Finite Time Blow-up Of Solutions Research Articles

Critical mass for Keller–Segel systems with supercritical nonlinear sensitivity

This paper is concerned with the following radially symmetric Keller–Segel systems with nonlinear sensitivity [Formula: see text] and [Formula: see text], posed on [Formula: see text] [Formula: see text] and subjected to homogeneous Neumann boundary conditions. It is well-known that [Formula: see text] is the critical exponent of the systems in the sense that all solutions exist globally if [Formula: see text] and there exist finite-time blowup solutions if [Formula: see text]. Here we consider the supercritical case [Formula: see text] and show a critical mass phenomenon. Precisely, we prove that there exists a critical mass [Formula: see text] such that (1) for arbitrary nonincreasing nonnegative initial data [Formula: see text] with [Formula: see text] and [Formula: see text], the corresponding solution blows up in finite time if [Formula: see text], and if [Formula: see text] we can only prove that the solution blows up in finite time or infinite time; (2) for some nonincreasing nonnegative initial data with [Formula: see text], the corresponding solutions are globally bounded. Our results extend that of Winkler’s paper [M. Winkler, How unstable is spatial homogeneity in Keller–Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic–elliptic cases, Math. Ann. 373 (2019) 1237–1282], where he proved similar results for the system with [Formula: see text].

Local Solvability, Blow-Up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: III. Cauchy Problems

Three Cauchy problems for Sobolev-type equations with a common linear part from the theory of ion acoustic and drift waves in a plasma are considered. The problems are reduced to equivalent integral equations. We prove the existence of unextendable solutions for two problems and the existence of a local-in-time solution for the third problem. For one of the problems, by applying a modified method of Kh.A. Levin, sufficient conditions for finite time blow-up of solutions are obtained and an upper bound for the solution blow-up time is found. For another problem, S.I. Pohozaev’s nonlinear capacity method is used to obtain a finite time blow-up result and two results concerning the nonexistence of even local solutions, and an upper bound for the solution blow-up time is obtained as well.

Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric Euler Equations Using Neural Networks.

Whether there exist finite-time blow-up solutions for the 2D Boussinesq and the 3D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks, that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate physics-informed neural networks could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.

Sharp scattering for focusing intercritical NLS on high-dimensional waveguide manifolds

We study the focusing intercritical NLS NLS $$\begin{aligned} i\partial _t u+\Delta _{x,y}u=-|u|^\alpha u \end{aligned}$$ on the semiperiodic waveguide manifold $$\mathbb {R}^d_x\times \mathbb {T}_y$$ with $$d\ge 5$$ and $$\alpha \in (\frac{4}{d},\frac{4}{d-1})$$ . In the case $$d\le 4$$ , with the aid of the semivirial vanishing theory (Luo in Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022), the author was able to construct a sharp threshold, which being uniquely characterized by the ground state solutions, that sharply determines the bifurcation of global scattering and finite time blow-up solutions in dependence of the sign of the semivirial functional. As the derivative of the nonlinear potential is no longer Lipschitz in $$d\ge 5$$ and the underlying domain possesses an anisotropic nature, the proof in Luo (Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022), which makes use of the concentration compactness principle, can not be extended to higher dimensional models. In this paper, we exploit a well-tailored adaptation of the interaction Morawetz–Dodson–Murphy (IMDM) estimates, which were only known to be applicable on Euclidean spaces, into the waveguide setting, in order to prove that the large data scattering result formulated in Luo (Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022) continues to hold for all $$d\ge 5$$ . Together with Tzvetkov–Visciglia (Rev. Mat. Iberoam. 32:1163–1188, 2016) and the author (Luo in Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022), we thus give a complete characterization of the large data scattering for (NLS) in both defocusing and focusing case and in arbitrary dimension.

Qualitative properties of solution to a viscoelastic Kirchhoff-like plate equation

This paper is concerned with the initial boundary value problem for viscoelastic Kirchhoff-like plate equations with rotational inertia, memory, p-Laplacian restoring force, weak damping, strong damping, and nonlinear source terms. We establish the local existence and uniqueness of the solution by linearization and the contraction mapping principle. Then, we obtain the global existence of solutions with subcritical and critical initial energy by applying potential well theory. Then, we prove the asymptotic behavior of the global solution with positive initial energy strictly below the depth of the potential well. Finally, we conduct a comprehensive study on the finite time blow-up of solutions with negative initial energy, null initial energy, and positive initial energy strictly below the depth of the potential well and arbitrary positive initial energy, respectively.

Pursuit-evasion dynamics for Bazykin-type predator-prey model with indirect predator taxis

We study a pursuit-evasion diffusive predator-prey model which combines prey-taxis in predators with evasive defense strategy of prey being capable to move in the opposite direction to the gradient of a chemical signal secreted by the predators (indirect predator taxis). The kinetic part of the model extends the Rosenzweig MacArthur predator-prey model by assuming an intraspecific competition among predators, as in the classical Bazykin model. The prey-taxis takes into account density-dependent velocity suppression of predators while chasing the prey. The assumptions enable us to prove the existence of global-in-time classical solutions for space dimension n≤3 which are not expected to exist for the Rosenzweig MacArthur model according to numerical simulations which depict a finite time blow-up of solutions for n=2.

Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial case

In this paper we consider the inhomogeneous nonlinear Schrödinger (INLS) equation $$\begin{aligned} i \partial _t u +\Delta u +|x|^{-b} |u|^{2\sigma }u = 0, \,\,\, x \in {\mathbb {R}}^N \end{aligned}$$with \(N\ge 3\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma }\) satisfies \(0<s_c<1\). In a previous work, for radial initial data in \(\dot{H}^{s_c}\cap \dot{H}^1\), we prove the existence of blow-up solutions and also a lower bound for the blow-up rate. Here we extend these results to the non-radial case. We also prove an upper bound for the blow-up rate and a concentration result for general finite time blow-up solutions in \(H^1\).

Construction of a Family of Stable Finite-Time Blowup Solutions for the Viscous Boussinesq System

Construction of a Family of Stable Finite-Time Blowup Solutions for the Viscous Boussinesq System

Well-posedness and dynamic properties of solutions to a class of fourth order parabolic equation with mean curvature nonlinearity

The well-posedness and dynamic properties of solutions to a class of fourth order parabolic equations with mean curvature nonlinearity are studied. More specifically, we first prove the existence and uniqueness of strong solutions by semigroup method. Then, we establish conditions for the global existence and finite time blow-up of solutions with subcritical, critical and supercritical initial energies. At the same time, we study the exponential decay and $ \omega $-limit set of the global solutions, and the upper bound of the blow-up time of the blow-up solutions. In addition, we also study the existence and regularity of ground state solutions for the steady state problems.

Global existence and finite time blow-up of solutions for a class of Dirichlet-to-Neumann operator heat flow equations with critical growth

Global existence and finite time blow-up of solutions for a class of Dirichlet-to-Neumann operator heat flow equations with critical growth

Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials

&lt;abstract&gt;&lt;p&gt;In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.&lt;/p&gt;&lt;/abstract&gt;

Small data finite-time blow-up solutions for the nonlinear Schrödinger equation in general dimensions

The purpose of this paper is to show the existence of finite-time blow-up solution for the nonlinear Schrödinger equation in general spatial dimension which starts with the initial data as small as we like. We construct a solution of the nonlinear Schrödinger equation defined in negative global time and blows-up at positive finite time. We derive decay estimates for the solutions in the negative time direction by using interpolation inequalities in the weighted Lebesgue spaces.

The blow-up solutions for fractional heat equations on torus and Euclidean space

We produce a finite time blow-up solution for nonlinear fractional heat equation (\(\partial _t u + (-\Delta )^{\beta /2}u=u^k\)) in modulation and Fourier amalgam spaces on the torus \({\mathbb {T}}^d\) and the Euclidean space \({\mathbb {R}}^d.\) This complements several known local and small data global well-posedness results in modulation spaces on \({\mathbb {R}}^d.\) Our method of proof rely on the formal solution of the equation. This method should be further applied to other non-linear evolution equations.

Finite-time blowup and existence of global solutions for a logarithmic semilinear hyperbolic equation

In this paper we consider the semilinear wave equation with the product of logarithmic and polynomial nonlinearities and establish the global existence and finite-time blowup of solutions by using the potential well method.

The 3D Nonlinear Schrödinger Equation with a Constant Magnetic Field Revisited

In this paper, we revisit the Cauchy problem for the three dimensional nonlinear Schrödinger equation with a constant magnetic field. We first establish sufficient conditions that ensure the existence of global in time and finite time blow-up solutions. In particular, we derive sharp thresholds for global existence versus blow-up for the equation with mass-critical and mass-supercritical nonlinearities. We next prove the existence and orbital stability of normalized standing waves which extend the previous known results to the mass-critical and mass-supercritical cases. To show the existence of normalized solitary waves, we present a new approach that avoids the celebrated concentration-compactness principle. Finally, we study the existence and strong instability of ground state standing waves which greatly improve the previous literature.

Blow-up for the Timoshenko-type equation with variable exponents

The finite time blow-up of solutions to a nonlinear Timoshenko-type equation with variable exponents is studied. More concretely, we prove that the solutions blow up in finite time with positive initial energy. Also, the existence of finite time blow-up solutions with arbitrarily high initial energy is established. Meanwhile, the upper and lower bounds of the blow-up time are derived. These results deepen and generalize the ones obtained in [Nonlinear Anal. Real World Appl., 61: Paper No. 103341, 2021].

Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations

Inspired by the numerical evidence of a potential 3D Euler singularity [54, 55], we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in [54, 55] for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in [54, 55] share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in [11] to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the $$C^\gamma $$ norm of the density $$\theta $$ with $$\gamma \approx 1/3$$ is uniformly bounded up to the singularity time.

Improved blowup time estimates for fourth-order damped wave equations with strain term and arbitrary positive initial energy

We propose a new differential inequality that improve the upper bound of the blowup time estimate for nonlinear fourth-order damped wave equations with strain term and arbitrary positive initial energy. We also give two new initial conditions to expand the range of the initial data leading to the finite time blowup of solutions. We obtain a sharp result of finite time blowup for the special case of the new differential inequality.We illustrate our results with some simulations.

Solvability of certain one-dimensional semilinear hyperbolic systems and hyperbolic Liouville equation

We study the initial-value problem for semilinear hyperbolic systems in one space dimension. When the power of nonlinearity is even , we have global existence of solution which is related with the repulsive Liouville equation. When n = 2k−1, we may have a finite time blow-up solution which is related with the attractive Liouville equation.

Critical curve for a two-species chemotaxis model with two chemicals in * *Supported by National Natural Science Foundation of China (12171218), LiaoNing Revitalization Talents Program (XLYC2007022) and Key Project of Education Department of Liaoning Province (LJKZ0083).

This paper deals with a two-species chemotaxis model with two chemicals in . In our previous work (2019 Nonlinearity 32 4762–78), the critical mass was obtained that the solutions exist globally if m 1 m 2 − 4π(m 1 + m 2) < 0, and the finite time blow-up of solutions may occur if m 1 m 2 − 4π(m 1 + m 2) > 0, where m 1 and m 2 describe the initial mass of the two species, respectively. In the present paper we furthermore determine that the critical situation belongs to the global existence case, namely, the system admits global solutions if m 1 m 2 − 4π(m 1 + m 2) = 0. To apply the key logarithmic Hardy–Littlewood–Sobolev inequality of vector form for the critical case, we should establish a positive lower bound for L 1-norms of the two species in exterior domains uniformly for t > 0.