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We study the initial boundary value problem for a fourth-order parabolic equation with nonstandard growth conditions. We establish the local existence of weak solutions and derive the finite time blow-up of solutions with nonpositive initial energy.
We study systems of partial differential equations of Briot–Bouquet type. The existence of holomorphic solutions to such systems largely depends on the eigenvalues of an associated matrix. For the noninteger case, we generalise the well-known result of Gérard and Tahara [‘Holomorphic and singular solutions of nonlinear singular first order partial differential equations’, Publ. Res. Inst. Math. Sci.26 (1990), 979–1000] for Briot–Bouquet type equations to Briot–Bouquet type systems. For the integer case, we introduce a sequence of blow-up like changes of variables and give necessary and sufficient conditions for the existence of holomorphic solutions. We also give some examples to illustrate our results.
The convergence and blow-up results are established for the evolution of non-simple closed curves in an area-preserving curvature flow. It is shown that the global solution starting from a locally convex curve converges to an m-fold circle if the enclosed algebraic area A0 is positive, and evolves into a point if A0 = 0.
We consider the second order nonlinear ordinary differential equation u″ (t) = u1+α (α > 0) with positive initial data u(0) = a0, u′(0) = a1, whose solution becomes unbounded in a finite time T. The finite time T is called the blow-up time. Since finite difference schemes with uniform meshes can not reproduce such a phenomenon well, adaptively-defined grids are applied. Convergence with mesh sizes of certain smallness has been considered before. However, more iterations are required to obtain an approximate blow-up time if smaller meshes are applied. As a consequence, we consider in this paper a finite difference scheme with a rather larger grid size and show the convergence of the numerical solution and the numerical blow-up time. Application to the nonlinear wave equation is also discussed.
In this paper, we study the positive solutions for a semilinear equation in hyperbolic space. Using the heat semigroup and by constructing subsolutions and supersolutions, a Fujita-type result is established.
The initial value problem for a semi-linear fractional heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.
We show that blow-up solutions of the critical generalized Korteweg–de Vries equation in H1() concentrate at least the mass of the ground state at the blow-up time. The I-method is used to prove a slightly weaker result in Hs() with 16/17 < s < 1. Under an assumption on the precise blow-up rate, we are able to use similar arguments to prove a more precise analogue of the H1() concentration result over the same range of s.
The paper deals with blow-up for the solutions of an evolution problem consisting in a semilinear wave equation posed in a bounded C1,1 open subset of ℝn, supplied with a Neumann boundary condition involving a nonlinear dissipation. The typical problem studied is
where ∂Ω = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = ∅, σ(Γ0) > 0, 2 < p ≤ 2(n − 1)/(n − 2) (when n ≥ 3), m > 1, α ∈ L∞(Γ1), α ≥ 0 and β ≥ 0. The initial data are posed in the energy space.The aim of the paper is to improve previous blow-up results concerning the problem.
This paper is concerned with the Cauchy problem for a nonlinear Schrödinger equation with a harmonic potential and exponential growth nonlinearity in two space dimensions. In the defocusing case, global well-posedness is obtained. In the focusing case, existence of nonglobal solutions is discussed via potential-well arguments.
In this paper we describe a new pseudo-spectral method to solve numerically two and three-dimensional nonlinear diffusion equations over unbounded domains, taking Hermite functions, sinc functions, and rational Chebyshev polynomials as basis functions. The idea is to discretize the equations by means of differentiation matrices and to relate them to Sylvester-type equations by means of a fourth-order implicit-explicit scheme, being of particular interest the treatment of three-dimensional Sylvester equations that we make. The resulting method is easy to understand and express, and can be implemented in a transparent way by means of a few lines of code. We test numerically the three choices of basis functions, showing the convenience of this new approach, especially when rational Chebyshev polynomials are considered.
It has been known for a long time that the equivariant $2+1$ wave map into the $2$-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.
In this paper, we consider the Gross-Pitaevskii equation for the trapped dipolar quantum gases. We obtain the sharp criterion for the global existence and finite time blow-up in the unstable regime by constructing a variational problem and the so-called invariant manifold of the evolution flow.
We consider the Euler equation for an incompressible fluid on a three dimensional torus,
and the construction of its solution as a power series in time. We point out some general
facts on this subject, from convergence issues for the power series to the role of
symmetries of the initial datum. We then turn the attention to a paper by Behr, Nečas and
Wu, ESAIM: M2AN 35 (2001) 229–238; here, the authors chose a
very simple Fourier polynomial as an initial datum for the Euler equation and analyzed the
power series in time for the solution, determining the first 35 terms by computer algebra.
Their calculations suggested for the series a finite convergence radius
τ3 in the H3 Sobolev space, with
0.32 < τ3 < 0.35; they regarded this as an indication
that the solution of the Euler equation blows up. We have repeated the calculations of E.
Behr, J. Nečas and H. Wu, ESAIM: M2AN 35 (2001) 229–238,
using again computer algebra; the order has been increased from 35 to 52, using the
symmetries of the initial datum to speed up computations. As for
τ3, our results agree with the original computations of E.
Behr, J. Nečas and H. Wu, ESAIM: M2AN 35 (2001) 229–238
(yielding in fact to conjecture that 0.32 < τ3 < 0.33).
Moreover, our analysis supports the following conclusions: (a) The finiteness of
τ3 is not at all an indication of a possible blow-up. (b)
There is a strong indication that the solution of the Euler equation does not blow up at a
time close to τ3. In fact, the solution is likely to exist, at
least, up to a time θ3 > 0.47. (c) There is a weak
indication, based on Padé analysis, that the solution might blow up at a later time.
This paper deals with the large-time behaviour of solutions to the fast diffusive Newtonian filtration equations coupled via the nonlinear boundary sources. A result of Fujita type is obtained by constructing various kinds of upper and lower solutions. In particular, it is shown that the critical global existence curve and the critical Fujita curve concide for the multi-dimensional system. This is quite different from the known results obtained in Wang, Zhou and Lou [‘Critical exponents for porous medium systems coupled via nonlinear boundary flux’, Nonlinear Anal.7(1) (2009), 2134–2140] for the corresponding one-dimensional problem.
We define and study the secondary Chern–Euler class for a general submanifold of a Riemannian manifold. Using this class, we define and study the index for a vector field with non-isolated singularities on a submanifold. As an application, we give conceptual proofs of a result of Chern.
We give a simpler and refined proof of some blow-up results of smooth solutions to the Cauchy problem for the Navier–Stokes equations of compressible, viscous and heat-conducting fluids in arbitrary space dimensions. Our main results reveal that smooth solutions with compactly supported initial density will blow up in finite time, and that if the initial density decays at infinity in space, then there is no global solution for which the velocity decays as the reciprocal of the elapsed time.
We consider a non-local filtration equation of the form
and a porous medium equation, in this case K(u) = um, with some boundary and initial data u0, where 0 < p < 1 and f, f′, f″ > 0. We prove blow-up of solutions for sufficiently large values of the parameter λ > 0 and for any u0 > 0, or for sufficiently large values of u0 > 0 and for any λ λ 0.