Blow-up Points Research Articles

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u t - ▵ u = | u | p - 1 u in R N . The initial data is of the form u 0 = λ ϕ , where ϕ ∈ C 0 ( R N ) is fixed and λ > 0 . We first take 1 < p < p f , where p f is the Fujita critical exponent, and ϕ ∈ C 0 ( R N ) ∩ L 1 ( R N ) with nonzero mean. We show that u ( t ) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1 < p < p s , where p s is the Sobolev critical exponent, and ϕ ( x ) decaying as | x | - σ at infinity, where p < 1 + 2 / σ . We also prove that u ( t ) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ϕ is not radial.

Blow-up behavior for a nonlinear heat equation with a localized source in a ball

In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: u t = Δ u + u p + u q ( x * , t ) in B ( R ) where p , q > 0 , B ( R ) = { x ∈ R N : | x | < R } and x * ≠ 0 . If max ( p , q ) > 1 , there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution u ( x , t ) = u ( r , t ) ( r = | x | ) of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q. (i) If q < p ( p > 1 ) or p = q > 2 , then single point blow-up occurs whenever solutions blow up. (ii) If 1 < p < q , both phenomena, total blow-up and single point blow-up, occur depending on the initial data. (iii) If p ⩽ 1 < q , total blow-up occurs whenever solutions blow up. (iv) If max ( p , q ) ⩽ 1 , every solution exists globally in time.

Analysis of singularities and development of shocks for a class of multidimensional hyperbolic systems of conservation laws

In this paper, we investigate the behaviour of singularities and the formation of shocks for a class of multidimensional quasilinear hyperbolic systems of conservation laws including the multidimensional Keyfitz–Kranzer system with a radial state function. Precisely speaking, for these kind of systems we give sharp estimates on the blow-up rate of the smooth solution, provide a complete description of the solution close to the blow-up point, and construct the entropy solution in a neighbourhood of the blow-up point, which contains a unique shock surface issuing from the blow-up point. Copyright © 2005 John Wiley & Sons, Ltd.

Quasilinear parabolic equations with localized reaction

In this paper, we study a nonnegative blow-up solution of the Dirichlet problem for a quasilinear parabolic equation $(u^{\alpha})_t=\Delta u +f(u) + g(u(x_0(t),t))$ in $B(R)$, where $B(R)=\{ x \in \mathbf{R^N}\,;\, |x| < R\}$, $0 < \alpha \le 1$, $x_0(t)\in C^{\infty}([0,\infty) ;B(R))$ satisfies $x_0(t)\not = 0$, and $f(\xi)$ and $g(\xi)$ satisfy some blow-up conditions. We consider radial blow-up solutions $u(r,t)$ ($r=|x|$) which are non-increasing in $r$, and give the classification between total blow-up and single point blow-up according to the growth orders of $f$ and $g$. Especially in the case $\alpha =1$ we completely classify blow-up phenomena except for $f\sim g$ as follows. (I) When $g=o(f)$, any blow-up solution blows up only at the origin (single point blow-up); (II) When $f=o(g)$, (i) a single point blow-up solution exists (ii) there exists an initial data such that the solution blows up in the whole domain $B(R)$ (total blow-up) (iii) there are no other blow-up phenomena except total blow-up and single point blow-up.

Blowup Behavior of Radial Solutions to Jaeger-Luckhaus System in High Dimensional Domains

In this paper, we will consider the blowup and the time-global existence of radial solutions to a simplified system of the so called Keller-Segel model in a domain of three or more dimensional Euclidean space. In the two dimensional case, the blowup solutions have a delta function singularity at each blowup point (see [13]). However, in the three dimensional case, Herrero, Medina and Velazquez [7] show that there exist self-similar blowup solutions and that the solutions have a singularity which is different from a delta function singularity. In this paper, we will consider the blowup criterion and the blowup self-similar solutions in the three or more dimensional cases.

Multiplicity of solutions to a scalar-field equation involving the Sobolev critical exponent with the Robin condition

A scalar-field type equation with the Robin condition is discussed. For some range of the parameter, the multiplicity of solutions is obtained. Moreover, one of the obtained solutions has a blowup property and the blowup point is determined.

Prescribing Q-curvature on higher dimensional spheres

We study the problem of prescribing a fourth order conformal invariant on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of such critical points at infinity, we prove some existence results.

Properties of blow-up solutions to a parabolic system with nonlinear localized terms

This paper deals with blow-up properties of the solution to a semi-linear parabolic system with nonlinear localized sources involved in a product with local terms, subject to the null Dirichlet boundary condition. We investigate the influence of localized sources and local terms on blow-up properties for this system. It will be proved that: (i) when $m, q\leq 1$ this system possesses uniform blow-up profiles. In other words, the localized terms play a leading role in the blow-up profile for this case. (ii) when $m, q>1$, this system presents single point blow-up patterns, or say that, in this time, local terms dominate localized terms in the blow-up profile. Moreover, the blow-up rate estimates in time and space are obtained, respectively.

Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data

AbstractIn this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

Blow-up behavior for a semilinear heat equation with a nonlinear boundary condition

We study the blow-up behaviors of solutions of a semilinear heat equation with a nonlinear boundary condition. Under certain conditions, we prove that the blow-up point occurs only at the boundary. Then, by applying the well-known method of Giga-Kohn, we derive the time asymptotic of solutions near the blow-up time. Finally, we prove that the blow-up is complete.

Conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary

This paper considers the prescribed zero scalar curvature and mean curvature problem on the n-dimensional Euclidean ball for n⩾3. We consider the limits of solutions of the regularization obtained by decreasing the critical exponent. We characterize those subcritical solutions which blow-up at the least possible energy level, determining the points at which they can concentrate, and their Morse indices. We show that when n=3 this is the only blow-up which can occur for solutions. We use this in combination with the Morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three-dimensional Euclidean ball. In the higher-dimensional case n⩾4, we give conditions on the function h to guarantee there is only one simple blow-up point.

Blowup properties for a class of nonlinear degenerate diffusion equation with nonlocal source

A nonlinear degenerate parabolic equation with nonlocal source was considered. It was shown that under certain assumptions the solution of the equation blows up in finite time and the set of blowup points is the whole region. The integral method is used to investigate the blowup properties of the solution.

Total versus single point blow-up of solutions of a semilinear parabolic equation with localized reaction

In this paper, we study the initial-boundary value problem of a semilinear parabolic equation with localized reaction, ( P) u t=Δu+u p+u q(x ∗,t), (x,t)∈B×(0,T), u(x,t)=0, (x,t)∈∂B×(0,T), u(x,0)=u 0(x), x∈B, where B is a unit ball in R N , x ∗∈B , and p, q>0. For the case x ∗=0 , we completely classify blow-up solutions of (P) into total blow-up cases and single point blow-up cases according to the values p and q. Moreover, we give the blow-up rates of solutions near the blow-up time. For the other case x ∗≠0 , we show total blow-up and single point blow-up of solutions of (P) in some cases depending on p and q.

Extended series analysis of full octahedral flow: numerical evidence for hydrodynamic blowup

The power series in time of a full-octahedral flow, a candidate for finite-time blowup of the incompressible Euler equations, is analyzed. Sixty-five Taylor coefficients are found with a precision of 512 bits per Fourier coefficient (154 digits). This work is an extension of Pelz and Gulak (Phys. Rev. Lett. 79 (1997) 1998), where 32 terms were found with half the precision. The solution is found in the form of a power series in time, Fourier series in space with the equation of motion written as a quadratic operation on the Fourier–Taylor coefficients of a single component of vorticity. Particular attention to precision and roundoff is given. In agreement with previous work, Padé approximants of enstrophy series consistently show an isolated, simple pole is located at a real time of about two. Similar findings exist for the series representation of vorticity derivatives at the origin, the purported blowup point. Despite the lack of convergence proofs for Padé resummation, consistent evidence typically yields a valid continuation. Higher-order Sobolev norms of vorticity, however, do not show a similar behavior. The results are also difficult to explain in terms of similarity scalings.

On quenching with logarithmic singularity

In this paper we study the quenching problem for the reaction diffusion equation u t − u xx = f( u) with Cauchy–Dirichlet data, in the case where we have only a logarithmic singularity, i.e., f( u)=ln( αu), α∈(0,1). We show that for sufficiently large domains of x quenching occurs, and that under certain assumptions on the initial function, the set of quenching points is finite. The main result of this paper concerns the asymptotic behavior of the solution in a neighborhood of a quenching point. This result gives the quenching rate for the problem. We also obtain new blow-up results for the equation v t − v xx = αve v − v x 2. These concern the occurrence of blow-up, the blow-up set and the asymptotics in a neighborhood of a blow-up point. The analysis is based on the equivalence between the quenching and the blow-up for these two equations.

An adaptive time step procedure for a parabolic problem with blow-up

In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time. We use standard linear elements with mass lumping for the space variable. For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then, we use explicit Runge-Kutta methods for this equivalent problem. In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case. We obtain necessary and sufficient conditions for the blowup of the numerical solution and prove that the numerical blow-up time converges to the continuous one. We also study, for the explicit Euler approximation, the localization of blow-up points for the numerical scheme.

Landau-Lifshitz equation of ferromagnetism with external magnetic field

AbstractIn this article, we prove the existence and uniqueness of solution for the Cauchy problem of the Landau-Lifshitz equation of ferromagnetism with external magnetic field. We also show that the solution is globally regular with the exception of at most finitely many blow-up points. An energy identity at blow-up points is presented.

CONES OF CURVES AND OF LINE BUNDLES ON SURFACES ASSOCIATED WITH CURVES HAVING ONE PLACE AT INFINITY

Let V be a pencil of curves in ${\bf P}^2$ with one place at infinity, and $X \longrightarrow {\bf P}^2$ the minimal composition of point blow-ups eliminating its base locus. We study the cone of curves and the cones of numerically effective and globally generated line bundles on X. It is proved that all of these cones are regular. In particular, this result provides a new class of rational projective surfaces with a rational polyhedral cone of curves. The surfaces in this class have non-numerically effective anticanonical sheaf if the pencil is neither rational nor elliptic. An application is a global version on X of Zariski's unique factorization theorem for complete ideals. We also define invariants of the semigroup of globally generated line bundles on X depending only on the topology of V at infinity.2000 Mathematical Subject Classification: primary 14C20; secondary 14E05.

Total versus Single Point Blow-Up for a Nonlocal Gaseous Ignition Model

In this paper we investigate an integroparabolic equation that may be considered as a mathematical model for the temperature within the ignition period of a gaseous fuel. For radially symmetric, non-increasing initial data, we determine where classical solutions become unbounded in finite time as well as describe the asymptotic behavior of these hot-spots. The method of analysis is based on maximum principle techniques and the method of stationary states.

Stability of Some Mechanisms of Chemotactic Aggregation

In this paper, the stability of a blow-up mechanism for a particular Keller--Segel model that yields chemotactic aggregation is considered. It is shown using matched asymptotic methods that the considered blow-up mechanism is stable. More precisely, small perturbations of the initial data produce a small shifting on the blow-up time and the blow-up point but not modification in the blow-up mechanism.