Incomplete Self‐Similar Blow‐Up in a Semilinear Fourth‐Order Reaction‐Diffusion Equation

Blow‐up behavior for the fourth‐order semilinear reaction‐diffusion equation (1) is studied. For the classic semilinear heat equation from combustion theory (2) various blow‐up patterns were investigated since 1970s, while the case of higher‐order diffusion was studied much less. Blow‐up self‐similar solutions of (1) of the form are constructed. These are shown to admit global similarity extensions for t > T : The continuity at t = T is preserved in the sense that This is in a striking difference with blow‐up for (2) , which is known to be always complete in the sense that the minimal (proper) extension beyond blow‐up is u(x, t) ≡+∞ for t > T . Difficult fourth‐order dynamical systems for extension pairs {f(y), F(y)} are studied by a combination of various analytic, formal, and numerical methods. Other nonsimilarity patterns for (1) with nongeneric complete blow‐up are also discussed.     

Single Point Gradient Blow‐Up and Nonuniqueness for a Third‐Order Nonlinear Dispersion Equation

A basic mechanism of a formation of shocks via gradient blow‐up from analytic solutions for the third‐order nonlinear dispersion PDE from compacton theory (1) is studied. Various self‐similar solutions exhibiting single point gradient blow‐up in finite time, as t → T^? < ∞ , with locally bounded final time profiles u(x, T^?) , are constructed. These are shown to admit infinitely many discontinuous shock‐type similarity extensions for t > T , all of them satisfying generalized Rankine–Hugoniot's condition at shocks. As a consequence, the nonuniqueness of solutions of the Cauchy problem after blow‐up is detected. This is in striking difference with general uniqueness‐entropy theory for the 1D conservation laws such as (a partial differential equation, PDE, Euler's equation from gas dynamics) (2) created by Oleinik in the middle of the 1950s. Contrary to (1) and not surprisingly, self‐similar gradient blow‐up for (2) is shown to admit a unique continuation. Bearing in mind the classic form (2) , the NDE (1) can be written as (3) with the standard linear integral operator (?D²_x)^?1 > 0 . However, because (3) is a nonlocal equation, no standard entropy and/or BV‐approaches apply (moreover, the x‐variations of solutions of (3) is increasing for BV data u₀(x) ).     

Very Singular Similarity Solutions and Hermitian Spectral Theory for Semilinear Odd-order PDEs

Asymptotic large- and short-time behavior of solutions of the linear dispersion equation μt = Uxxx in IR× IR＋, and its （2k＋l）th-order extensions are studied. Such a refined scattering is based on a ＂Hermitian＂ spectral theory for a pair {B,B＊} of non self-adjoint rescaled operators     

Blow Up in a Semilinear Wave Equation

We consider a semilinear wave equation of the form u_tt(x, t) - Δu(x, t) = - m(x, t)u_t(x, t) + ∇Φ(x) ⋅ ∇u(x, t ) + b(x)|u(x, t)|^{p-2}u(x, t) where p > 2. We show, under suitable conditions on m, Φ, b, that weak solutions break down in finite time if the initial energy is negative. This result improves an earlier one by the author [1].     

CR Structures on the 3‐Sphere and the Blow‐Up of the Projective Plane

Motivated by a problem of characterizing CR‐structures on the 3‐sphere, we give a geometric construction of formal deformations of a complex surface, which is the complement of a ball in the projective plane. They are described by cohomology groups of the blow‐up X of the projective plane. Moreover it will be shown that the space of these formal deformations is an infinite dimensional space with a natural stratification by finite dimensional subspaces. This stratification re ects algebro‐geometric properties of X. It is expected that our construction will clarify the complex geometric nature of the space of non‐embeddable CR‐structures on the 3‐sphere.     

一类四阶半线性方程的奇摄动解

讨论了一类四阶半线性方程奇摄动边值问题.利用上下解方法,研究了边值问题解的存在性和渐近性态.指出了在该文的情形下具有两参数的原奇摄动问题的解只有一个边界层.     

Blow‐up estimates for a quasi‐linear reaction–diffusion system

In this paper, some sufficient conditions under which the quasilinear elliptic system ‐div(∣?_u∣^p‐2?_u) = uv, ‐div(∣?_u∣^q‐2?_u) = uv in ?^N(N≥3) has no radially symmetric positive solution is derived. Then by using this non‐existence result, blow‐up estimates for a class of quasilinear reaction–diffusion systems u_t = div (∣?_u∣^p‐2?_u)+uv,vt = div(∣?_v∣^q‐2?_v) +uv with the homogeneous Dirichlet boundary value conditions are obtained. Copyright © 2003 John Wiley & Sons, Ltd.     

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

A spectral shifted Legendre Gauss–Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials \({P_{L,n}(x), \ x\in[0,L]}\) is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre–Gauss–Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge–Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.     

Blow‐up for a class of semilinear integro‐differential equations of parabolic type

In this paper, we study the following semilinear integro‐differential equation of the parabolic type that arise in the theory of nuclear reactor kinetics: under homogeneous Dirichlet boundary condition, where p, q⩾1. We first establish the local solvability of a large class of semilinear non‐local equations including the above equation. Next, we give the finite time blow‐up result by some modification of Kaplan's method and also the existence of global solutions by the comparison method. Copyright © 1999 John Wiley & Sons, Ltd.     

Exponential Attractor for a Nonlinear Boussinesq Equation

This paper is devoted to prove the existence of an exponential attractor for the semiflow generated by a nonlinear Boussinesq equation. We formulate the Boussinesq equation as an abstract equation in the Hilbert space H0^2（0, 1） × L^2（0, 1）. The main step in this research is to show that there exists an absorbing set for the solution semiflow in the Hilbert space H0^3（0, 1） × H0^1（0, 1）.     

高阶半线性椭圆型方程奇摄动广义Dirichlet边值问题

本文讨论了半线性椭圆型方程奇摄动广义边值问题,在适当的条件下研究了Dirichlet边值问题广义解的存在唯一性及其渐近性态。     

An Inverse Problem for a Semilinear Wave Equation

Doklady Mathematics - For the equation $${{u}_{{tt}}} - \Delta u - f(x,u) = 0, (x,t) \in {{\mathbb{R}}^{4}},$$ where $$f(x,u)$$ is a smooth function of its variables and is compact in x, the...     

一类半线性波动方程的Sobolev指数

Gaustavo Ponce与Thomas C.Sideris猜测：对一些具有特殊非线性项的半线性波动方程,如utt-△u=u^k(Du)^αx∈R^n,k∈Z^ ,ρ＝│α│≥2,其中Sobloev指数会在[n/2,n/2 1]中,他们在x∈R^3时回答了这一问题,本文在R^n(n≥4）中得到了半线性波动方程utt-△u=u^k(Du)^α（x∈R^n,k∈R^n,k∈Z^ ,p=│α│≥2）的Sobolev指数为max{n/2,(n/2-1)1-3/l-1 2},此数确实在区间[n/2,n/2 1]中,特别当ρ≤n-1时,我们得到了此半线性波动方程的Sobolev指数为n/2。     

半线性抛物方程各向异性最低阶R-T混合元超收敛分析

利用各向异性判别定理验证了最低阶数R-T混合元具有各向异性特征.利用积分恒等式技巧,得到了R-T元对半线性抛物方程的超逼近性质.通过构造新的插值后处理格式,导出了超收敛结果及后验误差估计.     

一类四阶半线性椭圆型方程非局部边值问题(英文)

本文研究了一类四阶半线性椭圆型方程奇摄动非局部问题.     

一类广义Boussinesq型方程的Cauchy问题

证明了一类广义Boussinesq型方程Cauchy问题整体解的存在性与唯一性,并给出解在有限时刻爆破的充分条件.     

Global Attractors for a Semilinear Hyperbolic Equation in Viscoelasticity

A semilinear partial differential equation of hyperbolic type with a convolution term describing simple viscoelastic materials with fading memory is considered. Regarding the past history (memory) of the displacement as a new variable, the equation is transformed into a dynamical system in a suitable Hilbert space. The dissipation is extremely weak, and it is all contained in the memory term. Longtime behavior of solutions is analyzed. In particular, in the autonomous case, the existence of a global attractor for solutions is achieved.     

一类奇异半线性椭圆方程解的不存在性

研究了一类奇异半线性椭圆方程解的不存在性,改进并补充了Arcoya等人的一个研究工作[Existence and nonexistence of solutions for singular quadratic quasilinear equations,J Differential Equations,2009,246:4006-4042].本文使用的方法非常简单,证明过程中用到了Poincare等式,Lusin定理以及Egoroff定理.