blow-up

摘要： In this paper,we give rigorous justification of the ideas put forward in§20,Chapter 4 of Schubert’s book;a section that deals with the enumeration of conics in space.In that section,Schubert introduced two degenerate conditions about conics,i.e.,the double line and the two intersection lines.Using these two degenerate conditions,he obtained all relations regarding the following three conditions:conics whose planes pass through a given point,conics intersecting with a given line,and conics which are tangent to a given plane.We use the language of blow-ups to rigorously treat the two degenerate conditions and prove all formulas about degenerate conditions stemming from Schubert’s idea.

摘要： In this paper,we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions.A non-existence result is established for the fan-shaped wave structure solution,including two shocks and one contact discontinuity which is a perturbation of plane waves.Therefore,unlike in the one-dimensional case,the multi-dimensional plane shocks are not stable globally.Moreover,a sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.

摘要： This paper deals with a homogeneous Neumann initial-boundary problem of a 4th-order parabolic equation modeling epitaxial growth of thin film. We determine the classification of initial energy on the existence of blow-up, global existence and extinction of solutions by using the potential well method and the auxiliary function method.Moreover, asymptotic estimates on global solution and extinction solution are studied,respectively.

摘要： In this paper,we consider the following nonlinear viscoelastic wave equation with variable exponents:utt-△u+∫_(0)^(t)g(t-τ)△u(x,τ)dτ+μut=|u|^(p(x)-2)-u,whereμis a nonnegative constant and the exponent of nonlinearity p(·)and g are given functions.Under arbitrary positive initial energy and specific conditions on the relaxation function g,we prove a finite-time blow-up result.We also give some numerical applications to illustrate our theoretical results.

摘要： In this paper,we are concerned with the existence of the positive bounded and blow-up radial solutions of the(k1,k2)-Hessian system■where G is a nonlinear operator,Ki=Ski(λ(D^(2) z_(i)))+ψ_(i)(|x|)|▽_(zi)|^(ki),i=1,2.Under the appropriate conditions on gi and g2,our main results are obtained by using the monotone iterative method and the Arzela-Ascoli theorem.Furthermore,our main results also extend the previous existence results for both the single equation and systems.

摘要： This paper presents a very short solution to the 4th Millennium problem about the Navier-Stokes equations. The solution proves that there cannot be a blow up in finite or infinite time, and the local in time smooth solutions can be extended for all times, thus regularity. This happily is proved not only for the Navier-Stokes equations but also for the inviscid case of the Euler equations both for the periodic or non-periodic formulation and without external forcing (homogeneous case). The proof is based on an appropriate modified extension in the viscous case of the well-known Helmholtz-Kelvin-Stokes theorem of invariance of the circulation of velocity in the Euler inviscid flows. This is essentially a 1D line density of (rotatory) momentum conservation. We discover a similar 2D surface density of (rotatory) momentum conservation. These conservations are indispensable, besides to the ordinary momentum conservation, to prove that there cannot be a blow-up in finite time, of the point vorticities, thus regularity.

摘要： In this paper,a fractional Laplacian mutualistic system under Neumann boundary conditions is studied.Using the method of upper and lower solutions,it is proven that the solutions of the fractional Laplacian strong mutualistic model with Neumann boundary conditions will blow up when the intrinsic growth rates of species are large.

摘要： In this paper,we study the Cauchy problem for the nonlinear Schrodinger equations with Coulomb potential i■tu+△u+k/|x|u=λ/|u|^(p-l)u with 10,and the scattering theory when the Coulomb potential is repulsive,i.e.,when K≤O.The argument is based on the newlyestablished interaction Morawetz-type inequalities and the equivalence of Sobolev norms for the Laplacian operator with the Coulomb potential.

摘要： In this article,we are concerned with analytical solutions for a model of inviscid liquid-gas two-phase flow.On the basis of Yuen’s works[25,27–29]on self-similar solutions for compressible Euler equations,we present some special self-similar solutions for a model of inviscid liquid-gas two-phase flow in radial symmetry with and without rotation,and in elliptic symmetry without rotation.Some blowup phenomena and the global existence of the solutions obtained are classified.

摘要： In this paper,we study the blow-up of solutions to a semilinear double-wave equation with nonlinearity of derivative type.By using the iteration method and the differential inequality techniques,we can get the estimates of the lifespan and the blow-up of solutions in the subcritical case under some assumptions.