On a class of nonlocal parabolic equations

We consider a boundary value problem for parabolic equations with nonlocal nonlinearity of such a form that favorably differs from other equations in that it leads to partial differential equations that have important properties of ordinary differential equations. Local solvability and uniqueness theorems are proved, and an analog of the Painlevé singular nonfixed points theorem is proved. In this case, there is an alternative—either a solution exists for all t ≥ 0 or it goes to infinity in a finite time t = T (blowup mode). Sufficient conditions for the existence of a blowup mode are given.     

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.     

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.     

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.     

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.     

A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar

An approximate Hamilton principle is established for the transverse vibration of a reinforced concrete pillar by considering the dissipation energy, and a generalized Boussinesq equation is obtained. The exp-function method is adopted to solve the equation, and its solution properties are discussed and elucidated, including solitary solution, blowup solution, and discontinuous solution. In order to study the effect of a porous structure on the vibration property, fractal calculus is used to derive the fractal Boussinesq equation, and a fractal variational principle is also established. The fractal model confers many attractive properties, which can not be revealed by the traditional protocol. The effect of the nanofiber-reinforced concrete structure on its wave morphology is discussed and illustrated. A blowup solution can be converted into a flat solution by adjusting the value of the fractal derivative order. The paper sheds new light on the design of reinforced concrete pillars to avoid vibration damage.     

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.     

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.     

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 OF RADIALLY SYMMETRIC SOLUTIONS FOR 2-D QUASILINEAR WAVE EQUATIONS WITH CUBIC NONLINEARITY

61.IntroductionInthispaper,weconsiderthefollowingtwodimensionalquasilinearwaveequationswiththenonlinearityofcubicform:wherex=(x1,x2),E>Oissmallenough,c'(otu,7u)=c'(otu,Oru)=l a,(otu)' a2Ofuoru a,(oru)' o(Iotul' lorul'),f(otu'Vu)=f(otu,o'u)=b,(otu)' b,(o,u)'oru b,otu(oru)' b,(oru)' O(Iotul' loruI'),a1-a2 a3/o,uo(x),ul(x)areCooradialfunctions(thatis,smoothfunctionsoflx1')andsupportedinaffeedba.llofradiusM.Moreoveruo(x)/Ooru1(x)*O.OuraimistostudythelifespanTeofsolutionsto(l.1)andthebreakdow…     

Blowup properties for nonlinear degenerate diffusion equations with nonlocal sources

This paper deals with degenerate diffusion equations with nonlocal sources. The local existence of a classical solution is given. By making use of super- and sub-solution method we show that the solution exists globally or blows up in finite time under some conditions. Furthermore, the blowup rates of the blowup solution are derived.     

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.     

On stable self‐similar blowup for equivariant wave maps

We consider corotational wave maps from (3 + 1) Minkowski space into the 3‐sphere. This is an energy supercritical model that is known to exhibit finite‐time blowup via self‐similar solutions. The ground state self‐similar solution f₀ is known in closed form, and according to numerics, it describes the generic blowup behavior of the system. We prove that the blowup via f₀ is stable under the assumption that f₀ does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blowup to a linear ODE spectral problem. Although we are unable at the moment to verify the mode stability of f₀ rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f₀. © 2011 Wiley Periodicals, Inc.     

The Nash sheaf of a complete resolution

Every three-dimensional complex algebraic variety with isolated singular point has a resolution factoring through the Nash blowup and the blowup of the maximal ideal over which the second Fitting ideal sheaf is locally principal. In such resolutions one can construct Hsiang–Pati coordinates and thus obtain generators for the Nash sheaf that are the differentials of monomial functions. Received: 9 April 2001 / Revised version: 26 July 2001     

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.     

Blowup of the solution to the Cauchy problem with arbitrary positive energy for a system of Klein–Gordon equations in the de Sitter metric

The ?⁴ model of a scalar (complex) field in the metric of an expanding universe, namely, in the de Sitter metric is considered. The initial energy of the system can have an arbitrarily high positive value. Sufficient conditions for solution blowup in a finite time are obtained. The existence of blowup is proved by applying H.A. Levine’s modified method is used.     

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.     

A note on gradient blowup rate of the inhomogeneous hamilton-jacobi equations

The gradient blowup of the equation u_t = Δu + a(x)|∇u|^p + h(x), where p > 2, is studied. It is shown that the gradient blowup rate will never match that of the self-similar variables. The exact blowup rate for radial solutions is established under the assumptions on the initial data so that the solution is monotonically increasing in time.     

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.     

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.