Blow-up Solutions Research Articles

A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations

In this paper, the tempered Ψ–Riemann–Liouville fractional derivative and the tempered Ψ–Caputo fractional derivative of order n−1<α<n∈N are introduced for Cn−1–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered Ψ–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered Ψ–Caputo fractional differential equations are proved. Illustrative examples are given.

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

An Adaptive Moving Mesh Method for Simulating Finite-time Blowup Solutions of the Landau-Lifshitz-Gilbert Equation

Global existence and blowup of solutions to a class of wave equations with Hartree type nonlinearity

Abstract In this paper, we consider a class of wave-Hartree equations on a bounded smooth convex domain with Dirichlet boundary condition. We prove the local existence of solutions in the natural energy space by using the standard Galërkin method. The results on global existence and nonexistence of solutions are obtained mainly by means of the potential well theory and concavity method.

The Blow-Up of the Local Energy Solution to the Wave Equation with a Nontrivial Boundary Condition

In this study, we examine the wave equation with a nontrivial boundary condition. The main target of this study is to prove the local-in-time existence and the blow-up in finite time of the energy solution. Through the construction of an auxiliary function and the imposition of appropriate conditions on the initial data, we establish the both lower and upper bounds for the blow-up time of the solution. Meanwhile, based on these estimates, we obtain the result of the local-in-time existence and the blow-up of the energy solution. This approach enhances our understanding of the dynamics leading to blow-up in the considered condition.

On Blow-Up of Solutions of the Cauchy Problems for a Class of Nonlinear Equations of Ferrite Theory

On Blow-Up of Solutions of the Cauchy Problems for a Class of Nonlinear Equations of Ferrite Theory

A special form of solution to complex valued Benjamin–Ono equations

We investigate a special form of solution to the complex valued Benjamin–Ono (cBO) equation. Under the special form of solution involving Hilbert transform, the complex valued (modified) Benjamin–Ono equations are equivalent to the quadratic (cubic) derivative Schrödinger equations. The soliton interaction dynamics is used to construct a finite time blowup solution of the cBO equation on R. We also find static solutions. Applying Cole–Hopf transformation to quadratic derivative Schrödinger equation, we derive a linear Schrödinger equation with a special form of initial data from which some solutions to cBO equation can be obtained by an inverse transform.

Concentration of blow-up solutions for the Gross-Pitaveskii equation

Abstract We consider the blow-up solutions for the Gross-Pitaveskii equation modeling the attractive Boes-Einstein condensate. First, a new variational characteristic is established by computing the best constant of a generalized Gagliardo-Nirenberg inequality. Then, a lower bound on blow-up rate and a new concentration phenomenon of blow-up solutions are obtained in the L 2 {L}^{2} supercritical case. Finally, in the L 2 {L}^{2} critical case, a delicate limit of blow-up solutions is analyzed.

Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production

In this paper, we consider the following two-dimensional chemotaxis system of attraction–repulsion with indirect signal production 𝜕tu=Δu−∇·χ1u∇v1+∇·(χ2u∇v2),x∈R2,t>0,0=Δvj−λjvj+w,x∈R2,t>0,(j=1,2),𝜕tw+δw=u,x∈R2,t>0,u(0,x)=u0(x),w(0,x)=w0(x),x∈R2, where the parameters χi≥0, λi>0(i=1,2) and non-negative initial data (u0(x),w0(x))∈L1(R2)∩L∞(R2). We prove the global bounded solution exists when the attraction is more dominant than the repulsion in the case of χ1≥χ2. At the same time, we propose that when the radial solution satisfies χ1−χ2≤2πδ∥u0∥L1(R2)+∥w0∥L1(R2), the global solution is bounded. During the proof process, we found that adding indirect signals can constrict the blow-up of the global solution.

Norm inflation for solutions of semi-linear one-dimensional Klein–Gordon equations

In space dimension larger than or equal to 2, the nonlinear Klein–Gordon equation with small, smooth, decaying initial data has global-in-time solutions. This no longer holds true in one space dimension, where examples of blowing-up solutions are known. On the other hand, it has been proved that if the nonlinearity satisfies a convenient compatibility condition, the “null condition”, one recovers global existence and that the solutions satisfy the same dispersive bounds as linear solutions. The goal of this paper is to show that, in the case of cubic semi-linear nonlinearities, this null condition is optimal, in the sense that, when it does not hold, one may construct small, smooth, decaying initial data giving rise to solutions that display inflation of their L^{\infty} and L^{2} -norms in finite time.

Blow-up of solutions to the Keller–Segel model with tensorial flux in high dimensions

Over the course of the last decade, there has been a significant level of interest in the analysis of Keller–Segel models incorporating tensorial flux. Despite this interest, the question of whether finite-time blowup solutions exist remains a topic of ongoing research. Our study provides evidence that solutions of this nature are indeed possible in dimensions n≥3, when utilizing a tensorial flux expressed in the form of A∇v, where A denotes a matrix with constant components.

L2 Concentration of Blow-Up Solutions for the Nonlinear Schrödinger Equation with an Inhomogeneous Combined Non-Linearity

This article studies the Schrödinger equation with an inhomogeneous combined term i∂tu−(−Δ)su+λ1|x|−b|u|pu+λ2|u|qu=0, where s∈(12,1),λ1,λ2=±1,0<b<{2s,N} and p,q>0. We study the limit behaviour of the infinite blow-up solution at the blow-up time. When the parameters p,q,λ1 and λ2 have different values, we obtain the nonexistence of a strong limit for the non-radial solution and the L2 concentration for the radial solution. Interestingly, we find that the mass of the finite time blow-up solutions are concentrated in different ways for different parameters.

Global Existence and Blow-up of Solutions for a Parabolic Equation with Nonlinear Memory and Absorption under Nonlinear Nonlocal Boundary Condition

Global Existence and Blow-up of Solutions for a Parabolic Equation with Nonlinear Memory and Absorption under Nonlinear Nonlocal Boundary Condition

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).

Finite point blowup for the critical generalized Korteweg-de Vries equation

In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-up dynamics and the residue are eventually uncoupled. Even at the formal level, there was no indication whether blowup at a finite point could occur for this equation. In this article, we answer this question by constructing solutions that blow up in finite time under the form of a single-bubble concentrating the ground state at a finite point with an unforeseen blow-up rate. Finding a blow-up rate intermediate between the self-similar rate and other rates previously known also reopens the question of which blow-up rates are actually possible for this equation.

On Fila-King conjecture in dimension four

We consider the following Cauchy problem for the four-dimensional energy critical heat equation{ut=Δu+u3 in R4×(0,∞),u(x,0)=u0(x) in R4. We construct a positive infinite time blow-up solution u(x,t) with the blow-up rate ‖u(⋅,t)‖L∞(R4)∼ln⁡t as t→∞ and establish the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King [15, Conjecture 1.1].

A pseudo‐parabolic equation with logarithmic nonlinearity: Global existence and blowup of solutions

This article deals with a fourth‐order pseudo‐parabolic partial differential equation with logarithmic nonlinearity. We adopt the Faedo–Galerkin method to analyze the global existence of weak solutions and discuss the existence of a solution in subcritical and critical initial energy situations. Further, applying the concavity approach, we have shown that the solution blows up when the initial energy is subcritical, critical, and supercritical. Moreover, we have provided an upper and lower bound for blow‐up time and a decay estimate for the global solutions.

On a Class of Stochastic Damped Wave Equation

The present work considers a wave equation with multiplicative Gaussian white noise and weak dissipative term on a bounded domain. We first give a theorem including the local existence of mild solutions. An energy bound and a differential inequality are used to give sufficient conditions that provide the blow-up of mild local solutions of the stochastic wave equation. The paper's main contribution comes from handling a multiplicative noise and a general source term contrary to the articles that exist in the literature.

Sharp thresholds of blowup and uniform bound for a Schrödinger system with second‐order derivative‐type and combined power‐type nonlinearities

AbstractConsidered herein is a Cauchy problem for a system of Schrödinger equations with second‐order derivative‐type and combined power‐type nonlinearities. Through the effective combination of potential well theory, conservation laws, and vector‐valued Gargliardo–Nirenberg inequality, we establish the uniform boundedness in ‐norm on and corresponding decay rate estimate. Moreover, we also prove the existence of corresponding ground‐state solutions for this problem. Finally, we mainly investigate three different sharp thresholds for blowup and uniform bound of solutions in ‐norm on by using potential well theory, variational method, and some transformation techniques.

Infinite-time blowing-up solutions to small perturbations of the Yamabe flow

In this paper, we examine a PDE aspect of the Yamabe flow as an energy-critical parabolic equation of the fast-diffusion type. It is well-known that under the validity of the positive mass theorem, the Yamabe flow on a smooth closed Riemannian manifold M exists for all time t and uniformly converges to a solution to the Yamabe problem on M as t→∞. We show that such results no longer hold if some arbitrarily small and smooth perturbation is imposed on it, by constructing solutions to the perturbed flow that blow up at multiple points on M in the infinite time. We also examine the stability of the blow-up phenomena under a negativity condition on the Ricci curvature at blow-up points.

On Blow-Up Solutions for the Fourth-Order Nonlinear Schrödinger Equation with Mixed Dispersions

In this paper, we consider blow-up solutions for the fourth-order nonlinear Schrödinger equation with mixed dispersions. We study the dynamical properties of blow-up solutions for this equation, including the H˙γc-concentration and limiting profiles, which extend and improve the existing results in the literature.