Finite Time Blow-up Of Solutions Research Articles

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.

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.

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).

Lower and upper bounds for the explosion times of a system of semilinear SPDEs

In this paper, we obtain lower and upper bounds for the blow-up times for a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of the explosive times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We provide lower and upper bounds for the probability of finite-time blow-up solution as well. The above obtained results are also extended for semilinear SPDEs forced by two-dimensional Brownian motions.

Global well-posedness of solutions for 2-D Klein–Gordon equations with exponential nonlinearity

This paper considers the global well-posedness of two-dimensional Klein–Gordon equations with exponential nonlinearity. By employing the potential well method, we conduct a comprehensive study on the global existence and finite time blowup of solutions by the requirement of the initial energy at three different initial energy levels.

On the Cauchy problem for the generalized double dispersion equation with logarithmic nonlinearity

Abstract In this paper, we investigate the global existence and finite time blow-up of solution for the Cauchy problem of one-dimensional fifth-order Boussinesq equation with logarithmic nonlinearity. Fist we prove the existence and uniqueness of local mild solutions in the energy space by means of the contraction mapping principle. Further under some restriction on the initial data, we establish the results on existence and uniqueness of global solutions and finite time blow-up of solutions by using the potential well method. Moreover, the sufficient and necessary conditions of global existence and finite time blow-up of solutions are given.

Global existence and blow-up in higher-dimensional Patlak-Keller-Segel system for multi populations

This work is concerned with a degenerate parabolic-elliptic Patlak-Keller-Segel system for multi populations in space dimensions d≥3. We first provide a sufficient condition for the global existence of weak solution to the Cauchy problem. Then the global results are obtained both for any large initial data in the sub-critical case and for small initial data in the super-critical case. Finally, the finite-time blow-up solutions are constructed for large initial data in the super-critical case.

A Globally Stable Self-Similar Blowup Profile in Energy Supercritical Yang-Mills Theory

This paper is concerned with the Cauchy problem for an energy-supercritical nonlinear wave equation in odd space dimensions that arises in equivariant Yang-Mills theory. In each dimension, there is a self-similar finite-time blowup solution to this equation known in closed form. It will be proved that this profile is stable in the whole space under small perturbations of the initial data. The blowup analysis is based on a recently developed coordinate system called hyperboloidal similarity coordinates and depends crucially on growth estimates for the free wave evolution, which will be constructed systematically for odd space dimensions in the first part of this paper. This allows to develop a nonlinear stability theory beyond the singularity.

Non-linear heat equation on the Hyperbolic space: Global existence and finite-time Blow-up

We consider the following Cauchy problem for the semi linear heat equation on the hyperbolic space : \begin{align} \left\{\begin{array}{ll} \partial_{t}u=\Delta_{\mathbb{H}^{n}} u+ f(u, t) & \hbox{ in }~ \mathbb{H}^{n}\times (0, T), \\ \quad u =u_{0} & \hbox{ in }~ \mathbb{H}^{n}\times \{0\}. \end{array}\right. \tag{0.1} \end{align} We study Fujita phenomena for the non-negative initial data $u_0$ belonging to $C(\mathbb{H}^{n}) \cap L^{\infty}(\mathbb{H}^{n})$ and for different choices of $f$ of the form $f(u,t) = h(t)g(u).$ It is well-known that for power nonlinearities in $u,$ the power weight $h(t) = t^q$ is sub-critical in the sense that non-negative global solutions exist for small initial data. On the other hand, (0.1) exhibits Fujita phenomena for the exponential weight $h(t) = e^{\mu t},$ i.e., there exists a critical exponent $ \mu^*$ such that if $\mu > \mu^*$, then all non-negative solutions blow-up in finite time and if $\mu \leq \mu^*$, there exists non-negative global solutions for small initial data. One of the main objectives of this article is to find an appropriate nonlinearity in $u$ so that (0.1) with the power weight $h(t) = t^q$ does exhibit Fujita phenomena. In the remaining part of this article, we study Fujita phenomena for exponential nonlinearity in $u.$ We further generalize some of these results to Cartan-Hadamard manifolds.

Finite time blow-up in a parabolic–elliptic Keller–Segel system with flux dependent chemotactic coefficient

A parabolic–elliptic Keller–Segel system ut=Δu−χ∇⋅(uf(|∇v|)∇v),0=Δv−M+u,with homogeneous Neumann boundary condition is considered in a radially symmetric domain Ω=BR(0)⊂RN(N≥3), where f(ξ)=(ξp−2(1+ξp)q−pp),ξ≥0,p≥2,1<q≤p<∞,and BR(0) is a N-dimensional ball of radius R>0. We assert that under a condition on the initial data, radial weak solutions blow-up in finite time when NN−1<q<2.

Optimal mass on the parabolic-elliptic-ODE minimal chemotaxis-haptotaxis in

In this paper, we study the Cauchy problem to a chemotaxis-haptotaxis model describing cancer invasion in . The main feature is to prove that 8π is the critical mass on initial data for distinguishing existence and blow-up of solutions to the model. Namely, when the initial mass is less than 8π, we prove global existence of solutions by constructing a proper free energy and using the Brezis-Merle type inequality. On the contrary, the finite time blow-up of solutions may occur if the initial mass is greater than 8π and the initial second moment is small enough. Comparing to the classical Keller-Segel model, we find that the haptotaxis has no effect on the critical mass of the model.

On Finite-Time Blow-Up Problem for Nonlinear Fractional Reaction Diffusion Equation: Analytical Results and Numerical Simulations

The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand on some previous blow-up results for the explicit values and numerical simulation of finite-time blow-up solutions for a semilinear fractional partial differential problem involving a positive power of the solution. We show the behavior solution of the fractional problem, and the numerical solution of the finite-time blow-up solution is also considered. Finally, some illustrative examples and comparisons with the classical problem with integer order are presented, and the validity of the results is demonstrated.

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 \(H^1\).