Finite Time Blow-up Of Solutions Research Articles

Boundedness and finite-time blow-up in a quasilinear parabolic–elliptic chemotaxis system with logistic source and nonlinear production

This paper deals with the quasilinear parabolic–elliptic chemotaxis system with logistic source and nonlinear production,{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+λu−μuκ,x∈Ω,t>0,0=Δv−Mf‾(t)+f(u),x∈Ω,t>0, where λ>0, μ>0, κ>1 andMf‾(t):=1|Ω|∫Ωf(u(x,t))dx, and D, S and f are functions generalizing the prototypesD(u)=(u+1)m−1,S(u)=u(u+1)α−1andf(u)=uℓ with m∈R, α>0 and ℓ>0. In the case m=α=ℓ=1, Fuest (2021) [5] obtained conditions for κ such that solutions blow up in finite time. However, in the above system boundedness and finite-time blow-up of solutions have been not yet established. This paper gives boundedness and finite-time blow-up under some conditions for m, α, κ and ℓ.

Exponential decay and blow-up results for a viscoelastic equation with variable sources

This work deals with a class of nonlinear viscoelastic wave equations with strong damping and variable exponent sources. The main interest of this paper compared to many previous works in the literature is able to use the idea of potential well method to study both the finite time blow-up for solutions starting from the unstable sets and decay estimate for global solutions starting in the potential wells.

On Well-Posedness and Concentration of Blow-Up Solutions for the Intercritical Inhomogeneous NLS Equation

We consider the focusing inhomogeneous nonlinear Schrödinger (INLS) equation in \({\mathbb {R}}^N\)$$\begin{aligned} i \partial _t u +\Delta u + |x|^{-b} |u|^{2\sigma }u = 0, \end{aligned}$$where \(N\ge 2\) and \(\sigma \), \(b>0\). We first obtain a small data global result in \(H^1\), which, in the two spatial dimensional case, improves the third author result in [22] on the range of b. For \(N\ge 3\) and \(\frac{2-b}{N}<\sigma <\frac{2-b}{N-2}\), we also study the local well posedness in \(\dot{H}^{s_c}\cap \dot{H}^1 \), where \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma }\). Sufficient conditions for global existence of solutions in \(\dot{H}^{s_c}\cap \dot{H}^1\) are also established, using a Gagliardo-Nirenberg type estimate. Finally, we study the \(L^{\sigma _c}\)-norm concentration phenomenon, where \(\sigma _c=\frac{2N\sigma }{2-b}\), for finite time blow-up solutions in \(\dot{H}^{s_c}\cap \dot{H}^1\) with bounded \(\dot{H}^{s_c}\)-norm. Our approach is based on the compact embedding of \(\dot{H}^{s_c}\cap \dot{H}^1\) into a weighted \(L^{2\sigma +2}\) space.

Minimal-mass blow-up solutions for nonlinear Schrödinger equations with an inverse potential

We consider the following nonlinear Schrödinger equation with an inverse potential: i∂u∂t+Δu+|u|4Nu±1|x|2σu=0in RN. From the classical argument, the solution with subcritical mass (‖u‖2<‖Q‖2) is global and bounded in H1(RN). Here, Q is the ground state of the mass-critical problem. Therefore, we are interested in the existence and behaviour of blow-up solutions for the threshold (‖u0‖2=‖Q‖2). Previous studies investigate the existence and behaviour of the critical-mass blow-up solution when the potential is smooth or unbounded but algebraically tractable. There exist no results when classical methods cannot be used, such as the inverse power type potential. However, in this paper, we construct a critical-mass finite-time blow-up solution. Moreover, we show that the blow-up solution converges to a certain blow-up profile in the virial space.

Nondispersive solutions to the mass critical half-wave equation in two dimensions

We consider the half-wave equation with mass critical in two dimensions First, we prove the existence of a family of traveling solitary waves. We then show the existence of finite-time blowup solutions with ground state mass where Q is the ground state solution of equation

Blow-up phenomenon for a semilinear pseudo-parabolic equation involving variable source

We consider a semilinear pseudo-parabolic equation with nonlinearities of variable exponent subject to Dirichlet boundary conditions We prove that the nonnegative classical solutions blow-up in finite time with arbitrary positive initial energy and suitable large initial values. Also, employing a differential inequality technique, we obtain an upper bound for blow-up time if and the initial value satisfies some conditions.

Relaxed parameter conditions for chemotactic collapse in logistic-type parabolic\u2013elliptic Keller\u2013Segel systems

We study the finite-time blow-up in two variants of the parabolic–elliptic Keller–Segel system with nonlinear diffusion and logistic source. In n-dimensional balls, we consider JLut=∇·((u+1)m-1∇u-u∇v)+λu-μu1+κ,0=Δv-1|Ω|∫Ωu+u\\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}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\nabla \\cdot ((u+1)^{m-1}\\nabla u -u \\nabla v) + \\lambda u - \\mu u^{1+\\kappa }, \\\\ 0 = \\Delta v - \\frac{1}{|\\Omega |} \\int \\limits _\\Omega u + u \\end{array}\\right. } \\end{aligned}$$\\end{document}and PEut=∇·((u+1)m-1∇u-u∇v)+λu-μu1+κ,0=Δv-v+u,\\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}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u_t = \\nabla \\cdot ((u+1)^{m-1}\\nabla u -u \\nabla v) + \\lambda u - \\mu u^{1+\\kappa }, \\\\ 0 = \\Delta v - v + u, \\end{array}\\right. } \\end{aligned}$$\\end{document}where lambda and mu are given spatially radial nonnegative functions and m, kappa > 0 are given parameters subject to further conditions. In a unified treatment, we establish a bridge between previously employed methods on blow-up detection and relatively new results on pointwise upper estimates of solutions in both of the systems above and then, making use of this newly found connection, provide extended parameter ranges for m,kappa leading to the existence of finite-time blow-up solutions in space dimensions three and above. In particular, for constant lambda , mu > 0, we find that there are initial data which lead to blow-up in (JL) if 0≤κ<min12,n-2n-(m-1)+ifm∈2n,2n-2nor0≤κ<min12,n-1n-m2ifm∈0,2n,\\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}$$\\begin{aligned} 0 \\le \\kappa&< \\min \\left\\{ \\frac{1}{2}, \\frac{n - 2}{n} - (m-1)_+ \\right\\}&\\quad \\text {if } m\\in \\left[ \\frac{2}{n},\\frac{2n-2}{n}\\right) \\\\ \\text { or }\\quad 0 \\le \\kappa&<\\min \\left\\{ \\frac{1}{2},\\frac{n-1}{n}-\\frac{m}{2}\\right\\}&\\quad \\text {if } m\\in \\left( 0,\\frac{2}{n}\\right) , \\end{aligned}$$\\end{document}and in (PE) if m in [1, frac{2n-2}{n}) and 0≤κ<min(m-1)n+12(n-1),n-2-(m-1)nn(n-1).\\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}$$\\begin{aligned} 0 \\le \\kappa < \\min \\left\\{ \\frac{(m-1) n + 1}{2(n-1)}, \\frac{n - 2 - (m-1) n}{n(n-1)} \\right\\} . \\end{aligned}$$\\end{document}

Initial boundary value problem for a strongly damped wave equation with a general nonlinearity

&lt;p style='text-indent:20px;'&gt;In this paper, a strongly damped semilinear wave equation with a general nonlinearity is considered. With the help of a newly constructed auxiliary functional and the concavity argument, a general finite time blow-up criterion is established for this problem. Furthermore, the lifespan of the weak solution is estimated from both above and below. This partially extends some results obtained in recent literatures and sheds some light on the similar effect of power type nonlinearity and logarithmic nonlinearity on finite time blow-up of solutions to such problems.&lt;/p&gt;

Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary

Inspired by the numerical evidence of a potential 3D Euler singularity by Luo-Hou [30,31] and the recent breakthrough by Elgindi [11] on the singularity formation of the 3D Euler equation without swirl with $C^{1,\alpha}$ initial velocity, we prove the finite time singularity for the 2D Boussinesq and the 3D axisymmetric Euler equations with boundary and $C^{1,\alpha}$ initial data for the velocity (and density in the case of Boussinesq equations). Our finite time blowup solution for the 3D Euler equations and the singular solution considered in [30,31] share many essential features, including the symmetry properties of the solution, the flow structure, and the sign of the solution in each quadrant, except that we use $C^{1,\alpha}$ initial velocity. We use a dynamic rescaling formulation and follow the general framework of analysis developed in [11]. We also use some strategy proposed in our recent joint work with Huang in [7] and adopt several methods of analysis in [11] to establish the linear and nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the 3D Euler or the 2D Boussinesq equations with $C^{1,\alpha}$ initial data will develop a finite time singularity. Moreover, the velocity field has finite energy before the singularity time. In the previous version of this paper, we proved the blowup results for the 3D axisymmetric Euler equations with initial data $(u_0^{\theta})^2, u_0^r, u_0^z \in C^{1,\alpha}$. Though the velocity $u^r, u^z$ in the axisymmetric setting is $C^{1,\alpha}$, our interpretation that the velocity is $C^{1,\alpha}$ is not correct since the velocity in 3D also depends on $u^{\theta}$, which is not $C^{1,\alpha}$. This oversight can be fixed easily with a minor change in the construction of the approximate steady state and a minor modification to localize the approximate steady state.

Finite-time blowup and ill-posedness in Sobolev spaces of the inviscid primitive equations with rotation

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. We construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order s>1, and suggests that a suitable space for the well-posedness is Gevrey class of order s=1, which is exactly the space of analytic functions.

Lower Bounds of Finite-Time Blow-Up of Solutions to a Two-Species Keller-Segel Chemotaxis Model

Lower Bounds of Finite-Time Blow-Up of Solutions to a Two-Species Keller-Segel Chemotaxis Model

Stable and unstable sets for damped nonlinear wave equations with variable exponent sources

In this paper, we study a class of nonlinear wave equations with variable exponent sources. By introducing a family of potential wells, we first prove the global existence of solutions with initial data in the potential wells and the finite time blow-up for solutions starting in the unstable sets. The boundedness and asymptotic behavior of global solutions are also concerned. Finally, we obtain the finite time blow-up with high energy initial data.

Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source

&lt;p style='text-indent:20px;'&gt;This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.&lt;/p&gt;

Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data

&lt;p style='text-indent:20px;'&gt;The Cauchy problem of one dimensional generalized Boussinesq equation is treated by the approach of variational method in order to realize the control aim, which is the control problem reflecting the relationship between initial data and global dynamics of solution. For a class of more general nonlinearities we classify the initial data for the global solution and finite time blowup solution. The results generalize some existing conclusions related this problem.&lt;/p&gt;

Blow-up of solutions for a parabolic Kirchhoff type equation with logarithmic nonlinearity

This study deals with the parabolic type Kirchhoff equation with logarithmic nonlinearity in a bounded domain. We obtain the finite time blow-up of solutions. This improves and extends some previous studies.

Blow-up and global existence for solutions to the porous medium equation with reaction and fast decaying density

We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density ρ(x) and a power-like reaction term up with p>1. The density decays fast at infinity, in the sense that ρ(x)∼|x|−q as |x|→+∞ with q≥2. In the case when q=2, if p is bigger than m, we show that, for large enough initial data, solutions blow-up in finite time and for small initial datum, solutions globally exist. On the other hand, in the case when q>2, we show that existence of global in time solutions always prevails. The case of slowly decaying density at infinity, i.e. q∈[0,2), is examined in Meglioli and Punzo (2020).

Existence of global solutions and blow-up of solutions for coupled systems of fractional diffusion equations

We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in \(\mathbb{R}_{+}\times \mathbb{R}^N\). Under appropriate conditions on the exponents and the orders of the fractional time derivatives, we present a critical value of the dimension N, for which global solutions with small data exist, otherwise solutions blow-up in finite time. Furthermore, the large time behavior of global solutions is discussed. For more information see https://ejde.math.txstate.edu/Volumes/2020/110/abstr.html

Global existence and blow-up of solutions for a parabolic equation involving the fractional p(x)-Laplacian

In this paper, we consider a nonlocal diffusion equation involving the fractional -Laplacian with nonlinearities of variable exponent type. Employing the subdifferential approach we establish the existence of local solutions. By combining the potential well theory with the Nehari manifold, we obtain the existence of global solutions and finite time blow-up of solutions. Moreover, we study the asymptotic stability of global solutions as time goes to infinity in some variable exponent Lebesgue spaces.

Global existence and blow-up of solutions of nonlocal diffusion problems with free boundaries

In this paper, we investigate some nonlocal diffusion problems with free boundaries. We first give the existence and uniqueness of local solution by the ODE basic theory and the contraction mapping principle. Then we provide a complete classification for the global existence and finite time blow-up of solutions. Moreover, estimates of blow-up rate and blow-up time are also obtained for the blow-up solution.

Blow-Up Phenomena of a Cancer Invasion Model with Nonlinear Diffusion and Haptotaxis Term

In this paper, we consider a nonlinear cancer invasion mathematical model with proliferation, growth and haptotaxis effects. We obtain lower bounds for the finite-time blow-up of solutions of the considered system with nonlinear diffusion operator when blow-up occurs. We have assumed both the Dirichlet and Neumann boundary conditions in $${\mathbb {R}}^n, n\ge 2$$ to attain the desire result.