Blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions

This paper concerns with blow-up behaviors for semilinear parabolic systems coupled in equations and boundary conditions in half space. We establish the rate estimates for blow-up solutions and prove that the blow-up set is under proper conditions on initial data. Furthermore, for N=1, more complete conclusions about such two topics are given.     

Blow-up of solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and boundary condition

Semilinear hyperbolic and parabolic initial–boundary value problems are studied. Criteria for solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and general boundary condition to blow up in finite time are obtained.     

Finite-dimensional controller design for semilinear parabolic systems

This paper considers the stabilization of steady-state solutions of a semilinear parabolic system using finite-dimensional feedback controllers with support in an arbitrary open subset and which are active in one equation only. It is shown that such a controller, with dimension given by the largest algebraic multiplicity of the unstable eigenvalues of the linearized system, exponentially stabilizes the steady-state solution. An optimal design methodology for these types of controllers, which is based on the finite element approximation of the semilinear parabolic system, is introduced and illustrated by numerical simulation examples.     

On a semilinear parabolic equation

We introduce a general class of potentials so that the semilinear parabolic equation in , , has global positive continuous solutions. These results extend the recent ones proved by Zhang to a more general class of potentials.

     

Blow-up for a degenerate parabolic equation with a nonlocal source

In this paper, we investigate the blowup properties of the positive solutions to the following nonlocal degenerate parabolic equation

     

Lower bounds for the blow-up time in a semilinear parabolic problem involving a variable source

This letter is concerned with the blow-up of the solutions to a semilinear parabolic problem with a reaction given by a variable exponent. Lower bounds for the time of blow-up are derived if the solutions blow up.     

Blow-up rates for semilinear parabolic systems with nonlinear boundary conditions

This paper deals with the blow-up rate estimates of positive solutions for semilinear parabolic systems with nonlinear boundary conditions. The upper and lower bounds of blow-up rates are obtained.     

Blow-up problems for a semilinear heat equation with large diffusion

We consider the blow-up problem of a semilinear heat equation,

     

二维空间上一类半线性波动方程的爆破问题

本文用类似于[1]中解决爆破问题的方法，对二维空间上一类半线性波动方程的初值问题证得了：当非线性项F（u）∈C2（R）和初值g（x）∈CO（R2）且满足一定条件时，初值问题不存在全局C2-解．     

Blow-up time for solutions to some nonlinear Volterra integral equations

The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model.     

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.     

Blow-up of solutions to semilinear wave equations with variable coefficients and boundary

This paper is devoted to studying initial-boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions. We will establish blow-up results for the initial-boundary value problems. It is proved that there can be no global solutions no matter how small the initial data are, and also we give the life span estimate of solutions for the problems.     

Blow-up phenomena for a family of Burgers-like equations

By introducing a stress multiplier we derive a family of Burgers-like equations. We investigate the blow-up phenomena of the equations both on the real line R and on the circle S to get a comparison with the Degasperis-Procesi equation. On the line R, we first establish the local well-posedness and the blow-up scenario. Then we use conservation laws of the equations to get the estimate for the L^∞-norm of the strong solutions, by which we prove that the solutions to the equations may blow up in the form of wave breaking for certain initial profiles. Analogous results are provided in the periodic case. Especially, we find differences between the Burgers-like equations and the Degasperis-Procesi equation, see Remark 4.1.     

Blow-up for semilinear parabolic equations with nonlinear memory

In this paper, we consider the semilinear parabolic equation with homogeneous Dirichlet boundary conditions, where p, q are nonnegative constants. The blowup criteria and the blowup rate are obtained.     

Blow-up properties for a degenerate parabolic system with nonlinear localized sources

This paper deals with blow-up properties for a degenerate parabolic system with nonlinear localized sources subject to the homogeneous Dirichlet boundary conditions. The main aim of this paper is to study the blow-up rate estimate and the uniform blow-up profile of the blow-up solution. Our conclusions extend the results of [L.L. Du, Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources, J. Math. Anal. Appl. 324 (2006) 304-320]. At the end, the blow-up set and blow up rate with respect to the radial variable is considered when the domain Ω is a ball.     

Blow-up phenomena for a class of fourth order parabolic equation

A class of fourth order parabolic equation is studied in this paper. Some related blow-up results are obtained by applying the potential well theory, the concavity method and a series of differential-integral inequality techniques. More precisely, under some proper assumptions, the upper and lower bounds of the blow-up time and the growth rate for blow-up solutions are estimated. Moreover, a new blow-up condition independent of the depth of the potential well is found. These results complement the recent results obtained in Han (2018).