Blow-up Phenomena Research Articles

Bound/positivity preserving SAV schemes for the Patlak-Keller-Segel-Navier-Stokes system

The Patlak-Keller-Segel-Navier-Stokes system describes the biological chemotaxis phenomenon in the fluid environment. It is a coupled nonlinear system with unknowns being the cell density, the concentration of chemoattractants, the fluid velocity and the pressure, and it satisfies an energy dissipation law, preserves the bound/positivity and mass of the cell density. We develop in this paper a class of scalar auxiliary variable (SAV) schemes with relaxation which preserve these properties unconditionally at the discrete level, and only require solving decoupled linear systems with constant coefficients at each time step. We present ample numerical results to validate these schemes, simulate the chemotactic non-aggregation and aggregation with a saturation concentration, as well as investigate the blow-up phenomenon.

Blow-up data for a two-component Camassa-Holm system with high order nonlinearity

This paper is concerned with the Cauchy problem for a two-component Camassa-Holm system with high order nonlinearity, which is a multi-component extension of the Fokas-Olver-Rosenau-Qiao equation. Firstly, we state the local well-posedness for the Cauchy problem of the system in the framework of Sobolev-Besov spaces. Then, we establish the precise blow-up mechanism for the strong solutions by means of the transport equation theory. As is well-known, the H1-norm conservation law of the velocity component is crucial to study blow-up phenomena of the single Camassa-Holm type equation. However, it is no longer available for our nonlinear coupled system. To overcome this difficulty, we derive some suitable conservation laws by sufficiently exploiting the fine structure of the system. Based on which, we finally construct several new blow-up strong solutions with certain initial profiles in finite time.

Qualitative Properties of Solutions of Equations and Inequalities with KPZ-Type Nonlinearities

For quasilinear partial differential and integrodifferential equations and inequalities containing nonlinearities of the Kardar—Parisi—Zhang type, various (old and recent) results on qualitative properties of solutions (such as the stabilization of solutions, blow-up phenomena, long-time decay of solutions, and others) are presented. Descriptive examples demonstrating the Bitsadze approach (the technique of monotone maps) applied in this research area are provided.

A reliable multi-resolution collocation algorithm for nonlinear Schrödinger equation with wave operator

The solution of a nonlinear hyperbolic Schrödinger equation (NHSE) is proposed in this paper using the Haar wavelet collocation technique (HWCM). The central difference technique is applied to handle the temporal derivative in the NHSE and the finite Haar functions are introduced to approximate the space derivatives. After linearizing the NHSE, it is transformed into full algebraic form with the help of finite difference and Haar wavelets approximation. Solving this well-conditional system of the algebraic equation, we obtained the required solution. Theoretical convergence and stability analysis of HWCM is also performed for two-dimensional NHSE which is supported by the experimental rate of convergence. The numerical findings for the moving soliton wave in the form of are explored in depth using Haar wavelets. The propagation of soliton waves is captured accordingly and the time blow-up phenomenon has also been handled by the proposed HWCM because of the well-conditional behaviour of the transformed algebraic equations. Several examples are presented to demonstrate the proposed method and the results are found correct and efficient. In the last example, we have considered a practical case that has no exact solution.

A blow-up phenomenon for a non-local Liouville-type equation

We consider the non-local Liouville equation $${\left( { - \Delta } \right)^{{1 \over 2}}}u = {h_\varepsilon }{e^u} - 1\,\,\,\,\,{\rm{in}}\,\,{\mathbb{S}^1},$$ corresponding to the prescription of the geodesic curvature on the circle. We build a family of solutions which blow up, when hε approaches a function h as ε → 0, at a critical point of the harmonic extension of h provided some generic assumptions are satisfied.

Blow-up phenomena and global existence for nonlinear parabolic problems under nonlinear boundary conditions

<abstract><p>In this paper, we consider an initial-boundary value parabolic problem under nonlinear Neumann boundary conditions. By virtue of the modified differential inequality, lower bounds for the blow-up time of the solution are derived in higher dimensional spaces. An upper bound for the blow-up time are specified under appropriate assumptions on the functions $ a, b, f, g, h $ and $ u_0 $.</p></abstract>

High-order finite difference approximation of the Keller-Segel model with additional self- and cross-diffusion terms and a logistic source

<abstract><p>In this paper, we consider the Keller-Segel chemotaxis model with self- and cross-diffusion terms and a logistic source. This system consists of a fully nonlinear reaction-diffusion equation with additional cross-diffusion. We establish some high-order finite difference schemes for solving one- and two-dimensional problems. The truncation error remainder correction method and fourth-order Padé compact schemes are employed to approximate the spatial and temporal derivatives, respectively. It is shown that the numerical schemes yield second-order accuracy in time and fourth-order accuracy in space. Some numerical experiments are demonstrated to verify the accuracy and reliability of the proposed schemes. Furthermore, the blow-up phenomenon and bacterial pattern formation are numerically simulated.</p></abstract>

Blow-up phenomena for a p(x)-biharmonic heat equation with variable exponent

In this paper, we deal with a p(x)-biharmonic heat equation with variable exponent under Dirichlet boundary and initial condition. We prove the blow up of solutions under suitable conditions.

Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity

<abstract><p>In this work, the initial-boundary value problem for the global dynamical properties of solutions to a class of finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied. With the help of the Nehari flow and Levine's concavity method, we establish some sharp-like threshold classifications of the initial data under sub-critical, critical and supercritical initial energy levels, that is, we describe the size of an initial data set. It requires the presumption that the initial data starting from one region of phase space have uniform global dynamical behavior, which means that the solution exists globally and decays via energy estimates that ultimately result in the solution tending to zero in the forward time. For the case in which the initial data corresponds to another region, we prove that the solutions related to these initial data are subject to blow-up phenomena in a finite time. In addition, we estimate the corresponding upper bound of the lifespan of the blow-up solution.</p></abstract>

Blow-up solution and analyticity to a generalized Camassa-Holm equation

<abstract><p>A generalized Camassa-Holm equation with higher order nonlinear term was studied. First, we give a new blow-up criterion and a new blow-up phenomenon to the Cauchy problem for the equation under some conditions. Then, we focus on the analytical solutions for the equation. Finally, we prove the analyticity of solution for the equation.</p></abstract>

Blow-up phenomenon to a generalized Camassa-Holm equation

Blow-up phenomenon to a generalized Camassa-Holm equation

Blow-up dynamic of solution to the semilinear Moore-Gibson-Thompson equation with memory terms

<abstract><p>This article is mainly concerned with the formation of singularity for a solution to the Cauchy problem of the semilinear Moore-Gibson-Thompson equation with general initial values and different types of nonlinear memory terms $ N_{\gamma, \, q}(u) $, $ N_{\gamma, \, p}(u_{t}) $, $ N_{\gamma, \, p, \, q}(u, \, u_{t}) $. The proof of the blow-up phenomenon for the solution in the whole space is based on the test function method ($ \psi(x, t) = \varphi_{R}(x)D_{t|T}^{\alpha}(w(t)) $). It is worth pointing out that the Moore-Gibson-Thompson equation with memory terms can be regarded as an approximation of the nonlinear Moore-Gibson-Thompson equation when $ \gamma\rightarrow 1^{-} $. To the best of our knowledge, the results in Theorems 1.1–1.3 are new.</p></abstract>

Discontinuous Galerkin method for blow-up solutions of nonlinear wave equations

e develop and study an explicit time-space discrete discontinuous Galerkin finite element method to approximate the solution of one-dimensional nonlinear wave equations. We show that the numerical scheme is stable if a nonuniform time mesh is considered. We also investigate the blow-up phenomena and we prove that under weak convergence assumptions, the numerical blow-up time tends toward the theoretical one. The validity of our results is confirmed throughout several examples and benchmarks.

Finite-time blow-up in a two-species chemotaxis-competition model with single production

This paper is concerned with blow-up of solutions to a two-species chemotaxis-competition model with production from only one species. In previous papers there are a lot of studies on boundedness for a two-species chemotaxis-competition model with productions from both two species. On the other hand, finite-time blow-up was recently obtained under smallness conditions for competitive effects. Now, in the biological view, the production term seems to promote blow-up phenomena; this implies that the lack of the production term makes the solution likely to be bounded. Thus, it is expected that there exists a solution of the system with single production such that the species which does not produce the chemical substance remains bounded, whereas the other species blows up. The purpose of this paper is to prove that this conjecture is true.

Existence and blowup of solutions for non-divergence polytropic variation-inequality in corn option trading

<abstract><p>This paper focuses on a class of variation-inequality problems involving non-divergence polytropic parabolic operators. The penalty method is employed, along with the Leray Schauder fixed point theory and limit progress, to determine the existence of solutions. The study also delves into the blow-up phenomena of the solution, revealing that under certain conditions, the solution will blow up in finite time.</p></abstract>

Blowup phenomena for a fourth order wave equation at high initial energy level

Blowup phenomena for a fourth order wave equation at high initial energy level

Blow-up Phenomena for a Reaction-diffusion Equation with Nonlocal Gradient Terms

In this paper, we investigate blow-up phenomena of the solution to a reaction-diffusion equation with nonlocal gradient absorption terms under Robin boundary condition on a bounded star-shaped region. Based on the method of auxiliary function and the technique of modified differential inequality, we establish some conditions on the nonlinearities for which the solution exists globally or blows up at finite time, when the sign of the constant $\sigma$ is either positive or negative. Moreover, upper and lower bounds for a blow-up time are derived under appropriate measure in higher dimensional spaces.

Blow-up phenomena for a damped plate equation with logarithmic nonlinearity

This paper is concerned with the finite time blow-up phenomena for a damped plate equation with logarithmic nonlinearity under the Navier boundary condition. Combining potential well method and modified differential inequality technique, we establish the finite time blow-up properties of solutions with low initial energy E(0)≤d and high energy. In particular, we consider the case where the initial displacement and initial velocity have the same (∫Ωu0u1dx>0) or different signs (∫Ωu0u1dx>−‖u0‖∗22) in the sense of L2 inner product at high energy level. Moreover, the lower bounds for the blow-up time in subcritical, critical, supercritical and ultra-supercritical cases are also derived.

Singular boundary behaviour and large solutions for fractional elliptic equations.

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equationsposed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equationwith zero boundarydata. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equationsdriven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Greenfunction.

Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel

We consider the semilinear diffusion equation ∂ t u = Au + |u| α u in the half-space R N + := R N −1 × (0, +∞), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ′ , 0) = 0 for x ′ ∈ R N −1. We prove that if the symbol of the operator A is of order a|ξ| β near the origin ξ = 0, for some β ∈ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 β/(N + 1). Notice that in the case of the half-space, the exponent β/(N + 1) is smaller than the so-called Fujita exponent β/N in R N. As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of R N , which are odd in the x N direction (and thus sign changing).