A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method

The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup when the underlying continuous problem does. In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to infinity as the number of mesh points increases. The inadequacy of a uniform mesh solution is clearly demonstrated.     

Blowup Solutions to the Semilinear Wave Equation with a Stylized Pyramid as a Blowup Surface

We consider the semilinear wave equation with subconformal power nonlinearity in two space dimensions. We construct a finite‐time blowup solution with an isolated characteristic blowup point at the origin and a blowup surface that is centered at the origin and has the shape of a stylized pyramid, whose edges follow the bisectrices of the axes in ℝ². The blowup surface is differentiable outside the bisectrices. As for the asymptotic behavior in similarity variables, the solution converges to the classical one‐dimensional soliton outside the bisectrices. On the bisectrices outside the origin, it converges (up to a subsequence) to a genuinely two‐dimensional stationary solution, whose existence is a by‐product of the proof. At the origin, it behaves like the sum of four solitons localized on the two axes, with opposite signs for neighbors. This is the first example of a blowup solution with a characteristic point in higher dimensions, showing a really two‐dimensional behavior. Moreover, the points of the bisectrices outside the origin give us the first example of noncharacteristic points where the blowup surface is nondifferentiable. © 2018 Wiley Periodicals, Inc.     

带局部非线性反应项的退化抛物方程解的爆破性质

本文研究带局部非线性反应项的退化抛物方程解的爆破性质ut=△um+up(x0,t)-kuq(x,t),其中p≥q>0,p>1,01),x0是有界区域Ω内的固定点,Ω(?)RN.在一定的假设条件下,证明了解在有限时刻爆破并且爆破点集是整个区域Ω.另外,如果解u(·,t)是径向对称函数且ur≤0,则解在接近爆破时刻的爆破速率在区域Ω上是一致的.在解是非径向对称的情况下,我们用其他技巧证得解的整体爆破性.     

The relationships between some types of partial differential equations and ordinary differential equations as well as their applications

In this article, we establish some relationships between several types of partial differential equations and ordinary differential equations. One application of these relationships is that we can get the exact values of the blowup time and the blowup rate of the solution to a partial differential equation by solving an ordinary differential equation. Another application of these relationships is that we can give the estimates for the spatial integration (or mean value) of the solution to a partial differential equation. We also obtain the lower and upper bounds for the blowup time of the solution to a parabolic equation with weighted function and space‐time integral in the nonlinear term.     

Computation of large shear deformations of a thermoplastic material

A specialized finite difference method with grid refinement and variable time steps is created to approximate the deformation velocity and the temperature in a simple model of the shearing of a thermoplastic material. A specific problem where the solution exhibits “blowup” in the adiabatic case is considered. The numerical method retains this property and is used to study the shape of the “blowup” function. The code is then used to investigate the solution in the closely related case where thermal conduction is included with a small conductivity coefficient. The computations indicate that the solution does not “blowup” in the nonadiabatic case. © 1996 John Wiley & Sons, Inc.     

On local existence and blowup of solutions for a time‐space fractional diffusion equation with exponential nonlinearity

In this paper, we are concerned with local existence and blowup of a unique solution to a time‐space fractional evolution equation with a time nonlocal nonlinearity of exponential growth. At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, the blowup result of the solution in finite time is established by the test function method with a judicious choice of the test function.     

Construction of a Multisoliton Blowup Solution to the Semilinear Wave Equation in One Space Dimension

We consider the semilinear wave equation with power nonlinearity in one space dimension. Given a blowup solution with a characteristic point, we refine the blowup behavior first derived by Merle and Zaag. We also refine the geometry of the blowup set near a characteristic point and show that, except for perhaps one exceptional situation, it is never symmetric with respect to the characteristic point. Then, we show that all blowup modalities predicted by those authors do occur. More precisely, given any integer k ≥ 2 and $\zeta _0 \in {\cal R}$ , we construct a blowup solution with a characteristic point a such that the asymptotic behavior of the solution near (a,T(a)) shows a decoupled sum of k solitons with alternate signs whose centers (in the hyperbolic geometry) have ζ₀ as a center of mass for all times. © 2013 Wiley Periodicals, Inc.     

The blowup mechanism for 3-D quasilinear wave equations with small data

For a class of special three-dimensional quasilinear wave equations, we study the blowup mechanism of classical solutions. More precisely, under the nondegenerate conditions, any radially symmetric solution with small initial data is shown to develop singularities in the second order derivatives while the first order derivatives and itself remain continuous, moreover the blowup of solution is of “cusp type”.     

Nonexistence of Type Ⅱ Blowup for Heat Equation with Exponential Nonlinearity

This paper deals with the blowup behavior of the radially symmetric solution of the nonlinear heat equation ut = ?u + e~u in R~N. The authors show the nonexistence of type II blowup under radial symmetric case in the lower supercritical range 3 ≤ N ≤ 9,and give a sufficient condition for the occurrence of type I blowup. The result extends that of Fila and Pulkkinen(2008) in a finite ball to the whole space.     

GLOBAL BLOWUP AND BLOWUP RATE ESTIMATES OF SOLUTIONS FOR A CLASS OF NONLINEAR NON-LOCAL REACTION-DIFFUSION PROBLEMS

In this paper, the global blowup properties of solutions for a class of non-linear non-local reaction-diffusion problems are investigated by the methods of the priorestimates. Moreover, the blowup rate estimate of the solution is given.     

Sufficient close-to-necessary conditions for the blowup of solutions to a strongly nonlinear generalized Boussinesq equation

An initial boundary value problem for the generalized Boussinesq equation with allowance for linear dissipation and free electron sources is considered. The strong generalized time-local solvability of the problem is proved. Sufficient conditions are obtained for the blowup of the solution and for time-global solvability. Two-sided estimates of the blowup time are derived.     

An initial-boundary value problem for a Sobolev-type strongly nonlinear dissipative equation

An initial-boundary value problem is considered for a model equation governing waves in crystalline semiconductors with allowance for strong spatial dispersion, linear dissipation, and sources of free charges. The weak generalized local-in-time solvability of the problem is proved. Sufficient conditions are obtained for the blowup of the solution and for global-in-time solvability. Two-sided estimates for the blowup time are derived.     

三维空间中带非线性阻尼项的等熵欧拉方程组轴对称解的爆破

研究了三维空间中带非线性阻尼项的可压缩等熵欧拉方程组Dirichlet初边值问题.采用泛函方法,定义几种不同的泛函,当初始速度足够大时分别得到了经典解在某一时间内必定爆破的结论.由于出现了非线性阻尼项,较之线性阻尼的情形,经典解爆破的难度随之增加.     

带逆平方势的非线性Schr?dinger方程的有限时间性态

在关于非相对论分子物理中磁性粒子捕获电子的研究中,带逆平方势的非线性Schr?dinger方程起着重要的作用.我们重点关注该系统有限时间内的存在性和性态,并导出了该系统解爆破的一个显示精确门槛标准.进一步,证明了该系统径对称爆破解的集中性.     

Global well-posedness of coupled parabolic systems

The initial boundary value problem of a class of reaction-diffusion systems(coupled parabolic systems)with nonlinear coupled source terms is considered in order to classify the initial data for the global existence,finite time blowup and long time decay of the solution.The whole study is conducted by considering three cases according to initial energy:the low initial energy case,critical initial energy case and high initial energy case.For the low initial energy case and critical initial energy case the sufficient initial conditions of global existence,long time decay and finite time blowup are given to show a sharp-like condition.In addition,for the high initial energy case the possibility of both global existence and finite time blowup is proved first,and then some sufficient initial conditions of finite time blowup and global existence are obtained,respectively.     

拟线性抛物型方程解的爆破

本文讨论了一类拟线性抛物型方程具有某种非线性边界条件的解的爆破性质,证明了其解在有限时间T_0~-内爆破。     

Fujita type critical exponent for a free boundary problem with spatial–temporal source

In this paper, we investigate a nonlinear free boundary problem incorporating with nontrivial spatial and exponential temporal weighted source. To portray the asymptotic behavior of the solution, we first derive some sufficient conditions for finite time blowup. Furthermore, the global vanishing solution is also obtained for a class of small initial data. Finally, a sharp threshold trichotomy result is provided in terms of the size of the initial data to distinguish the blowup solution, the global vanishing solution, and the global transition solution. In particular, our results show that such a problem always possesses a Fujita type critical exponent whenever the spatial source is just equivalent to a trivial constant, or is an extreme one, such as “very negative” one in the sense of measure or integral.     

Numerical simulation of blowup in nonlocal reaction–diffusion equations using a moving mesh method

In this paper we implement the moving mesh PDE method for simulating the blowup in reaction–diffusion equations with temporal and spacial nonlinear nonlocal terms. By a time-dependent transformation, the physical equation is written into a Lagrangian form with respect to the computational variables. The time-dependent transformation function satisfies a parabolic partial differential equation — usually called moving mesh PDE (MMPDE). The transformed physical equation and MMPDE are solved alternately by central finite difference method combined with a backward time-stepping scheme. The integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach. The monitor function in MMPDEs plays a key role in the performance of the moving mesh PDE method. The dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle. A variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests.     

Global existence and blowup of a nonlocal problem in space with free boundary

This paper concerns a double fronts free boundary problem for the reaction–diffusion equation with a nonlocal nonlinear reaction term in space. For such a problem, we mainly study the blowup property and global existence of the solutions. Our results show that if the initial value is sufficiently large, then the blowup occurs, while the global fast solution exists for a sufficiently small initial data, and the intermediate case with a suitably large initial data gives the existence of the global slow solution.     

Blowup,Global Fast and Slow Solutions for a Semilinear Combustible System

In this paper, we investigate a semilinear combustible system $u_t-du_{xx}=v^p, v_t-dv_{xx}=u^q$ with double fronts free boundary, where p ≥ 1, q ≥ 1. For such a problem, we use the contraction mapping theorem to prove the local existence and uniqueness of the solution. Also we study the blowup and global existence property of the solution. Our results show that when pq › 1 blowup occurs if the initial datum is large enough and the solution is global and slow, whose decay rate is at most polynomial if the initial value is suitably large, while when p › 1, q › 1 there is a global and fast solution, which decays uniformly at an exponential rate if the initial datum is small.