, then comparison with the original 2D system is performed. Some numerical examples are also presented.

Blow-up of ground states of fractional Choquard equations

We investigate the blow-up behavior of ground states of the fractional Choquard equation (−Δ)su+u=(Kα∗|u|pϵ)|u|pϵ−2uin RN,with s∈(0,1), N>4s, Kα Riesz potential of order α∈(0,N), as the exponent pϵ approaches the upper critical growth regime in Hardy–Littlewood–Sobolev’s inequality. We prove that the ground state uϵ blows up in the sense that ‖uϵ‖L∞(RN)=oϵ−N−2s4s as ϵ→0+.

On Cauchy problem for fractional parabolic-elliptic Keller-Segel model

Abstract In this paper, we concern about a modified version of the Keller-Segel model. The Keller-Segel is a system of partial differential equations used for modeling Chemotaxis in which chemical substances impact the movement of mobile species. For considering memory effects on the model, we replace the classical derivative with respect to time variable by the time-fractional derivative in the sense of Caputo. From this modification, we focus on the well-posedness of the Cauchy problem associated with such the model. Precisely, when the spatial variable is considered in the space R d {{\mathbb{R}}}^{d} , a global solution is obtained in a critical homogeneous Besov space with the assumption that the initial datum is sufficiently small. For the bounded domain case, by using a discrete spectrum of the Neumann Laplace operator, we provide the existence and uniqueness of a mild solution in Hilbert scale spaces. Moreover, the blowup behavior is also studied. To overcome the challenges of the problem and obtain all the aforementioned results, we use the Banach fixed point theorem, some special functions like the Mainardi function and the Mittag-Leffler function, as well as their properties.

Chirped self-similar pulses and envelope solutions for a nonlinear Schrödinger's in optical fibers using Lie group method

In this work, we present an application of Lie group analysis to study the generalized derivative nonlinear Schrödinger equation, which governs the evolution of a nonlinear wave and plays an important role in the propagation of short pulses in optical fiber systems. To construct Lie group reductions, we study the symmetry properties and introduce various infinitesimal operators. Further, we obtain self-similar solutions and periodic soliton solutions of the generalized derivative nonlinear Schrödinger equation. This type of solution plays a vital role in the study of the blow-up and asymptotic behavior of non-global solutions. And at the end, we present graphs for each solution by considering the physical meaning of the solutions.