Blow-up Solutions Research Articles

On the blow-up for a Kuramoto–Velarde type equation

It is known that the Kuramoto-Velarde equation is globally well-posed on Sobolev spaces in the case when the parameters γ1 and γ2 involved in the non-linear terms verify γ2=γ12 or γ2=0. In the complementary case of these parameters, the global existence or blow-up of solutions is a completely open (and hard) problem. Motivated by this fact, in this work we consider a non-local version of the Kuramoto-Velarde equation. This equation allows us to apply a Fourier-based method and, within the framework γ2≠γ12 and γ2≠0, we show that large values of these parameters yield a blow-up in finite time of solutions in the Sobolev norm. As a complement to it, we address an alternative result on the finite-time blow-up of smooth solutions by considering a virial-type estimate.

Blow-up of the critical norm for a supercritical semilinear heat equation

We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation u_{t}=\Delta u+|u|^{p-1}u in an arbitrary C^{2+\alpha} domain of \mathbf{R}^{n} . In the range p>p_{S}:=(n+2)/(n-2) , we show that the critical norm must be unbounded near the blow-up time, where the type I blow-up condition is not imposed. The range p>p_{S} is optimal in view of the existence of type II blow-up solutions with bounded critical norm for p=p_{S} .

On Type I Blowup of Some Nonlinear Heat Equations With a Potential

In this paper, we are concerned with the following initial-boundary value problem{ut=Δu+Q(|x|)|u|p−1u,x∈BR,t>0u(x,t)=0,x∈∂BR,t>0u(x,0)=u0(x),x∈BR, where p≥ps:=N+2N−2, u0∈L∞(BR), and Q(r)∈C1([0,R]), 0<C_≤Q(r)≤C‾<∞,Q′(r)≤0. We extend the asymptotic behavior results, which is well-known when Q is constant according to Matano-Merle (cf. [26]), for the blow-up solutions. More precisely, we show that when ps≤p<p⁎, the blowup of radial solution to this problem is always of Type I, and hence partially generalize the conclusions in [26] for Q≡1. This extension is nontrivial due to the appearance of Q. The quasi-monotonicity formula established by the third author and Cheng in [8] allows us to use an energy method to get a priori estimates on the rescaled solutions. The contraction mapping principle shows the existence of singular stationary solutions to an associated elliptic equation with a potential. In the end, the properties of zero number for solutions lead to the nonexistence of type II singularity for the problem.

On the lifespan of nonzero background solutions to a class of focusing nonlinear Schrödinger equations

The global solvability in time and the potential for blow-up of solutions to non-integrable focusing nonlinear Schrödinger equations with nonzero boundary conditions at infinity present challenges that are less explored and understood compared to the case of zero boundary conditions. In this work, we address these questions by establishing estimates on the lifespan of solutions to non-integrable equations involving a general class of nonlinearities. These estimates depend on the size of the initial data, the growth of the nonlinearity, and relevant quantities associated with the amplitude of the background. The estimates provide quantified upper bounds for the minimum guaranteed lifespan of solutions. Qualitatively, for small initial data and background, these upper bounds suggest long survival times consistent with global existence of solutions. On the other hand, for larger initial data and background, the estimates indicate the potential for the intriguing phenomenon of instantaneous collapse in finite time. These qualitative theoretical results are illustrated via numerical simulations. Furthermore, importantly, the numerical findings motivate the proof of improved theoretical upper bounds that provide excellent quantitative agreement with the order of the numerically identified lifespan of solutions.

Blowup dynamics for smooth equivariant solutions to energy critical Landau-Lifschitz flow

In this paper, we study the energy critical 1-equivariant Landau-Lifschitz flow mapping R2 to S2 with arbitrary given coefficients ρ1∈R,ρ2>0. We prove that there exists a codimension one smooth well-localized set of initial data arbitrarily close to the ground state which generates type-II finite-time blowup solutions, and give a precise description of the corresponding singularity formation. In our proof, both the Schrödinger part and the heat part play important roles in the construction of approximate solutions and the mixed energy/Morawetz functional. However, the blowup rate is independent of the coefficients.

Critical exponent to a cancer invasion model with nonlinear diffusion

This paper is concerned with a cancer invasion model that incorporates porous medium diffusion (Δum) and extracellular matrix remodeling effects [ηω(1 − u − ω)] in a bounded domain of RN (N ≥ 2). Rich achievements have been achieved for the case η = 0 in the past ten years for the nonlinear diffusion case, but there is no any progress for η &amp;gt; 0. In this paper, we pay our attention to the global existence of solutions of the case η &amp;gt; 0, and establish the critical exponent m*=2N−2N of global solvability. More precisely, if m &amp;gt; m*, the solution will always exist globally, while if m &amp;lt; m*, there exist blow-up solutions. In this system, the remodeling effect of extracellular matrix [ηω(1 − u − ω)] bring some essential difficulties to the estimation of the haptotactic term, so the main technique we used is completely different from the case of η = 0.

Global existence and blow-up of solutions for mixed local and nonlocal hyperbolic equations

In this paper, we consider the following mixed local and nonlocal hyperbolic equation: u t t − Δ u + μ ( − Δ ) s u = | u | p − 2 u , in Ω × R + , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , in Ω , u ( x , t ) = 0 , in ( R N ∖ Ω ) × R 0 + , where s ∈ ( 0 , 1 ), N &gt; 2, p ∈ ( 2 , 2 s ∗ ], μ is a nonnegative real parameter, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω, Δ is the Laplace operator, ( − Δ ) s is the fractional Laplace operator. By combining the Galerkin approach with the modified potential well method, we obtain the global existence, vacuum isolating, and blow-up of solutions for the aforementioned problem, provided certain assumptions are fulfilled. Specifically, we study the existence of global solutions for the above problem in the cases of subcritical and critical initial energy levels, as well as the finite time blow-up of solutions. Then, we investigate the blow-up of solutions for the above problem in the case of supercritical initial energy level, as well as upper and lower bounds of blow-up time of solutions.

Finite time blowup of solutions of the hydrostatic Euler equations

Finite time blowup of solutions of the hydrostatic Euler equations

Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.

On the local existence and blow-up solutions to a quasi-linear bi-hyperbolic equation with dynamic boundary conditions

This note aims to study the existence of the local solutions and derive a blow-up result for a quasi-linear bi-hyperbolic equation with dynamic boundary conditions. We use the maximal monotone operator theory to demonstrate the solution’s local well-posedness, and a concavity method to establish the blow-up result.

The Classical Boundary Blow-Up Solutions for a Class of Gaussian Curvature Equations

The Classical Boundary Blow-Up Solutions for a Class of Gaussian Curvature Equations

On Blow-Up and Explicit Soliton Solutions for Coupled Variable Coefficient Nonlinear Schrödinger Equations

On Blow-Up and Explicit Soliton Solutions for Coupled Variable Coefficient Nonlinear Schrödinger Equations

Complex-plane singularity dynamics for blow up in a nonlinear heat equation: analysis and computation

Abstract Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on tracking the dynamics of the singularities in the complexified space domain all the way from the initial time until the blow-up time, which occurs when the singularities reach the real axis. This widely applicable approach gives forewarning of the possibility of blow up and an understanding of the influence of singularities on the solution behaviour on the real axis, aiding the (perhaps surprisingly involved) asymptotic analysis of the real-line behaviour. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales over which blow up is approached. The solution to the nonlinear heat equation in the complex spatial plane is shown to be related asymptotically to a nonlinear ordinary differential equation. This latter equation is studied in detail, including its computation on multiple Riemann sheets, providing further insight into the singularities of blow-up solutions of the nonlinear heat equation when viewed as multivalued functions in the complex space domain and illustrating the potential intricacy of singularity dynamics in such (non-integrable) nonlinear contexts.

Optimal decay rate and blow-up of solution for a classical thermoelastic system with viscoelastic damping and nonlinear sources

Optimal decay rate and blow-up of solution for a classical thermoelastic system with viscoelastic damping and nonlinear sources

Self-similar blow-up solutions in the generalised Korteweg-de Vries equation: spectral analysis, normal form and asymptotics

In the present work we revisit the problem of the generalised Korteweg–de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter p, here at p = 5. We provide a normal form of the associated collapse dynamics, and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterise the linearisation spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterise. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schrödinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for p > 5 in the co-exploding frame.

Blow-Up of Solutions for the Fourth-Order Schrödinger Equation with Combined Power-Type Nonlinearities

Blow-Up of Solutions for the Fourth-Order Schrödinger Equation with Combined Power-Type Nonlinearities

Equatorial blowup and polar caps in drop electrohydrodynamics

We illuminate effects of surface-charge convection intrinsic to leaky-dielectric electrohydrodynamics by analyzing the symmetric steady state of a circular drop in an external field at arbitrary electric Reynolds number ReE. In formulating the problem, we identify an exact factorization that reduces the number of dimensionless parameters from four—ReE and the conductivity, permittivity and viscosity ratios—to two: a modified electric Reynolds number Rẽ and a charging parameter ϖ. In the case ϖ&lt;0, where charge relaxation in the drop phase is slower than in the suspending phase, and, as a consequence, the interface polarizes antiparallel to the external field, we find that above a critical Rẽ value the solution exhibits a blowup singularity such that the surface-charge density diverges antisymmetrically with the −1/3 power of distance from the equator. We use local analysis to uncover the structure of that blowup singularity, wherein surface charges are convected by a locally induced flow towards the equator where they annihilate. To study the blowup regime, we devise a numerical scheme encoding that local structure where the blowup prefactor is determined by a global charging-annihilation balance. We also employ asymptotic analysis to construct a universal problem governing the blowup solutions in the regime Rẽ≫1, far beyond the blowup threshold. In the case ϖ&gt;0, where charge relaxation is faster in the drop phase and the interface polarizes parallel to the external field, we numerically observe and asymptotically characterize the formation at large Rẽ of stagnant, perfectly conducting surface-charge caps about the drop poles. The cap size grows and the cap voltage decreases monotonically with increasing conductivity or decreasing permittivity of the drop phase relative to the suspending phase. The flow in this scenario is nonlinearly driven by electrical shear stresses at the complement of the caps. In both polarization scenarios, the flow at large Rẽ scales linearly with the magnitude of the external field, contrasting the familiar quadratic scaling under weak fields. Published by the American Physical Society 2024

Prevention of infinite-time blowup by slightly super-linear degradation in a Keller–Segel system with density-suppressed motility

An initial-Neumann boundary value problem for a Keller–Segel system with density-suppressed motility and source terms is considered. Infinite-time blowup of the classical solution was previously observed for its source-free version when dimension N⩾2 . In this work, we prove that with any source term involving a slightly super-linear degradation effect on the density, of a growth order of slog⁡s at most, the classical solution is uniformly-in-time bounded when N⩽3 , thus preventing the infinite-time explosion detected in the source-free counter-part. The cornerstone of our proof lies in an improved comparison argument and a construction of an entropy inequality.

Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential

We consider the magnetic nonlinear inhomogeneous Schrödinger equationi∂tu−(−i∇+α|x|2(−x2,x1))2u=±|x|−ϱ|u|p−1u,(t,x)∈R×R2, where α∈R∖Z,ϱ>0,p>1. We establish a dichotomy between global existence and scattering versus finite-time blow-up for energy solutions below the ground state threshold in the inter-critical regime. The scattering result is obtained by employing the novel approach developed by Dodson and Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Am. Math. Soc. (2017)). We employ Tao's scattering criteria and Morawetz estimates as the foundation of this method. The novelty of our approach lies in two aspects: firstly, we explore the case of nonzero ϱα, and secondly, we consider general energy initial data that need not be radially symmetric. The specific instance of α=0, known as INLS, has garnered significant attention in recent years. Additionally, X. Gao and C. Xu recently investigated the case of ϱ=0 in the homogeneous regime, establishing scattering theory for spherically symmetric data in their work (Scattering theory for NLS with inverse-square potential in 2D, J. Math. Anal. Appl. (2020)). In the radial framework, the aforementioned problem can be translated to the INLS with an inverse square potential, which has been extensively studied in higher-dimensional spaces. The Hardy inequality ‖|x|−1f‖L2(RN)≤2N−2‖∇f‖L2(RN), which establishes the norm equivalence ‖f‖H1≃‖f‖H1+‖|x|−1f‖L2, does not hold in two space dimensions. Consequently, it remains unclear how to handle the NLS with an inverse square potential in the H1 framework for two-dimensional space. This article stands out as the first one to address the NLS with Aharonov-Bohm magnetic potential within the inhomogeneous regime, specifically when ϱ≠0.

On a Viscoelastic Plate Equation with Logarithmic Nonlinearity and Variable-Exponents: Global Existence, General Decay and Blow-Up of Solutions

On a Viscoelastic Plate Equation with Logarithmic Nonlinearity and Variable-Exponents: Global Existence, General Decay and Blow-Up of Solutions