Existence Of Blow-up Solutions Research Articles

Simple blow-up solutions of singular Liouville equations

In a recent series of important works Wei-Zhang [Adv. Math. 380 (2021), Paper No. 107606, 45; Proc. Lond. Math. Soc. (3) 124 (2022), pp. 106–131; Laplacian vanishing theorem for quantized singular Liouville equation, Preprint, arXiv:2202.10825, 2022] proved several vanishing theorems for non-simple blow-up solutions of singular Liouville equations. It is well known that a non-simple blow-up situation happens when the spherical Harnack inequality is violated near a quantized singular source. In this article, we further strengthen the conclusions of Wei-Zhang by proving that if the spherical Harnack inequality does hold, there exist blow-up solutions with non-vanishing coefficient functions.

Upper estimates for blow-up solutions of a quasi-linear parabolic equation

In this paper, we consider a quasi-linear parabolic equation u_t=u^p(x_{xx}+u). It is known that there exist blow-up solutions and some of them develop Type II singularity. However, only a few results are known about the precise behavior of Type II blow-up solutions for p>2. We investigated the blow-up solutions for the equation with periodic boundary conditions and derived upper estimates of the blow-up rates in the case of 2<p<3 and in the case of p=3, separately. In addition, we assert that if 2 le p le 3 then lim _{t nearrow T}(T-t)^{frac{1}{p}+varepsilon }max u(x,t)=0 z for any varepsilon >0 under some assumptions.

Comparative analysis on the blow-up occurrence of solutions to Hadamard type fractional differential systems

The main intention of this paper is to deal with the weak singularity of solutions to some nonlinear Hadamard type fractional differential systems (HTFDSs) which could be viewed as the generalization of classic Hadamard fractional settings. Resorting to the lower and upper solutions technique and constructing the compatible weighted Banach space, the completely continuous operator described by Hadamard type fractional versions as well as a sufficient criterion for the existence of blow-up solutions to some nonlinear HTFDSs is established. Additionally, the comparative analysis on the blow-up rate affected by critical system parameters is also presented, and several examples clearly illustrate the effectiveness and efficiency of the proposed results.

Stability of the standing waves of the concentrated NLSE in dimension two

In this paper we will continue the analysis of two dimensional Schr\"odinger equation with a fixed, pointwise, nonlinearity started in [2, 13]. In this model, the occurrence of a blow-up phenomenon has two peculiar features: the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves; there is no critical power nonlinearity, i.e., for every power there exist blow-up solutions. Here we study the stability properties of stationary states to verify whether the anomalies mentioned before have any counterpart on the stability features.

Blow-Up Solution of a Porous Medium Equation with Nonlocal Boundary Conditions

In this paper, we devote to studying the blow-up phenomena for a porous medium equation under nonlocal boundary conditions. Based on auxiliary functions and differential inequality technique, we derive the sufficient conditions to guarantee the existence of blow-up solutions under different measures and obtain an upper bound for blow-up time. Moreover, we demonstrate the lower bounds for blow-up time under some appropriate measures in R3 and in the higher-dimensional space Rnn≥3, respectively. At last, two examples are given to illustrate the applications of main results.

Global existence and blow-up solutions of the radial Schrödinger maps

This paper studies the Cauchy problem of radial inhomogeneous Schrödinger maps (ISM) which arises from the integrable model of the inhomogeneous spherically symmetric Heisenberg ferromagnetic spin system. Through a complex transformation the radial ISM is equivalent to an integro-differential Schrödinger equation. A new weighted Sobolev space [Formula: see text] is introduced and the well-posedness of integro-differential Schrödinger equations, including the integral radial IMS, with small spherically symmetric initial data in one-dimensional energy space [Formula: see text] is established. Furthermore, for [Formula: see text], we prove the existence of blow-up solutions for the integral radial ISM.

A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained.

Blow-up for the pointwise NLS in dimension two: Absence of critical power

We consider the Schrödinger equation in dimension two with a fixed, pointwise, focusing nonlinearity and show the occurrence of a blow-up phenomenon with two peculiar features: first, the energy threshold under which all solutions blow up is strictly negative and coincides with the infimum of the energy of the standing waves. Second, there is no critical power nonlinearity, i.e. for every power there exist blow-up solutions. This last property is uncommon among the conservative Schrödinger equations with local nonlinearity.

Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one

The paper discusses nonlinear singular delta-type perturbations of the fractional Schrödinger equation , with , in dimension one. In particular, we investigate local and global well-posedness (in a strong sense), conservation laws and the existence of blow-up solutions and standing waves.

Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension

This paper deals with positive solutions of the fully parabolic system{ut=Δu−χ∇⋅(u∇v)inΩ×(0,∞),τ1vt=Δv−v+winΩ×(0,∞),τ2wt=Δw−w+uinΩ×(0,∞) under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain Ω⊂R4 with positive parameters τ1,τ2,χ>0 and nonnegative smooth initial data (u0,v0,w0).Global existence and boundedness of solutions were shown if ‖u0‖L1(Ω)<(8π)2/χ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ‖u0‖L1(Ω)>(8π)2/χ. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has 8π/χ-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in R4.

Dynamics of blow-up solutions for the Schr\xf6dinger\u2013Choquard equation

In this paper, we study the dynamics of blow-up solutions for the nonlinear Schrödinger–Choquard equation \t\t\tiψt+Δψ=λ1|ψ|p1ψ+λ2(Iα∗|ψ|p2)|ψ|p2−2ψ.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$i\\psi_{t}+\\Delta \\psi =\\lambda_{1}\\vert \\psi \\vert ^{p_{1}}\\psi +\\lambda_{2}\\bigl(I _{\\alpha }\\ast \\vert \\psi \\vert ^{p_{2}}\\bigr)\\vert \\psi \\vert ^{p_{2}-2}\\psi. $$\\end{document} We first show existence of blow-up solutions and obtain a sharp threshold mass of global existence and blow-up for this equation with lambda_{1}>0, lambda_{2}<0, 0< p_{1}<frac{4}{N} and p_{2}=1+frac{2+alpha }{N}. Then we obtain some dynamical properties of blow-up solutions by the corresponding ground state of this equation with lambda_{1}=0.

On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

This paper is devoted to the analysis of blow-up solutions for the fractional nonlinear Schr\"odinger equation with combined power-type nonlinearities \[ i\partial_t u-(-\Delta)^su+\lambda_1|u|^{2p_1}u+\lambda_2|u|^{2p_2}u=0, \] where $0<p_1<p_2<\frac{2s}{N-2s}$. Firstly, we obtain some sufficient conditions about existence of blow-up solutions, and then derive some sharp thresholds of blow-up and global existence by constructing some new estimates. Moreover, we find the sharp threshold mass of blow-up and global existence in the case $0<p_1<\frac{2s}{N}$ and $p_2=\frac{2s}{N}$. Finally, we investigate the dynamical properties of blow-up solutions, including $L^2$-concentration, blow-up rate and limiting profile.

Blow-up of p-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following $p$-Laplacian evolution equation with variable source power $$u_t(x,t)=\textrm{div}(| \nabla u|^{p-2} \nabla u)+u^{q(x)} \quad\textrm{in} \quad\Omega \times (0,T),$$ where $ \Omega$ is either a bounded domain or the whole space $\mathbb{R}^N$, and $ q(x)$ is a positive and continuous function defined in $ \Omega$ with $0＜q_-=\inf q(x)\leq q(x)\leq \sup q(x)=q_+＜\infty$. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of $q(x)$ and the structure of spatial domain $ \Omega$, compared with the case of constant source power. For the case that $\Omega$ is a bounded domain, the exponent $p-1$ plays a crucial role. If $q_+>p-1$, there exist blow-up solutions, while if $q_+＜p-1$, all the solutions are global. If $q_->p-1$, there exist global solutions, while for given $q_-＜p-1＜q_+$, there exist some function $q(x)$ and $\Omega$ such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case $\Omega= \mathbb{R}^N$, the Fujita phenomenon occurs if $1＜q_-\leq q_+\leq p-1+p/N$, while if $q_->p-1+p/N$, there exist global solutions.

Self-similar slip instability on interfaces with rate- and state-dependent friction.

We examine the development of a frictional instability, with diverging sliding rate, at the interface of elastic bodies in contact. Evolution of friction is determined by a slip rate and state dependence. Following Viesca (2016 Phys. Rev. E93, 060202(R). (doi:10.1103/PhysRevE.93.060202)), we show through an appropriate change of variable, the existence of blow-up solutions that are fixed points of a dynamical system. The solutions show self-similarity of the simple variety: separable dependence of time and space. For an interface with uniform frictional properties, there is a single-problem parameter. We examine the linear stability of these fixed points, as this problem parameter is varied. Specifically, we consider two archetypical elastic settings of the slip surface, in which interactions between points on the surface are either local or non-local. We show that, independent of the nature of elastic interactions, the fixed-points lose stability in the same matter as the parameter is increased towards a limit value: an apparently infinite sequence of Hopf bifurcations. However, for any value of the parameter, the nonlinear development of the instability is attraction, if not asymptotic convergence, towards these fixed points, owing to the existence of stable eigenmodes. For comparison, we perform numerical solutions of the original evolution equations and find precise agreement with the results of the analysis.

A new blow-up criterion for Gross–Pitaevskii equation

This paper is devoted to the study of the Gross–Pitaevskii equation, which describes the attractive Bose–Einstein condensate under a magnetic trap. By establishing an invariant set and applying the Gagliardo–Nirenberg type inequality, we then get the existence of blow-up solutions with arbitrary large initial energy.

Blow-up properties in the parabolic problems with anisotropic nonstandard growth conditions

In this paper, we study the parabolic problems with anisotropic nonstandard growth nonlinearities. We first give the existence and uniqueness of weak solutions in variable Sobolev spaces. Second, we use the energy methods to show the existence of blow-up solutions with negative or positive initial energy, respectively. Both the variable exponents and the coefficients make important roles in Fujita blow-up phenomena. Moreover, asymptotic properties of the blow-up solutions are determined.

Notes on global existence for the nonlinear Schrödinger equation involves derivative

In this paper, we consider the scattering for the nonlinear Schrodinger equation with small, smooth, and localized data. In particular, we prove that the solution of the quadratic nonlinear Schrodinger equation with nonlinear term |u|2 involving some derivatives in two dimension exists globally and scatters. It is worth to note that there exist blow-up solutions of these equations without derivatives. Moreover, for radial data, we prove that for the equation with p-order nonlinearity with derivatives, the similar results hold for \(p \geqslant \tfrac{{2d + 3}} {{2d - 1}} \) and d ≥ 2, which is lower than the Strauss exponents.

Boundedness and blow-up of solutions for a nonlinear elliptic system

The main objective in this paper is to obtain the existence results for bounded and unbounded solutions of some quasilinear elliptic systems. Related results as obtained here have been established recently in [C. O. Alves and A. R. F. de Holanda, Existence of blow-up solutions for a class of elliptic systems, Differ. Integral Eqs.26(1/2) (2013) 105–118]. Also, we present some references to give the connection between these types of problems with probability and stochastic processes, hoping that these are interesting for the audience of analysts likely to read this paper.

Blow-up criteria for the inhomogeneous nonlinear Schrödinger equation

In this paper, using the variational characteristic of the virial identity and a new estimate of the kinetic energy, we obtain a new sufficient condition for the existence of blow-up solutions.

Blow-Up Solutions and Global Existence for Quasilinear Parabolic Problems with Robin Boundary Conditions

We study the blow-up and global solutions for a class of quasilinear parabolic problems with Robin boundary conditions. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of blow-up solution, an upper bound for the “blow-up time,” an upper estimate of the “blow-up rate,” the sufficient conditions for the existence of global solution, and an upper estimate of the global solution are specified.