Properties of non-simultaneous blow-up solutions in nonlocal parabolic equations

This paper deals with blow-up solutions in parabolic equations coupled via nonlocal nonlinearities, subject to homogeneous Dirichlet conditions. Firstly, some criteria on non-simultaneous and simultaneous blow-up are given, including four kinds of phenomena: (i) the existence of non-simultaneous blow-up; (ii) the coexistence of non-simultaneous and simultaneous blow-up; (iii) any blow-up must be simultaneous; (iv) any blow-up must be non-simultaneous. Next, total versus single point blow-up are classified completely. Moreover, blow-up rates are obtained for both non-simultaneous and simultaneous blow-up solutions.     

Asymptotic properties of singular solutions in degenerate parabolic equation with boundary flux

ABSTRACT

This paper deals with blow-up and quenching solutions of degenerate parabolic problem involving m-Laplacian operator and nonlinear boundary flux. The blow-up and quenching criteria are classified under the conditions on the initial data but with less conditions on the relationship among the exponents, respectively. Moreover, asymptotic properties including singular rates, set and time estimates are determined for the blow-up solutions and the quenching solutions, respectively.     

Global existence and blow-up of solutions to a degenerate parabolic system with nonlocal sources and nonlocal boundaries

This paper deals with a nonlinear degenerate parabolic system with nonlocal source and nonlocal boundaries. By super-solution, sub-solution and auxiliary functions, a criteria for nonnegative solution of global existence and blow-up in finite time is obtained for this degenerate nonlocal problem. Finally, the blow-up rates of blow-up solutions are also estimated.     

Blow-up analysis for a nonlocal reaction-diffusion system with time-dependent coefficients

ABSTRACT

A blow-up analysis for a nonlocal reaction-diffusion system with time-dependent coefficients is investigated under null Dirichlet boundary conditions. Based on the Kaplan's method, comparison principle and modified differential inequality technique, we establish a blow-up criteria and derive the bounds for the blow-up time under the appropriate measures in whole-dimensional space.     

Multiple absorption-related blow-up rates to a coupled heat system

This paper deals with asymptotic behavior of solutions to a heat system with absorptions and coupling positive multi-nonlinearities. It is known that although absorption mechanisms may affect such as blow-up criteria, blow-up time, and initial data required for blow-up solutions, they cannot change blow-up rates of solutions in general. It has been reported in the current literature that blow-up rates for scalar equations with absorptions are all absorption-independent. In a previous paper of the authors, four absorption-independent simultaneous blow-up rates were obtained already for the same problem under weak absorptions. The present paper will furthermore prove that if the absorptions are unbalanced in the model (i.e., the absorption is stronger for one component and weaker for another), then there are in addition eight possible absorption-related blow-up rates for the model, besides the four absorption-independent ones. This exposes a significant difference between scalar and coupled nonlinear parabolic equations with absorptions.     

A degenerate parabolic system with nonlocal boundary condition

In this article, we investigate the blow-up properties of the positive solutions to a degenerate parabolic system with nonlocal boundary condition. We give the criteria for finite time blow-up or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate for small weighted nonlocal boundary.     

Global boundedness and blow-up for a parabolic system with positive Dirichlet boundary value

This paper deals with a parabolic system coupled via nonlocal sources, subjecting to positive Dirichlet boundary value conditions. By using the super-, sub-solution methods and techniques, and piecewise functions, the blow-up criteria and global boundedness of nonnegative solutions are determined. The results show the positive boundary value ε ₀ plays an important role in the case of blow-up.     

Global blow-up for a localized quasilinear parabolic system with nonlocal boundary conditions

In this article, we investigate the positive solution of a localized quasilinear parabolic system with nonlocal boundary conditions. Under certain conditions, the global existence and finite time blow-up criteria are established, and the global blow-up behaviour is also obtained.     

Multiple blow-up rates in a coupled heat system with mixed-type nonlinearities

This paper studies heat equations with inner absorptions and coupled boundary fluxes of mixed-type nonlinearities. At first, the critical exponent is obtained, and simply described via a characteristic algebraic system introduced by us. Then, as the main results of the paper, three blow-up rates are established under different dominations of nonlinearities for the one-dimensional case, and represented in another characteristic algebraic system. In particular, it is observed that unlike those in previous literature on parabolic models with absorptions, two of the multiple blow-up rates obtained here do depend on the absorption exponents. In the known works, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates. To our knowledge, this is the first example of absorption-dependent blow-up rates, exploiting the significant interactions among diffusions, inner absorptions and nonlinear boundary fluxes in the coupled system. It is also proved that the blow-up of solutions in the model occurs on the boundary only.     

On the Behaviors of Solutions Near Possible Blow-Up Time in the Incompressible Euler and Related Equations

We study behaviors of scalar quantities near the possible blow-up time, which is made of smooth solutions of the Euler equations, Navier–Stokes equations and the surface quasi-geostrophic equations. Integrating the dynamical equations of the scaling invariant norms, we derive the possible blow-up behaviors of the above quantities, from which we obtain new type of blow-up criteria and some necessary conditions for the finite time blow-up.     

On the blow-up of solutions to a nonlinear dispersive rod equation

Using a variational approach we prove an optimal nonlinear convolution inequality. This result is then applied to give criteria for finite-time blow-up of solutions to a nonlinear model equation in elasticity, improving considerably upon recent blow-up results.     

Blow-up for a weakly dissipative modified two-component Camassa–Holm system

In this paper, we study the Cauchy problem of a weakly dissipative modified two-component Camassa–Holm (MCH2) system. We first derive the precise blow-up scenario and then give several criteria guaranteeing the blow-up of the solutions. We finally discuss the blow-up rate of the blowing-up solutions.     

THE SIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP CRITERIA FOR A DIFFUSION SYSTEM

This paper investigates the finite time blow-up of nonnegative solutions for a nonlinear diffusion system with a more complicated source term,which is a product of localized source,local source,and weight function,and complemented by homogeneous Dirichlet boundary conditions.The criteria are proposed to identify simultaneous and nonsimultaneous blow-up solutions.Moreover,the related classification for the four parameters in the model is optimal and complete.The results extend those in Zhang and Yang [12].     

The Cauchy problem for a generalized Camassa–Holm equation with the velocity potential

In this paper, we discuss a generalized Camassa–Holm equation whose solutions are velocity potentials of the classical Camassa–Holm equation. By exploiting the connection between these two equations, we first establish the local well-posedness of the new equation in the Besov spaces and deduce several blow-up criteria and blow-up results. Then, we investigate the existence of global strong solutions and present a class of cuspon weak solutions for the new equation.     

On the Cauchy problem for the two-component Euler–Poincaré equations

In the paper, we first use the energy method to establish the local well-posedness as well as blow-up criteria for the Cauchy problem on the two-component Euler–Poincaré equations in multi-dimensional space. In the case of dimensions 2 and 3, we show that for a large class of smooth initial data with some concentration property, the corresponding solutions blow up in finite time by using Constantin–Escher Lemma and Littlewood–Paley decomposition theory. Then for the one-component case, a more precise blow-up estimate and a global existence result are also established by using similar methods. Next, we investigate the zero density limit and the zero dispersion limit. At the end, we also briefly demonstrate a Liouville type theorem for the stationary weak solution.     

Nonlinear Volterra integral equations and the Schröder functional equation

We show an interesting connection between a special class of Volterra integral equations and the famous Schröder equation. The basic results provide criteria for the existence of nontrivial as well as blow-up solutions of the Volterra equation, expressed in terms of the convergence of some integrals. Examples related to Volterra equations with power and exponential nonlinearities are presented.     

Profiles of blow-up solutions for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of the Cauchy problem for Gross-Pitaevskii equation.In terms of Merle and Raphёel＇s arguments as well as Carles＇ transformation,the limiting profiles of blow-up solutions are obtained.In addition,the nonexistence of a strong limit at the blow-up time and the existence of L2 profile outside the blow-up point for the blow-up solutions are obtained.     

Blow-up criteria for the three dimensional nonlinear dissipative system modeling electro-hydrodynamics

In this paper, we investigate some sufficient conditions for the breakdown of local smooth solutions to the three dimensional nonlinear nonlocal dissipative system modeling electro-hydrodynamics. This model is a strongly coupled system by the well-known incompressible Navier–Stokes equations and the classical Poisson–Nernst–Planck equations. We show that the maximum of the vorticity field alone controls the breakdown of smooth solutions, which reveals that the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system. Moreover, some Prodi–Serrin type blow-up criteria are also established.     

Boussinesq方程组在Besov空间中局部解的存在性和延拓准则

本文研究二维无粘性Boussinesq方程组在超临界Besov空间B_(p,q)~s(R~2),s>1+2/p,1相似文献   

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.